diff --git a/-tAzT4oBgHgl3EQfFfrV/content/tmp_files/2301.01013v1.pdf.txt b/-tAzT4oBgHgl3EQfFfrV/content/tmp_files/2301.01013v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..df5c60acdff9774b56ad4033c338ee2a8fb685e8 --- /dev/null +++ b/-tAzT4oBgHgl3EQfFfrV/content/tmp_files/2301.01013v1.pdf.txt @@ -0,0 +1,2053 @@ +Generalised (non-singular) entropy functions with applications to cosmology and +black holes +Sergei D. Odintsov1,2 , Tanmoy Paul3 +1) ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain +2) Institute of Space Sciences (ICE, CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain +3) Department of Physics, Chandernagore College, Hooghly - 712 136, India. +The growing interest of different entropy functions proposed so far (like the Bekenstein-Hawking, +Tsallis, R´enyi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies) towards +black hole thermodynamics as well as towards cosmology lead to the natural question that whether +there exists a generalized entropy function that can generalize all these known entropies. With this +spirit, we propose a new 4-parameter entropy function that seems to converge to the aforementioned +known entropies for certain limits of the entropic parameters. The proposal of generalized entropy +is extended to non-singular case, in which case , the entropy proves to be singular-free during the +entire cosmological evolution of the universe. The hallmark of such generalized entropies is that it +helps us to fundamentally understand one of the important physical quantities namely “entropy”. +Consequently we address the implications of the generalized entropies on black hole thermodynamics +as well as on cosmology, and discuss various constraints of the entropic parameters from different +perspectives. +I. +INTRODUCTION +One of the most important discoveries in theoretical physics is the black body radiation of a black hole, which +is described by a certain temperature and by a Bekenstein-Hawking entropy function [1, 2] (see [3, 4] for extensive +reviews). On contrary to classical thermodynamics where the entropy is proportional to volume of the system under +consideration, the Bekenstein-Hawking entropy is proportional to the area of the black hole horizon. Such unusual +behaviour of the black hole entropy leads to the proposals of different entropy functions, such as, the Tsallis [5], +R´enyi [6], Barrow [7], Sharma-Mittal [8], Kaniadakis [9] and the Loop Quantum Gravity entropies [10] are well known +entropy functions proposed so far. All of these known entropies have the common properties like – (1) they seem +to be the monotonic increasing function with respect to the Bekenstein-Hawkinh entropy variable, (2) they obey the +third law of thermodynamics, in particular, all of these entropies tend to zero as S → 0 (where S represents the +Bekenstein-Hawking entropy) and (3) they converge to the Bekenstein-Hawking entropy for suitable choices of the +respective entropic parameter, for example, the Tsallis entropy goes to the Bekenstein-Hawking entropy when the +Tsallis exponent tends to unity. Furthermore, these entropies have rich consequences towards cosmology, particularly +in describing the dark energy era of the universe [16–49]. The growing interest of such known entropies and due to +their common properties lead to a natural question that whether there exists some generalized entropy function which +is able to generalize all the known entropies proposed so far for suitable limits of the parameters. +The entropy functions are extensively applied in the realm of black hole thermodynamics and cosmological evolution +of the universe. Recently we showed that the entropic cosmology corresponding to different entropy functions can be +equivalently represented by holographic cosmology where the equivalent holographic cut-offs come in terms of either +particle horizon and its derivative or the future horizon and its derivative. One of the mysteries in today’s cosmology +is to explain the acceleration of the universe in the high as well as in the low curvature regime, known as inflation and +the dark energy era respectively. These eras are well described by entropic cosmology or equivalently by holographic +cosmology [16–51, 53–60], and more interestingly, the entropic cosmology proves to be useful to unify the early inflation +and the late dark energy era of the universe in a covariant manner [61]. Apart from the inflation, the holographic +cosmology turns out to be useful in describing the bouncing scenario [62, 63]. In regard to the bounce scenario, the +energy density sourced from the holographic principle or from some entropy function under consideration helps to +violate the null energy condition at a finite time, which in turn triggers a non-singular bouncing universe. However +here it deserves mentioning that all the known entropies mentioned above (like Tsallis , R´enyi, Barrow, Sharma- +Mittal, Kaniadakis and the Loop Quantum Gravity entropies) become singular (or diverge) at a certain cosmological +evolution of the universe, particularly in the context of bounce cosmology. Actually such entropies contain a factor +that is proportional to 1/H2 (where H is the Hubble parameter), and thus they diverge at the instant when the +Hubble parameter vanishes, i.e, at the instant of a bounce in bouncing cosmology. This makes such known entropies +ill-defined in describing a non-singular bounce scenario. +Based on the above arguments, the questions that naturally arise are following: +• Does there exist a generalized entropy function that generalizes all the known entropies proposed so far ? +arXiv:2301.01013v1 [gr-qc] 3 Jan 2023 + +2 +• If so, then what is its implications on black hole thermodynamics as well as on cosmology ? +• Similar to the known entropies, is the generalized entropy becomes singular at the instant when the Hubble +parameter of the universe vanishes, for instance, in the bounce cosmology ? If so, then does there exist an +entropy function that generalizes all the known entropies, and at the same time, also proves to be singular-free +during the entire cosmic evolution of the universe ? +The present article, based on some of our previous works [50–52], gives a brief review in answering the above +questions. The notations or conventions in this article are following: we will follow the (−, +, +, +) signature of the +spacetime metric, and κ2 = 8πG = +1 +M 2 +Pl where G is the Newton’s constant or MPl denotes the four dimensional Planck +mass. In regard to the cosmological evolution, a(t) and H(t) are the scale factor and the Hubble parameter of the +universe respectively, N being the e-folding number, an overprime will denote +d +dη where η is the conformal time, an +overdot will symbolize +d +dt with t being the cosmic time, otherwise an overprime with some argument will represent +the derivative of the function with respect to that argument. +II. +POSSIBLE GENERALIZATIONS OF KNOWN ENTROPIES +Here we will propose a generalized four-parameter entropy function which can lead to various known entropy +functions proposed so far for suitable choices of the parameters. +Let us start with the Bekenstein-Hawking entropy, the very first proposal of thermodynamical entropy of black hole +physics [1, 2], +S = A +4G , +(1) +where A = 4πr2 +h is the area of the horizon and rh is the horizon radius. Consequently, different entropy functions have +been introduced depending on the system under consideration. Let us briefly recall some of the entropy functions +proposed so far: +• For the systems with long range interactions where the Boltzmann-Gibbs entropy is not applied, one needs to +introduce the Tsallis entropy which is given by [5], +ST = A0 +4G +� A +A0 +�δ +, +(2) +where A0 is a constant and δ is the exponent. +• The R´enyi entropy is given by [6], +SR = 1 +α ln (1 + αS) , +(3) +where S is identified with the Bekenstein-Hawking entropy and α is a parameter. +• The Barrow entropy is given by [7], +SB = +� A +APl +�1+∆/2 +, +(4) +where A is the usual black hole horizon area and APl = 4G is the Planck area. The Barrow entropy describes +the fractal structures of black hole that may generate from quantum gravity effects. +• The Sharma-Mittal entropy is given by [8], +SSM = 1 +R +� +(1 + δ S)R/δ − 1 +� +, +(5) +where R and δ are two parameters. The Sharma-Mittal entropy can be regarded as a possible combination of +the Tsallis and R´enyi entropies. + +3 +• The Kaniadakis entropy function is of the following form [9]: +SK = 1 +K sinh (KS) , +(6) +where K is a phenomenological parameter. +• In the context of Loop Quantum Gravity, one may get the following entropy function [10]: +Sq = +1 +(1 − q) +� +e(1−q)Λ(γ0)S − 1 +� +, +(7) +where q is the exponent and Λ(γ0) = ln 2/ +�√ +3πγ0 +� +with γ0 being the Barbero-Immirzi parameter. The γ0 +generally takes either γ0 = +ln 2 +π +√ +3 or γ0 = +ln 3 +2π +√ +2. However with γ0 = +ln 2 +π +√ +3, Λ(γ0) becomes unity and Sq resembles +with the Bekenstein-Hawking entropy for q → 1. +All the above entropies – (1) obeys the generalized third law of thermodynamics, i.e the entropy function(s) vanishes +at the limit S → 0; (2) monotonically increases with respect to the Bekenstein-Hawking variable and (3) converges to +the Bekenstein-Hawking entropy for suitable limit of the entropic parameter, for example, the Tsallis entropy tends +to S at δ = 1. +In [50, 51], we proposed two different entropy functions containing 6-parameters and 4-parameters respectively, +which can generalize all the known entropies mentioned from Eq.(2) to Eq.(7). In particular, the generalized entropies +are given by, +6 parameter entropy : +S6 [α±, β±, γ±] = +1 +α+ + α− +�� +1 + α+ +β+ +Sγ+ +�β+ +− +� +1 + α− +β− +Sγ− +�−β−� +, +(8) +4 parameter entropy : +Sg [α+, α−, β, γ] = 1 +γ +�� +1 + α+ +β +S +�β +− +� +1 + α− +β +S +�−β� +, +(9) +where the respective parameters are given in the argument and they are assumed to be positive. +Here S is the +Bekenstein-Hawking entropy. Below we prove the generality of the above generalized entropy functions, in particular, +we show that both the generalized entropies reduce to the known entropies mentioned in Eqs. (2), (3), (4), (5), (6), +and (7) for suitable choices of the respective parameters. Here we establish it particularly for the 4-parameter entropy +function, while the similar calculations hold for the 6-parameter entropy as well [50]. +• For α+ → ∞ and α− = 0, one gets +Sg = 1 +γ +�α+ +β +�β +Sβ . +If we further choose γ = (α+/β)β, then the generalized entropy reduces to +Sg = Sβ . +Therefore with β = δ or β = 1 + ∆, the generalized entropy resembles with the Tsallis entropy or with the +Barrow entropy respectively. +• For α− = 0, β → 0 and α+ +β → finite – Eq. (9) leads to, +Sg = 1 +γ +�� +1 + α+ +β +S +�β +− 1 +� += 1 +γ +� +exp +� +β ln +� +1 + α+ +β +S +�� +− 1 +� +≈ +1 +(γ/β) ln +� +1 + α+ +β +S +� +. +Further choosing γ = α+ and identifying α+ +β = α, we can write the above expression as, +Sg = 1 +α ln (1 + α S) , +(10) +i.e., Sg reduces to the R´enyi entropy. + +4 +• In the case when α− = 0, the generalized entropy becomes, +Sg = 1 +γ +�� +1 + α+ +β +S +�β +− 1 +� +. +(11) +Thereby identifying γ = R, α+ = R and β = R/δ, the generalized entropy function Sg gets similar to the +Sharma-Mittal entropy. +• For β → ∞, α+ = α− = γ +2 = K, we may write Eq. (9) as, +Sg = 1 +2K lim +β→∞ +�� +1 + K +β S +�β +− +� +1 + K +β S +�−β� += 1 +2K +� +eKS − e−KS� += 1 +K sinh (KS) → Kaniadakis entropy . +(12) +• Finally, with α− = 0, β → ∞ and γ = α+ = (1 − q), Eq. (9) immediately yields, +Sg = +1 +(1 − q) +� +e(1−q)S − 1 +� +, +which is the Loop Quantum Gravity entropy with Λ(γ0) = 1 or equivalently γ0 = +ln 2 +π +√ +3. +Furthermore, the generalized entropy function in Eq. (9) shares the following properties: (1) Sg → 0 for S → 0. +(2) The entropy Sg [α+, α−, β, γ] is a monotonically increasing function with S because both the terms +� +1 + α+ +β S +�β +and − +� +1 + α− +β +S +�−β +present in the expression of Sg increase with S. (3) Sg [α+, α−, β, γ] seems to converge to the +Bekenstein-Hawking entropy at certain limit of the parameters. In particular, for α+ → ∞, α− = 0, γ = (α+/β)β +and β = 1, the generalized entropy function in Eq. (9) becomes equivalent to the Bekenstein-Hawking entropy. +Here it deserves mentioning that beside the entropy function proposed in Eq. (9) which contains four parameters, +one may consider a three parameter entropy having the following form: +S3[α, β, γ] = 1 +γ +�� +1 + α +β S +�β +− 1 +� +, +(13) +where α, β and γ are the parameters. The above form of S3[α, β, γ] satisfies all the properties, like – (1) S3[α, β, γ] → 0 +for S → 0, (2) S3 is an increasing function with S and (3) S3 has a Bekenstein-Hawking entropy limit for the choices: +α → ∞, γ = (α/β)β and β = 1 respectively. However S3[α, β, γ] is not able to generalize all the known entropies +mentioned from Eq. (2) to Eq. (7), in particular, S3[α, β, γ] does not reduce to the Kaniadakis entropy for any +possible choices of the parameters. +Conjecture - I: Based on our findings, we propose the following postulate in regard to the generalized entropy +function – “The minimum number of parameters required in a generalized entropy function that can generalize all +the known entropies mentioned from Eq. (2) to Eq. (7) is equal to four”. +Below we will address the possible implications of such generalized entropies on black hole thermodynamics as well +as on cosmology. +III. +BLACK HOLE THERMODYNAMICS WITH 3-PARAMETER GENERALIZED ENTROPY +It is interesting to see what happens when the generalized entropy (13) is ascribed to the prototypical black hole, +given by the Schwarzschild geometry [50] +ds2 = −f(r) dt2 + dr2 +f(r) + r2dΩ2 +(2) , +f(r) = 1 − 2GM +r +, +(14) + +5 +where M is the black hole mass and dΩ2 +(2) = dϑ2 + sin2 ϑ dϕ2 is the line element on the unit two-sphere. The black +hole event horizon is located at the Schwarzschild radius +rH = 2GM . +(15) +Studying quantum field theory on the spacetime with this horizon, Hawking discovered that the Schwarzschild black +hole radiates with a blackbody spectrum at the temperature +TH = +1 +8πGM . +(16) +As explained in general below, if we assume that the mass M coincides with the thermodynamical energy, then the +temperature obtained from the thermodynamical law is different from the Hawking temperature, a contradiction for +observers detecting Hawking radiation. Alternatively, if the Hawking temperature TH is identified with the physical +black hole temperature, the obtained thermodynamical energy differs from the Schwarzschild mass M even for the +Tsallis entropy or the R´enyi entropy, which seems to imply a breakdown of energy conservation. +If the mass M coincides with the thermodynamical energy E of the system due to energy conservation, as in, +in order for this system to be consistent with the thermodynamical equation dSG = dE/T one needs to define the +generalized temperature TG as +1 +TG +≡ dSG +dM +(17) +which is, in general, different from the Hawking temperature TH. For example, in the case of the entropy (13), one +has +1 +TG += α +γ +� +1 + α +β S +�β−1 dS +dM = α +γ +� +1 + α +β S +�β−1 1 +TH +, +(18) +where +S = A +4G = 4πGM 2 = +1 +16πGTH +2 . +(19) +Because α +γ +� +1 + α +β S +�β−1 +̸= 1, it is necessarily TG ̸= TH. Since the Hawking temperature (16) is the temperature +perceived by observers detecting Hawking radiation, the generalized temperature TG in (18) cannot be a physically +meaningful temperature. +In Eq. (17), assuming that the thermodynamical energy E is the black hole mass M leads to an unphysical result. +As an alternative, assume that the thermodynamical temperature T coincides with the Hawking temperature TH +instead of assuming E = M. This assumption leads to +dEG = TH dSG = dSG +dS +dS +√ +16πGS +(20) +which, in the case of Eq. (13), yields +dEG = α +γ +� +1 + α +β S +�β−1 +dS +√ +16πGS += +α +γ +√ +16πG +� +S−1/2 + α (β − 1) +β +S1/2 + O +� +S3/2�� +. +(21) +The integration of Eq. (21) gives +EG = +α +γ +√ +16πG +� +2S1/2 + 2α (β − 1) +3β +S3/2 + O +� +S5/2�� += α +γ +� +M + 4πGα (β − 1) +3β +M 3 + O +� +M 5�� +, +(22) +where the integration constant is determined by the condition that EG = 0 when M = 0. Even when α = γ, due to +the correction 4πGα(β−1) +3β +M 3, the expression (22) of the thermodynamical energy ER obtained differs from the black +hole mass M, EG ̸= E, which seems unphysical. + +6 +IV. +COSMOLOGY WITH THE 4-PARAMETER GENERALIZED ENTROPY +Here we consider the 4-parameter generalized entropy (9), which is indeed more generalized compared to the 3- +parameter entropy function of Eq.(13), to describe the cosmological behaviour of the universe [51]. In particular, we +examine whether the 4-parameter entropy function results to an unified scenario of early inflation and the late dark +energy era of the universe. +The Friedmann-Lemaˆıtre-Robertson-Walker space-time with flat spacial part will serve our purpose, in particular, +ds2 = −dt2 + a2(t) +� +i=1,2,3 +� +dxi�2 . +(23) +Here a(t) is called as a scale factor. +The radius rH of the cosmological horizon is given by +rH = 1 +H , +(24) +with H = ˙a/a is the Hubble parameter of the universe. Then the entropy contained within the cosmological horizon +can be obtained from the Bekenstein-Hawking relation [65]. Furthermore the flux of the energy E, or equivalently, +the increase of the heat Q in the region comes as +dQ = −dE = −4π +3 r3 +H ˙ρdt = − 4π +3H3 ˙ρ dt = 4π +H2 (ρ + p) dt , +(25) +where, in the last equality, we use the conservation law: 0 = ˙ρ + 3H (ρ + p). Then from the Hawking temperature +[66] +T = +1 +2πrH += H +2π , +(26) +and by using the first law of thermodynamics TdS = dQ, one obtains ˙H = −4πG (ρ + p). Integrating the expression +immediately leads to the first FRW equation, +H2 = 8πG +3 +ρ + Λ +3 , +(27) +where the integration constant Λ can be treated as a cosmological constant. +Instead of the Bekenstein-Hawking entropy of Eq. (1), we may use the generalized entropy in Eq. (9), in regard to +which, the first law of thermodynamics leads to the following equation: +˙H +�∂Sg +∂S +� += −4πG (ρ + p) . +(28) +With the explicit form of Sg from Eq. (9), the above equation turns out to be, +1 +γ +� +α+ +� +1 + πα+ +βGH2 +�β−1 ++ α− +� +1 + πα− +βGH2 +�−β−1� +˙H = −4πG (ρ + p) +(29) +where we use S = A/(4G) = π/(GH2). Using the conservation relation of the matter fields, i.e., ˙ρ + 3H (ρ + p) = 0, +Eq. (29) can be written as, +2 +γ +� +α+ +� +1 + πα+ +βGH2 +�β−1 ++ α− +� +1 + πα− +βGH2 +�−β−1� +H dH = +�8πG +3 +� +dρ , +on integrating which, we obtain, +GH4β +πγ +� +1 +(2 + β) +�GH2β +πα− +�β +2F1 +� +1 + β, 2 + β, 3 + β, −GH2β +πα− +� ++ +1 +(2 − β) +�GH2β +πα+ +�−β +2F1 +� +1 − β, 2 − β, 3 − β, −GH2β +πα+ +�� += 8πGρ +3 ++ Λ +3 , +(30) +where Λ is the integration constant (known as the cosmological constant) and 2F1(arguments) denotes the Hypergeo- +metric function. Eq. (29) and Eq. (30) represent the modified Friedmann equations corresponding to the generalized +entropy function Sg. In the next section, we aim to study the cosmological implications of the modified Friedmann +Eq. (29) and Eq. (30). + +7 +A. +Early universe cosmology from the 4-parameter generalized entropy +During the early stage of the universe we consider the matter field and the cosmological constant (Λ) to be absent, +i.e., ρ = p = Λ = 0. During the early universe, the cosmological constant is highly suppressed with respect to the +entropic energy density and thus we can safely neglect the Λ in studying the early inflationary scenario of the universe. +Therefore during the early universe, Eq. (30) becomes, +� +1 +(2 + β) +�GH2β +πα− +�β +2F1 +� +1 + β, 2 + β, 3 + β, −GH2β +πα− +� ++ +1 +(2 − β) +�GH2β +πα+ +�−β +2F1 +� +1 − β, 2 − β, 3 − β, −GH2β +πα+ +�� += 0 . +(31) +Here it may be mentioned that the typical energy scale during early universe is of the order ∼ 1016GeV (= 10−3MPl +where recall that MPl is the Planck mass and MPl = 1/ +√ +16πG). This indicates that the condition GH2 ≪ 1 holds +during the early phase of the universe. Owing to such condition, we can safely expand the Hypergeometric function +of Eq. (31) as the Taylor series with respect to the argument containing GH2, and as a result, the above equation +provides a constant Hubble parameter as the solution: +H = 4πMPl +�α+ +β +� +(3 − β) +(2 − β)(1 − β) +� +. +(32) +For α+ +β ∼ 10−6 and β ≲ O(1), the constant Hubble parameter can be fixed at H ∼ 10−3MPl which can be identified +with typical inflationary energy scale. Therefore the entropic cosmology corresponding to the generalized entropy +function Sg leads to a de-Sitter inflationary scenario during the early universe. However, a de-Sitter inflation has no +exit mechanism, and moreover, the primordial curvature perturbation gets exactly scale invariant in the context of a +de-Sitter inflation, which is not consistent with the recent Planck data [75] at all. This indicates that the constant +Hubble parameter obtained in Eq. (32) does not lead to a good inflationary scenario of the universe. Thus in order +to achieve a viable quasi de-Sitter inflation in the present context, we consider the parameters of Sg to be slowly +varying functions with respect to the cosmic time. In particular, we consider the parameter γ to vary and the other +parameters (i.e., α+, α− and β) remain constant with t. In particular, +γ(N) = +� +γ0 exp +� +− +� Nf +N +σ(N) dN +� +; N ≤ Nf +γ0 +; N ≥ Nf , +(33) +where γ0 is a constant and N denotes the inflationary e-folding number with Nf being the total e-folding number of +the inflationary era. The function σ(N) has the following form, +σ(N) = σ0 + e−(Nf −N) , +(34) +where σ0 is a constant. The second term in the expression of σ(N) becomes effective only when N ≈ Nf, i.e., near +the end of inflation. The term e−(Nf −N) in Eq. (34) is actually considered to ensure an exit from inflation era and +thus proves to be an useful one to make the inflationary scenario viable. In such scenario where γ varies with N, the +Friedmann equation turns out to be, +− +�2π +G +� +� +�� +α+ +� +1 + α+ +β S +�β−1 ++ α− +� +1 + α− +β +S +�−β−1 +� +1 + α+ +β S +�β +− +� +1 + α− +β +S +�−β +� +�� H′(N) +H3 += σ(N) . +(35) +By using S = π/(GH2), or equivalently, 2HdH = − +π +GS2 dS, one can integrate Eq.(35) to get H(N) as, +H(N) = 4πMPl +�α+ +β +� +���� +21/(2β) exp +� +− 1 +2β +� N +0 σ(N)dN +� +� +1 + +� +1 + 4 (α+/α−)β exp +� +−2 +� N +0 σ(N)dN +��1/(2β) +� +���� . +(36) + +8 +The above solution of H(N) allows an exit from inflation at finite e-fold number which can be fixed at Nf = 58 for +suitable choices of the entropic parameters [51]. Moreover we determine the spectral index for curvature perturbation +(ns) and the tensor-to-scalar ratio (r) in the present context of entropic cosmology, and they are given by [51]: +ns = 1 − +2σ0 +� +1 + 4 (α+/α−)β exp [−2 (1 + σ0Nf)] +(1 + σ0) +� +1 + 4 (α+/α−)β +− 8σ0 (α+/α−)β +1 + 4 (α+/α−)β , +(37) +and +r = +16σ0 +� +1 + 4 (α+/α−)β exp [−2 (1 + σ0Nf)] +(1 + σ0) +� +1 + 4 (α+/α−)β +(38) +respectively. It turns out that the theoretical expectations of ns and r get simultaneously compatible with the Planck +data for the following ranges of the parameters: +σ0 = [0.013, 0.017] , +(α+/α−)β ≥ 7.5 , +β = (0, 0.4] and (α+/β) ≈ 10−6 , +(39) +for Nf = 58. The consideration of α+ +β ∼ 10−6 leads to the energy scale at the onset of inflation as H ∼ 10−3MPl. +B. +Dark energy era from the 4-parameter generalized entropy +In this section we will concentrate on late time cosmological implications of the generalized entropy function (Sg), +where the cosmological constant Λ is considered to be non-zero. During the late time, the parameter γ becomes +constant, in particular γ = γ0, as we demonstrated in Eq. (33). As a result, the entropy function at the late time +takes the following form, +Sg = 1 +γ0 +�� +1 + α+ +β +S +�β +− +� +1 + α− +β +S +�−β� +, +(40) +with S = π/(GH2). Consequently, the energy density and pressure corresponding to the Sg are given by, +ρg = 3H2 +8πG +� +1 − +α+ +γ0(2 − β) +�GH2β +πα+ +�1−β� +, +pg = − +˙H +4πG +� +1 − α+ +γ0 +�GH2β +πα+ +�1−β +− +�α+ +γ0 +� �α+ +α− +�β �GH2β +πα+ +�1+β� +− ρg . +(41) +Therefore the dark energy density (ρD) is contributed from the entropic energy density (ρg) as well as from the +cosmological constant. In particular +ρD = ρg + +3 +8πG +�Λ +3 +� +, +ρD + pD = ρg + pg . +(42) +Consequently, the dark energy EoS parameter comes with the following expression: +ωD = pD/ρD = −1 − +� +2 ˙H +3H2 +� � +�� +1 − α+ +γ0 +� +GH2β +πα+ +�1−β +− +� +α+ +γ0 +� � +α+ +α− +�β � +GH2β +πα+ +�1+β +1 − +α+ +γ0(2−β) +� +GH2β +πα+ +�1−β ++ +Λ +3H2 +� +�� . +(43) +In presence of the cosmological constant, the Friedmann equations are written as, +H2 = 8πG +3 +(ρm + ρD) = 8πG +3 +(ρm + ρg) + Λ +3 , + +9 +˙H = −4πG [ρm + (ρD + pD)] = −4πG [ρm + (ρg + pg)] . +(44) +As usual, the fractional energy density of the pressureless matter and the dark energy satisfy Ωm + ΩD = 1 which +along with ρm = ρm0 +� a0 +a +�3 (with ρm0 being the present matter energy density) result to the Hubble parameter in +terms of the red shift factor (z) as follows, +H(z) = H0 +� +Ωm0(1 + z)3 +√1 − ΩD +. +(45) +Plugging the expression of ρg from Eq. (41) into ΩD = +� 8πG +3H2 +� +ρg + Λ +3 , and using the above form of H(z), we obtain, +ΩD(z) = 1 − +� +α+ +γ0(2−β) +� +1 +2−β � +GH2 +0β +πα+ +Ωm0(1 + z)3� 1−β +2−β +� +1 + +Λ +3H2 +0Ωm0(1+z)3 +�1/(2−β) +. +(46) +By using the above expressions, we determine the DE EoS parameter from Eq. (43) as follows (see [51]), +ωD(z) = −1 + +1 +(2 − β) +� +1 + +Λ +3H2 +0Ωm0(1+z)3 +� +�N +D +� +, +(47) +where N (the numerator) and D (the denominator) have the following forms, +N = 1 − Ωm0(2 − β) (1 + z) +3(1−β) +(2−β) +� � +1 + +Λ +3H2 +0Ωm0 +[f(Λ, Ωm0, H0, z)]1−β +� ++ +�α+ +α− +�β +[Ωm0(2 − β)γ0/α+] +2β +1−β (1 + z) +6β +2−β +× +� +� +� +� +1 + +Λ +3H2 +0Ωm0 +�(1+β)/(1−β) +[f(Λ, Ωm0, H0, z)]1+β +� +� +� +� +, +and +D = 1 − Ωm0 (1 + z) +3(1−β) +(2−β) +� +1 + +Λ +3H2 +0Ωm0 +[f(Λ, Ωm0, H0, z)]1−β +� ++ +Λ +3H2 +0 +�f(Λ, Ωm0, H0, z) +(1 + z)3/(2−β) +� +(48) +respectively. +Therefore ωD depends on the parameters: β, (α+/α−)β, γ0 and α+. +Recall that the inflationary +quantities are found to be simultaneously compatible with the Planck data if some of the parameters like α+, α− and +β get constrained according to Eq. (39), while the parameter γ0 remains free from the inflationary requirement. With +the aforementioned ranges of α+, α− and β, ωD(0) becomes compatible with the Planck observational data, provided +γ0 lies within a small window as follows, +1.5 × 10−4 ≤ +γ0 +(8πGH2 +0)1−β ≤ 2 × 10−4 . +(49) +Furthermore the deceleration parameter (symbolized by q) at present universe is obtained as, +q = −1 + +3 +2(2 − β) +� +1 + +Λ +3H2 +0Ωm0 +� . +(50) +Therefore for γm = [1.5×10−4, 2×10−4], the theoretical expression of q lies within q = [−0.56, −0.42] which certainly +contains the observational value of q = −0.535 from the Planck data [76]. In particular, q = −0.535 occurs for +γm = 1.8 × 10−4. Considering this value of γm and by using Eq.(47), we give the plot of ωD(z) vs. z, see Fig. 1. +The figure reveals that that the theoretical expectation of the DE EoS parameter at present time acquires the value: +ωD(0) = −0.950 which is well consistent with the Planck observational data [76]. +As a whole, we may argue that the entropic cosmology from the generalized entropy function Sg can unify the early +inflation to the late dark energy era of the universe, for suitable ranges of the parameters given by: +σ0 = [0.013, 0.017] , +(α+/α−)β ≥ 7.5 , + +10 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-3.0 +-2.5 +-2.0 +-1.5 +-1.0 +z +ωD(z) +FIG. 1: ωD(z) vs. z for a particular set of values of the parameters from their viable ranges as per Eq.(39) and Eq.(49), say +β = 0.35, (α+/α−)β = 10, α+/β = 10−6 and γm = 1.8 × 10−4. +β = (0, 0.4] and γm = [1.5 × 10−4, 2 × 10−4] . +(51) +Despite these successes, here it deserves mentioning that the entropy function Sg seems to be plagued with singularity +for certain cosmological evolution of the universe, in particular, in the context of bounce cosmology. Due to the reason +that the Bekenstein-Hawking entropy can be expressed as S = π/ +� +GH2� +, the generalized entropy Sg contains factor +that is proportional to 1/H2 which diverges at H = 0, for instance at the instant of bounce in the context of bounce +cosmology. Therefore in a bounce scenario, the generalized entropy function shown in Eq.(9) is not physical, and +thus, we need to search for a different generalized entropy function which can lead to various known entropy functions +for suitable choices of the parameters, and at the same time, proves to be non-singular for the entire cosmological +evolution of the universe even at H = 0. +V. +SEARCH FOR A SINGULAR-FREE GENERALIZED ENTROPY +With this spirit, we propose a new singular-free entropy function given by [52], +Sns [α±, β, γ, ϵ] = 1 +γ +� � +1 + 1 +ϵ tanh +�ϵα+ +β +S +��β +− +� +1 + 1 +ϵ tanh +�ϵα− +β +S +��−β � +, +(52) +where α±, β, γ and ϵ are the parameters which are considered to be positive, S symbolizes the Bekenstein-Hawking +entropy and the suffix ’ns’ stands for ’non-singular’. In regard to the number of parameters, we propose a conjecture at +the end of this section. First we demonstrate that the above entropy function remains finite, and thus is non-singular, +during the whole cosmological evolution of a bouncing universe. In particular, the Sg takes the following form at the +instant of bounce: +Sns [α±, β, γ, ϵ] = 1 +γ +� � +1 + 1 +ϵ +�β +− +� +1 + 1 +ϵ +�−β � +. +(53) +Having demonstrated the non-singular behaviour of the entropy function, we now show that Sns of Eq.(52), for suitable +choices of the parameters, reduces to various known entropies proposed so far. +• For ϵ → 0, α+ → ∞ and α− = 0 along with the identification γ = (α+/β)β, Sns converges to the Tsallis entropy +or to the Barrow entropy respectively. +• The limit ϵ → 0, α− = 0, β → 0 and α+ +β → finite results to the similarity between the non-singular generalized +entropy Sg and the R´enyi entropy. + +11 +• For ϵ → 0 and α− → 0, the non-singular generalized entropy converges to the following form, +Sns = 1 +γ +�� +1 + α+ +β +S +�β +− 1 +� +(54) +Therefore with γ = R, α+ = R and β = R/δ, the above form of Sns becomes similar to the Sharma-Mittal +entropy. +• For ϵ → 0, β → ∞, α+ = α− = γ +2 = K – the generalized entropy converges to the form of Kaniadakis entropy, +• Finally, ϵ → 0, α− → 0, β → ∞ and γ = α+ = (1 − q), the generalized entropy of Eq. (52) gets resemble with +the Loop Quantum Gravity entropy. +Furthermore, the generalized entropy function in Eq. (52) shares the following properties: (1) the non-singular +generalized entropy satisfies the generalized third law of thermodynamics. +(2) Sns [α±, β, γ, ϵ] turns out to be a +monotonically increasing function of S. (3) Sns [α±, β, γ, ϵ] proves to converge to the Bekenstein-Hawking entropy at +certain limit of the parameters. +At this stage it deserves mentioning that we have proposed two different generalized entropy functions in Eq.(9) +and in Eq.(52) respectively – the former entropy function contains four independent parameters while the latter +one has five parameters. +Furthermore both the entropies are able to generalize the known entropies for suitable +choices of the respective parameters. However as mentioned earlier that the entropy with four parameters becomes +singular at H = 0 (for instance, in a bounce scenario when the Hubble parameter vanishes at the instant of bounce), +while the entropy function having five parameters proves to be singular-free during the whole cosmological evolution +of the universe. Based on these findings, we give a second conjecture regarding the number of parameters in the +non-singular generalized entropy function: +Conjecture - II: “The minimum number of parameters required in a generalized entropy function that can gen- +eralize all the known entropies, and at the same time, is also singular-free during the universe’s evolution – is equal +to five”. +VI. +COSMOLOGY WITH THE NON-SINGULAR GENERALIZED ENTROPY +Applying the thermodynamic laws to the non-singular generalized entropy function Sns and by following the same +procedure as of Sec.[IV], one gets the cosmological field equations corresponding to the Sgns [52]: +1 +γ +� +α+ sech2 +� ϵπα+ +βGH2 +� � +1 + 1 +ϵ tanh +� ϵπα+ +βGH2 +��β−1 ++ α− sech2 +� ϵπα− +βGH2 +� � +1 + 1 +ϵ tanh +� ϵπα− +βGH2 +��−β−1 � +˙H = −4πG (ρ + p) +. +(55) +Owing to the conservation equation of matter fields, in particular ˙ρ + 3H (ρ + p) = 0, the above expression can be +integrated to get +f (H; α±, β, γ, ϵ) = 8πGρ +3 ++ Λ +3 . +(56) +Here the integration constant is symbolized by Λ and the function f has the following form: +f (H; α±, β, γ, ϵ) = 2 +γ +� +� +α+ sech2 +� ϵπα+ +βGH2 +� � +1 + 1 +ϵ tanh +� ϵπα+ +βGH2 +��β−1 ++ α− sech2 +� ϵπα− +βGH2 +� � +1 + 1 +ϵ tanh +� ϵπα− +βGH2 +��−β−1 � +H dH . +(57) +In regard to the functional form of f (H; α±, β, γ, ϵ), we would like to mention that the integration in Eq.(57) may not +be performed in a closed form, unless certain conditions are imposed. For example, we consider GH2 ≪ 1 which is, in + +12 +fact, valid during the universe’s evolution (i.e the Hubble parameter is less than the Planck scale). With GH2 ≪ 1, +the functional form of f turns out to be, +f (H; α±, β, γ, ϵ) = 4 +γ H2 +� +α+ +� +1 + 1 +ϵ +�β−1 � +exp +� +−2ϵπα+ +βGH2 +� ++ +�2ϵπα+ +βGH2 +� +Ei +� +−2ϵπα+ +βGH2 +�� ++ α− +� +1 + 1 +ϵ +�−β−1 � +exp +� +−2ϵπα− +βGH2 +� ++ +�2ϵπα− +βGH2 +� +Ei +� +−2ϵπα− +βGH2 +�� � +. +(58) +Therefore as a whole, Eq. (55) and Eq. (56) are the cosmological field equations corresponding to the generalized +entropy Sg. +A. +Non-singular entropy on bounce cosmology +In this section, we will address the implications of the generalized entropy Sns on non-singular bounce cosmology, +in particular, we will investigate whether the entropic energy density can trigger a viable bounce during the early +stage of the universe that is consistent with the observational constraints. For this purpose, we take the matter field +and the cosmological constant to be absent, i.e., ρ = p = Λ = 0. In effect, Eq. (55) becomes, +1 +γ +� +α+ sech2 +� ϵπα+ +βGH2 +� � +1 + 1 +ϵ tanh +� ϵπα+ +βGH2 +��β−1 ++ α− sech2 +� ϵπα− +βGH2 +� � +1 + 1 +ϵ tanh +� ϵπα− +βGH2 +��−β−1 � +˙H = 0 . +(59) +The parameters (α±, β, γ, ϵ) are positive, and thus the solution of the above equation is given by: ˙H = 0 or equivalently +H = constant. Clearly H = constant does not lead to the correct evolution of the universe. Thus similar to the +previous case, we consider the parameters of Sns[α±, β, γ, ϵ] vary with time. In particular, we consider the parameter +γ to vary with time, and all the other parameters remain fixed, i.e. +γ = γ(N) , +(60) +with N being the e-fold number of the universe. In such scenario where γ(N) varies with time, the Friedmann equation +corresponds to Sns[α±, β, γ, ϵ] gets modified compared to Eq.(59), and is given by: +� +�� +α+ sech2 � +ϵα+ +β S +� � +1 + 1 +ϵ tanh +� +ϵα+ +β S +��β−1 ++ α− sech2 � +ϵα− +β S +� � +1 + 1 +ϵ tanh +� +ϵα− +β S +��−β−1 +� +1 + 1 +ϵ tanh +� +ϵα+ +β S +��β +− +� +1 + 1 +ϵ tanh +� +ϵα− +β S +��−β +� +�� dS = γ′(N) +γ(N) dN (61) +where an overprime denotes +d +dη. Eq.(61) can be integrated to get, +tanh +� ϵπα +βGH2 +� += +� +γ(N) + +� +γ2(N) + 4 +2 +�1/β +− 1 . +(62) +where we take α+ = α− = α (say, without losing any generality) in order to extract an explicit solution of H(N). +Due to the appearance of quadratic power of H, Eq.(62) allows a positive branch as well as a negative branch of the +Hubble parameter. This leads to a natural possibility of symmetric bounce in the present context of singular free +generalized entropic cosmology. Moreover Eq.(62) also demonstrates that the explicit evolution of H(N) does depend +on the form of γ(N). In the following, we will consider two cases where we will determine the form of γ(N) such that +it gives two different symmetric bounce scenarios respectively. +1. The exponential bounce described by the scale factor, +a(t) = exp +� +a0t2� +. +(63) +This results to a symmetric bounce at t = 0. Here a0 is a constant having mass dimension [+2] – this constant +is related with the entropic parameters of Sns and thus, without losing any generality, we take a0 = ϵπα +4Gβ . Such +an exponential bounce can be achieved from singular free entropic cosmology provided the γ(N) is given by, +γ(N) = +� +1 + 1 +ϵ tanh +� 1 +N +��β +− +� +1 + 1 +ϵ tanh +� 1 +N +��−β +. +(64) + +13 +2. The quasi-matter bounce is described by, In this case, the scale factor is, +a(t) = +� +1 + a0 +� t +t0 +�2�n +(65) +which is symmetric about t = 0 when the bounce happens. The n, a0 and t0 considered in the scale factor are +related to the entropic parameters, and we take it as follows: +n = √α +, +a0 = π +4β +and +t0 = +� +G/ϵ , +(66) +with G being the gravitational constant. The relation between (n, a0, t0) with the entropic parameters can be +considered in a different way compared to the Eq.(66), however for a simplified expression of γ(N) we consider +the relations as of Eq.(66). Consequently the γ(N) which leads to such quasi-matter bounce, comes as, +γ(N) = +� +1 + 1 +ϵ tanh +� +e−N/√α � +eN/√α − 1 +� 1 +2 ��β +− +� +1 + 1 +ϵ tanh +� +e−N/√α � +eN/√α − 1 +� 1 +2 ��−β +. +(67) +Here it deserves mentioning that in the case of exponential bounce, the comoving Hubble radius asymptotically goes +to zero and thus the perturbation modes remain at the super-Hubble regime at the distant past. This may results to +the “horizon problem” in the exponential bounce scenario. On contrary, the comoving Hubble radius in the case of +quasi-matter bounce asymptotically diverges to infinity at both sides of the bounce, and thus the perturbation modes +lie within the deep sub-Hubble regime at the distant past – this resolves the horizon issue. Based on this arguments, +we will concentrate on the quasi-matter bounce to perform the perturbation analysis. +In regard to the perturbation analysis, we represent the present entropic cosmology with the ghost free Gauss- +Bonnet (GB) theory of gravity proposed in [67]. The motivation of such representation is due to the rich structure of +the Gauss-Bonnet theory in various directions of cosmology [68–71]. The action for f(G) gravity is given by [67], +S = +� +d4x√−g +� 1 +2κ2 R + λ +�1 +2∂µχ∂µχ + µ4 +2 +� +− 1 +2∂µχ∂µχ + h (χ) G − V (χ) +� +, +(68) +where µ is a constant having mass dimension [+1], λ represents the Lagrange multiplier, χ is a scalar field and V (χ) +is its potential. Moreover G = R2 − 4RµνRµν + RµναβRµναβ is the Gauss-Bonnet scalar and h(χ) symbolizes the +Gauss-Bonnet coupling with the scalar field. Moreover we consider such class of Gauss-Bonnet coupling functions +that satisfy ¨h = ˙hH. This condition actually leads to the speed of the gravitational wave as unity in the context of +GB theory and makes the model compatible with the GW170817 event. For a certain γ(N) in the context of entropic +cosmology, there exists an equivalent set of GB parameters in the side of Gauss-Bonnet cosmology that results to the +same cosmological evolution as of the generalized entropy. In particular, the equivalent forms of ˜V (χ) and λ(t) for a +certain γ(N) turn out to be, +˜V (χ) = −8πG F1 [γ(N), γ′(N)] +� 1 +κ2 + 8h0a(t)H(t) +� ���� +t=χ/µ2 , +(69) +µ4λ(t) = −8πG F2 [γ(N), γ′(N)] +� 1 +κ2 − 8h0a(t)H(t) +� +, +(70) +where the functions F1 [γ(N), γ′(N)] and F2 [γ(N), γ′(N)] are given by, +F1 [γ(N), γ′(N)] = − +� 3ϵα +4βG2 +� +� +����ln +� +� +� +� +� +� +� +� +� +1 +2 +� +2 +γ(N)+√ +γ2(N)+4 +�1/β +− 1 +� +� +� +� +� +� +� +� +� +� +���� +−1 ++ H4 +� γ′(N) +8π2γ(N) +� +× +� +� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��β +− +� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��−β +α sech2 � +ϵπα +βGH2 +� �� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��β−1 ++ +� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��−β−1� +� + +14 +and +F2 [γ(N), γ′(N)] = H4 +� γ′(N) +8π2γ(N) +� +� +��� +� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��β +− +� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��−β +α sech2 � +ϵπα +βGH2 +� �� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��β−1 ++ +� +1 + 1 +ϵ tanh +� +ϵπα +βGH2 +��−β−1� +� +��� +respectively. Based on Eq.(69) and Eq.(70), we may argue that the entropic cosmology of Sns can be equivalently +represented by Gauss-Bonnet cosmology. +As mentioned earlier that we consider the quasi-matter bounce scenario described by the scale factor (65) to analyze +the perturbation, where the perturbation modes generate during the contracting phase deep in the sub-Hubble regime, +which in turn ensures the resolution of the horizon problem. The important quantities that we will need are, +Qa = −8˙hH2 = −4n2(1 + 2n) +� +�R +πG +� �R +R0 +� 1 +2 −n +, +Qb = −16˙hH = 4n(1 + 2n) +πG +� �R +R0 +� 1 +2 −n +, +Qc = Qd = 0 +, +Qe = −32˙h ˙H = 8n(1 + 2n) +� +�R +πG +� �R +R0 +� 1 +2 −n +, +Qf = 16 +� +¨h − ˙hH +� += 0 , +(71) +respectively, where R0 = +1 +t2 +0 and �R(t) = +R(t) +12n(1−4n). In regard to curvature perturbation, the Mukhanov-Sasaki (MS) +equation in Fourier mode comes as, +d2vk(η) +dη2 ++ +� +k2 − σ +η2 +� +vk(η) = 0 , +(72) +here η symbolizes the conformal time coordinate and v(k, η) is the scalar MS variable. Moreover σ is given by, +σ = ξ(ξ − 1) +� +�1 + 24 +� +1 − 4n2� +� �R +R0 +� 1 +2 −n� +� , +(73) +which is approximately a constant during the generation era of the perturbation modes in the sub-Hubble regime +during the contracting phase, due to the condition n < 1/2 (required to solve the horizon problem). In effect of which +and considering the Bunch-Davies initial condition, the scalar power spectrum PΨ(k, η) in the super-horizon scale +becomes, +PΨ(k, η) = +�� 1 +2π +� +1 +z |η| +Γ(ν) +Γ(3/2) +�2 �k|η| +2 +�3−2ν +, +(74) +In regard to the tensor perturbation, the Mukhanov-Sasaki equation takes the following form, +d2vT (k, η) +dη2 ++ +� +k2 − σT +η2 +� +vT (k, η) = 0 , +(75) +where vT (k, η) being the Fourier mode for the tensor MS variable, and σT has the following form, +σT = ξ(ξ − 1) +� +�1 − 16(1 − 4n2) +� �R +R0 +� 1 +2 −n� +� . +(76) +Due to n < 1/2, the quantity σT can be safely considered to be a constant during the generation era of the perturbation +modes at the contracting phase of the universe. Here it may be mentioned that both the tensor polarization modes +(+ and × polarization modes) obey the same evolution Eq.(75) – this means that the two polarization modes equally +contribute to the energy density of the tensor perturbation variable, and thus we will multiply by the factor ’2’ in the +final expression of the tensor power spectrum. Similar to the curvature perturbation variable, the tensor perturbation +initiates from the Bunch-Davies vacuum at the distant past, i.e. vT (k, η), i.e limk|η|≫1 vT (k, η) = +1 +√ +2ke−ikη. With +such initial condition, we obtain the tensor power spectrum for kth mode in the super-Hubble regime as, +PT (k, τ) = 2 +� 1 +2π +1 +zT |η| +Γ(θ) +Γ(3/2) +�2 �k|η| +2 +�3−2θ +, +(77) + +15 +where θ = +� +σT + 1 +4. Having obtained the scalar and tensor power spectra, we determine ns and r, and they are +given by (the suffix ’h’ with a quantity represents the quantity at the instant of horizon crossing), +ns = 4 − +√ +1 + 4σh , +r = 2 +� z(ηh) +zT (ηh) +Γ(θ) +Γ(ν) +�2 +(k |ηh|)2(ν−θ) , +(78) +where the quantities have the following forms, +ν = +� +σh + 1 +4 ; +σh = ξ(ξ − 1) +� +�1 + 24 +� +1 − 4n2� +� �Rh +R0 +� 1 +2 −n� +� , +θ = +� +σT,h + 1 +4 ; +σT,h = ξ(ξ − 1) +� +�1 − 16(1 − 4n2) +� �Rh +R0 +� 1 +2 −n� +� , +z(ηh) = − +1 +√n +� +an +0 +κ �Rn +h +� � +�1 − 24n(1 + 2n) +� �Rh +R0 +� 1 +2 −n� +� , +zT (ηh) = 1 +√ +2 +� +an +0 +κ �Rn +h +� � +�1 + 16n(1 + 2n) +� �Rh +R0 +� 1 +2 −n� +� . +(79) +0.960 0.962 0.964 0.966 0.968 0.970 +0.01177 +0.01178 +0.01179 +0.01180 +0.01181 +0.01182 +ns +r +FIG. 2: Parametric plot of ns (along x-axis) vs. r (along y-axis) with respect to n. Here we take α = [0.0938, 0.0939] and +β = +π +16. +Here �Rh represents the Ricci scalar at the horizon crossing, and using the horizon crossing condition kηh = +2n +1−2n, +it comes as, +�Rh = +� +1 +26nan +0 +�2/(1−2n) +By−2 . +(80) +Therefore it is clear that ns and r in the present context depends on the parameters n and a0. Here we need to recall +that n and a0 are related to the entropic parameters as n = √α and a0 = π/ (4β) respectively. It turns out that the +theoretical predictions for ns and r get simultaneously compatible with the recent Planck data for a small range of +the entropic parameters given by: α = [0.0938, 0.0939] and β = π +16, see Fig.[2]. + +16 +VII. +CONCLUSION +In this short review article, we have proposed generalized entropic function(s) and have addressed their implications +on black hole thermodynamics as well as on cosmology. In the first half of the paper, a 4-parameter and a 3-parameter +generalized entropy functions are shown, which are able to generalize the known entropies proposed so far, like the +Tsallis, R´enyi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable choices of the +respective entropic parameters. However the 4-parameter entropy functions proves to be more general compared to +the 3-parameter entropy function, in particular, the 3-parameter entropy does not converge to the Kaniadakis entropy +for any choices of the parameters, unlike to the entropy having 4 parameters which generalizes all the known entropies +including the Kaniadakis one. Thus regarding to the number of parameters in a generalized entropy function, we have +provided a conjecture – “The minimum number of parameters required in a generalized entropy function that can +generalize all the known entropies mentioned above is equal to four”. Consequently the interesting implications of +3-parameter entropy on black hole thermodynamics and the 4-parameter entropy on cosmology have been addressed. +It turns out that the entropic cosmology corresponding to the 4-parameter generalized entropy results to an unified +cosmological scenario of early inflation and the late dark energy era of the universe, where the observable quantities +are found to be compatible with the recent Planck data for certain viable ranges of the entropic parameters. +Despite these successes, here it deserves mentioning that the 4-parameter entropy function (Sg) seems to be plagued +with singularity for certain cosmological evolution of the universe. In particular, Sg diverges at the instant when the +Hubble parameter vanishes, for instance at the instant of bounce in the context of bounce cosmology. With this +spirit, we have proposed a singular-free 5-parameter entropy function (Sns) which converges to all the known entropy +functions for particular limits of the entropic parameters, and at the same time, also proves to be non-singular for the +entire cosmological evolution of the universe even at H = 0 (where H represents the Hubble parameter). Regarding +to the non-singular entropy, a second conjecture has been given : “The minimum number of parameters required in +a generalized entropy function that can generalize all the known entropies, and at the same time, is also singular-free +during the universe’s evolution – is equal to five”. Such non-singular behaviour of Sns proves to be useful in describing +the bounce cosmology, in particular, the entropic cosmology corresponding to Sns naturally allows symmetric bounce +universe. With the perturbation analysis in the context of entropic bounce, it has been shown that the observable +quantities like the spectral tilt and the tensor-to-scalar ratio are simultaneously compatible with the Planck data in +the background of symmetric quasi-matter bounce scenario. +Finally we would like to mention that the proposals of generalized entropy functions (Sg or Sns) opens a new +directions in theoretical physics, and its vast consequences may hint some unexplored directions of black hole thermo- +dynamics as well as of cosmology. For example, it will be of utmost interest to study the aspects of the generalized +entropy functions on primordial black hole formation or primordial gravitational wave or the recently found astro- +physical black holes as well. With the recent and future advancements of different detectors (like the GW detectors +or regarding the black hole detection), we hope that these study can indirectly quantify the viable ranges of entropic +parameters. +Acknowledgments +This work was supported by MINECO (Spain), project PID2019-104397GB-I00 and also partially supported by the +program Unidad de Excelencia Maria de Maeztu CEX2020-001058-M, Spain (SDO). This research was also supported +in part by the International Centre for Theoretical Sciences (ICTS) for the online program - Physics of the Early +Universe (code: ICTS/peu2022/1) (TP). +[1] J. D. Bekenstein, Phys. Rev. D 7 (1973), 2333-2346 doi:10.1103/PhysRevD.7.2333 +[2] S. W. Hawking, Commun. Math. Phys. 43 (1975), 199-220 [erratum: +Commun. Math. Phys. 46 (1976), 206] +doi:10.1007/BF02345020 +[3] J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973), 161-170 doi:10.1007/BF01645742 +[4] R. M. Wald, Living Rev. 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Astrophys. 652 (2021), C4] +doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]]. + diff --git a/-tAzT4oBgHgl3EQfFfrV/content/tmp_files/load_file.txt b/-tAzT4oBgHgl3EQfFfrV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0dc6de7e4cf5b74b15db67c63d8f81d47abbd641 --- /dev/null +++ b/-tAzT4oBgHgl3EQfFfrV/content/tmp_files/load_file.txt @@ -0,0 +1,1121 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf,len=1120 +page_content='Generalised (non-singular) entropy functions with applications to cosmology and black holes Sergei D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Odintsov1,2 , Tanmoy Paul3 1) ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain 2) Institute of Space Sciences (ICE, CSIC) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Can Magrans s/n, 08193 Barcelona, Spain 3) Department of Physics, Chandernagore College, Hooghly - 712 136, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The growing interest of different entropy functions proposed so far (like the Bekenstein-Hawking, Tsallis, R´enyi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies) towards black hole thermodynamics as well as towards cosmology lead to the natural question that whether there exists a generalized entropy function that can generalize all these known entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With this spirit, we propose a new 4-parameter entropy function that seems to converge to the aforementioned known entropies for certain limits of the entropic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The proposal of generalized entropy is extended to non-singular case, in which case , the entropy proves to be singular-free during the entire cosmological evolution of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The hallmark of such generalized entropies is that it helps us to fundamentally understand one of the important physical quantities namely “entropy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Consequently we address the implications of the generalized entropies on black hole thermodynamics as well as on cosmology, and discuss various constraints of the entropic parameters from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' INTRODUCTION One of the most important discoveries in theoretical physics is the black body radiation of a black hole, which is described by a certain temperature and by a Bekenstein-Hawking entropy function [1, 2] (see [3, 4] for extensive reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' On contrary to classical thermodynamics where the entropy is proportional to volume of the system under consideration, the Bekenstein-Hawking entropy is proportional to the area of the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Such unusual behaviour of the black hole entropy leads to the proposals of different entropy functions, such as, the Tsallis [5], R´enyi [6], Barrow [7], Sharma-Mittal [8], Kaniadakis [9] and the Loop Quantum Gravity entropies [10] are well known entropy functions proposed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' All of these known entropies have the common properties like – (1) they seem to be the monotonic increasing function with respect to the Bekenstein-Hawkinh entropy variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) they obey the third law of thermodynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' all of these entropies tend to zero as S → 0 (where S represents the Bekenstein-Hawking entropy) and (3) they converge to the Bekenstein-Hawking entropy for suitable choices of the respective entropic parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' the Tsallis entropy goes to the Bekenstein-Hawking entropy when the Tsallis exponent tends to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Furthermore, these entropies have rich consequences towards cosmology, particularly in describing the dark energy era of the universe [16–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The growing interest of such known entropies and due to their common properties lead to a natural question that whether there exists some generalized entropy function which is able to generalize all the known entropies proposed so far for suitable limits of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The entropy functions are extensively applied in the realm of black hole thermodynamics and cosmological evolution of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Recently we showed that the entropic cosmology corresponding to different entropy functions can be equivalently represented by holographic cosmology where the equivalent holographic cut-offs come in terms of either particle horizon and its derivative or the future horizon and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' One of the mysteries in today’s cosmology is to explain the acceleration of the universe in the high as well as in the low curvature regime, known as inflation and the dark energy era respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' These eras are well described by entropic cosmology or equivalently by holographic cosmology [16–51, 53–60], and more interestingly, the entropic cosmology proves to be useful to unify the early inflation and the late dark energy era of the universe in a covariant manner [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Apart from the inflation, the holographic cosmology turns out to be useful in describing the bouncing scenario [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In regard to the bounce scenario, the energy density sourced from the holographic principle or from some entropy function under consideration helps to violate the null energy condition at a finite time, which in turn triggers a non-singular bouncing universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' However here it deserves mentioning that all the known entropies mentioned above (like Tsallis , R´enyi, Barrow, Sharma- Mittal, Kaniadakis and the Loop Quantum Gravity entropies) become singular (or diverge) at a certain cosmological evolution of the universe, particularly in the context of bounce cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Actually such entropies contain a factor that is proportional to 1/H2 (where H is the Hubble parameter), and thus they diverge at the instant when the Hubble parameter vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e, at the instant of a bounce in bouncing cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This makes such known entropies ill-defined in describing a non-singular bounce scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Based on the above arguments, the questions that naturally arise are following: Does there exist a generalized entropy function that generalizes all the known entropies proposed so far ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01013v1 [gr-qc] 3 Jan 2023 2 If so, then what is its implications on black hole thermodynamics as well as on cosmology ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Similar to the known entropies, is the generalized entropy becomes singular at the instant when the Hubble parameter of the universe vanishes, for instance, in the bounce cosmology ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' If so, then does there exist an entropy function that generalizes all the known entropies, and at the same time, also proves to be singular-free during the entire cosmic evolution of the universe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The present article, based on some of our previous works [50–52], gives a brief review in answering the above questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The notations or conventions in this article are following: we will follow the (−, +, +, +) signature of the spacetime metric, and κ2 = 8πG = 1 M 2 Pl where G is the Newton’s constant or MPl denotes the four dimensional Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In regard to the cosmological evolution, a(t) and H(t) are the scale factor and the Hubble parameter of the universe respectively, N being the e-folding number, an overprime will denote d dη where η is the conformal time, an overdot will symbolize d dt with t being the cosmic time, otherwise an overprime with some argument will represent the derivative of the function with respect to that argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' POSSIBLE GENERALIZATIONS OF KNOWN ENTROPIES Here we will propose a generalized four-parameter entropy function which can lead to various known entropy functions proposed so far for suitable choices of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Let us start with the Bekenstein-Hawking entropy, the very first proposal of thermodynamical entropy of black hole physics [1, 2], S = A 4G , (1) where A = 4πr2 h is the area of the horizon and rh is the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Consequently, different entropy functions have been introduced depending on the system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Let us briefly recall some of the entropy functions proposed so far: For the systems with long range interactions where the Boltzmann-Gibbs entropy is not applied, one needs to introduce the Tsallis entropy which is given by [5], ST = A0 4G � A A0 �δ , (2) where A0 is a constant and δ is the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The R´enyi entropy is given by [6], SR = 1 α ln (1 + αS) , (3) where S is identified with the Bekenstein-Hawking entropy and α is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The Barrow entropy is given by [7], SB = � A APl �1+∆/2 , (4) where A is the usual black hole horizon area and APl = 4G is the Planck area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The Barrow entropy describes the fractal structures of black hole that may generate from quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The Sharma-Mittal entropy is given by [8], SSM = 1 R � (1 + δ S)R/δ − 1 � , (5) where R and δ are two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The Sharma-Mittal entropy can be regarded as a possible combination of the Tsallis and R´enyi entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 3 The Kaniadakis entropy function is of the following form [9]: SK = 1 K sinh (KS) , (6) where K is a phenomenological parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In the context of Loop Quantum Gravity, one may get the following entropy function [10]: Sq = 1 (1 − q) � e(1−q)Λ(γ0)S − 1 � , (7) where q is the exponent and Λ(γ0) = ln 2/ �√ 3πγ0 � with γ0 being the Barbero-Immirzi parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The γ0 generally takes either γ0 = ln 2 π √ 3 or γ0 = ln 3 2π √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' However with γ0 = ln 2 π √ 3, Λ(γ0) becomes unity and Sq resembles with the Bekenstein-Hawking entropy for q → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' All the above entropies – (1) obeys the generalized third law of thermodynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e the entropy function(s) vanishes at the limit S → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) monotonically increases with respect to the Bekenstein-Hawking variable and (3) converges to the Bekenstein-Hawking entropy for suitable limit of the entropic parameter, for example, the Tsallis entropy tends to S at δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In [50, 51], we proposed two different entropy functions containing 6-parameters and 4-parameters respectively, which can generalize all the known entropies mentioned from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, the generalized entropies are given by, 6 parameter entropy : S6 [α±, β±, γ±] = 1 α+ + α− �� 1 + α+ β+ Sγ+ �β+ − � 1 + α− β− Sγ− �−β−� , (8) 4 parameter entropy : Sg [α+, α−, β, γ] = 1 γ �� 1 + α+ β S �β − � 1 + α− β S �−β� , (9) where the respective parameters are given in the argument and they are assumed to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here S is the Bekenstein-Hawking entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Below we prove the generality of the above generalized entropy functions, in particular, we show that both the generalized entropies reduce to the known entropies mentioned in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2), (3), (4), (5), (6), and (7) for suitable choices of the respective parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here we establish it particularly for the 4-parameter entropy function, while the similar calculations hold for the 6-parameter entropy as well [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For α+ → ∞ and α− = 0, one gets Sg = 1 γ �α+ β �β Sβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' If we further choose γ = (α+/β)β, then the generalized entropy reduces to Sg = Sβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Therefore with β = δ or β = 1 + ∆, the generalized entropy resembles with the Tsallis entropy or with the Barrow entropy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For α− = 0, β → 0 and α+ β → finite – Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) leads to, Sg = 1 γ �� 1 + α+ β S �β − 1 � = 1 γ � exp � β ln � 1 + α+ β S �� − 1 � ≈ 1 (γ/β) ln � 1 + α+ β S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Further choosing γ = α+ and identifying α+ β = α, we can write the above expression as, Sg = 1 α ln (1 + α S) , (10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=', Sg reduces to the R´enyi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 4 In the case when α− = 0, the generalized entropy becomes, Sg = 1 γ �� 1 + α+ β S �β − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (11) Thereby identifying γ = R, α+ = R and β = R/δ, the generalized entropy function Sg gets similar to the Sharma-Mittal entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For β → ∞, α+ = α− = γ 2 = K, we may write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) as, Sg = 1 2K lim β→∞ �� 1 + K β S �β − � 1 + K β S �−β� = 1 2K � eKS − e−KS� = 1 K sinh (KS) → Kaniadakis entropy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (12) Finally, with α− = 0, β → ∞ and γ = α+ = (1 − q), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) immediately yields, Sg = 1 (1 − q) � e(1−q)S − 1 � , which is the Loop Quantum Gravity entropy with Λ(γ0) = 1 or equivalently γ0 = ln 2 π √ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Furthermore, the generalized entropy function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) shares the following properties: (1) Sg → 0 for S → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) The entropy Sg [α+, α−, β, γ] is a monotonically increasing function with S because both the terms � 1 + α+ β S �β and − � 1 + α− β S �−β present in the expression of Sg increase with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (3) Sg [α+, α−, β, γ] seems to converge to the Bekenstein-Hawking entropy at certain limit of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, for α+ → ∞, α− = 0, γ = (α+/β)β and β = 1, the generalized entropy function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) becomes equivalent to the Bekenstein-Hawking entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here it deserves mentioning that beside the entropy function proposed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) which contains four parameters, one may consider a three parameter entropy having the following form: S3[α, β, γ] = 1 γ �� 1 + α β S �β − 1 � , (13) where α, β and γ are the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The above form of S3[α, β, γ] satisfies all the properties, like – (1) S3[α, β, γ] → 0 for S → 0, (2) S3 is an increasing function with S and (3) S3 has a Bekenstein-Hawking entropy limit for the choices: α → ∞, γ = (α/β)β and β = 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' However S3[α, β, γ] is not able to generalize all the known entropies mentioned from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (7), in particular, S3[α, β, γ] does not reduce to the Kaniadakis entropy for any possible choices of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Conjecture - I: Based on our findings, we propose the following postulate in regard to the generalized entropy function – “The minimum number of parameters required in a generalized entropy function that can generalize all the known entropies mentioned from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (7) is equal to four”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Below we will address the possible implications of such generalized entropies on black hole thermodynamics as well as on cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' BLACK HOLE THERMODYNAMICS WITH 3-PARAMETER GENERALIZED ENTROPY It is interesting to see what happens when the generalized entropy (13) is ascribed to the prototypical black hole, given by the Schwarzschild geometry [50] ds2 = −f(r) dt2 + dr2 f(r) + r2dΩ2 (2) , f(r) = 1 − 2GM r , (14) 5 where M is the black hole mass and dΩ2 (2) = dϑ2 + sin2 ϑ dϕ2 is the line element on the unit two-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The black hole event horizon is located at the Schwarzschild radius rH = 2GM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (15) Studying quantum field theory on the spacetime with this horizon, Hawking discovered that the Schwarzschild black hole radiates with a blackbody spectrum at the temperature TH = 1 8πGM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (16) As explained in general below, if we assume that the mass M coincides with the thermodynamical energy, then the temperature obtained from the thermodynamical law is different from the Hawking temperature, a contradiction for observers detecting Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Alternatively, if the Hawking temperature TH is identified with the physical black hole temperature, the obtained thermodynamical energy differs from the Schwarzschild mass M even for the Tsallis entropy or the R´enyi entropy, which seems to imply a breakdown of energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' If the mass M coincides with the thermodynamical energy E of the system due to energy conservation, as in, in order for this system to be consistent with the thermodynamical equation dSG = dE/T one needs to define the generalized temperature TG as 1 TG ≡ dSG dM (17) which is, in general, different from the Hawking temperature TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For example, in the case of the entropy (13), one has 1 TG = α γ � 1 + α β S �β−1 dS dM = α γ � 1 + α β S �β−1 1 TH , (18) where S = A 4G = 4πGM 2 = 1 16πGTH 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (19) Because α γ � 1 + α β S �β−1 ̸= 1, it is necessarily TG ̸= TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Since the Hawking temperature (16) is the temperature perceived by observers detecting Hawking radiation, the generalized temperature TG in (18) cannot be a physically meaningful temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (17), assuming that the thermodynamical energy E is the black hole mass M leads to an unphysical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' As an alternative, assume that the thermodynamical temperature T coincides with the Hawking temperature TH instead of assuming E = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This assumption leads to dEG = TH dSG = dSG dS dS √ 16πGS (20) which, in the case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (13), yields dEG = α γ � 1 + α β S �β−1 dS √ 16πGS = α γ √ 16πG � S−1/2 + α (β − 1) β S1/2 + O � S3/2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (21) The integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (21) gives EG = α γ √ 16πG � 2S1/2 + 2α (β − 1) 3β S3/2 + O � S5/2�� = α γ � M + 4πGα (β − 1) 3β M 3 + O � M 5�� , (22) where the integration constant is determined by the condition that EG = 0 when M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Even when α = γ, due to the correction 4πGα(β−1) 3β M 3, the expression (22) of the thermodynamical energy ER obtained differs from the black hole mass M, EG ̸= E, which seems unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 6 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' COSMOLOGY WITH THE 4-PARAMETER GENERALIZED ENTROPY Here we consider the 4-parameter generalized entropy (9), which is indeed more generalized compared to the 3- parameter entropy function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (13), to describe the cosmological behaviour of the universe [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, we examine whether the 4-parameter entropy function results to an unified scenario of early inflation and the late dark energy era of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The Friedmann-Lemaˆıtre-Robertson-Walker space-time with flat spacial part will serve our purpose, in particular, ds2 = −dt2 + a2(t) � i=1,2,3 � dxi�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (23) Here a(t) is called as a scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The radius rH of the cosmological horizon is given by rH = 1 H , (24) with H = ˙a/a is the Hubble parameter of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Then the entropy contained within the cosmological horizon can be obtained from the Bekenstein-Hawking relation [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Furthermore the flux of the energy E, or equivalently, the increase of the heat Q in the region comes as dQ = −dE = −4π 3 r3 H ˙ρdt = − 4π 3H3 ˙ρ dt = 4π H2 (ρ + p) dt , (25) where, in the last equality, we use the conservation law: 0 = ˙ρ + 3H (ρ + p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Then from the Hawking temperature [66] T = 1 2πrH = H 2π , (26) and by using the first law of thermodynamics TdS = dQ, one obtains ˙H = −4πG (ρ + p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Integrating the expression immediately leads to the first FRW equation, H2 = 8πG 3 ρ + Λ 3 , (27) where the integration constant Λ can be treated as a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Instead of the Bekenstein-Hawking entropy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (1), we may use the generalized entropy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9), in regard to which, the first law of thermodynamics leads to the following equation: ˙H �∂Sg ∂S � = −4πG (ρ + p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (28) With the explicit form of Sg from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9), the above equation turns out to be, 1 γ � α+ � 1 + πα+ βGH2 �β−1 + α− � 1 + πα− βGH2 �−β−1� ˙H = −4πG (ρ + p) (29) where we use S = A/(4G) = π/(GH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Using the conservation relation of the matter fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=', ˙ρ + 3H (ρ + p) = 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (29) can be written as, 2 γ � α+ � 1 + πα+ βGH2 �β−1 + α− � 1 + πα− βGH2 �−β−1� H dH = �8πG 3 � dρ , on integrating which, we obtain, GH4β πγ � 1 (2 + β) �GH2β πα− �β 2F1 � 1 + β, 2 + β, 3 + β, −GH2β πα− � + 1 (2 − β) �GH2β πα+ �−β 2F1 � 1 − β, 2 − β, 3 − β, −GH2β πα+ �� = 8πGρ 3 + Λ 3 , (30) where Λ is the integration constant (known as the cosmological constant) and 2F1(arguments) denotes the Hypergeo- metric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (29) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (30) represent the modified Friedmann equations corresponding to the generalized entropy function Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In the next section, we aim to study the cosmological implications of the modified Friedmann Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (29) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Early universe cosmology from the 4-parameter generalized entropy During the early stage of the universe we consider the matter field and the cosmological constant (Λ) to be absent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=', ρ = p = Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' During the early universe, the cosmological constant is highly suppressed with respect to the entropic energy density and thus we can safely neglect the Λ in studying the early inflationary scenario of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Therefore during the early universe, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (30) becomes, � 1 (2 + β) �GH2β πα− �β 2F1 � 1 + β, 2 + β, 3 + β, −GH2β πα− � + 1 (2 − β) �GH2β πα+ �−β 2F1 � 1 − β, 2 − β, 3 − β, −GH2β πα+ �� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (31) Here it may be mentioned that the typical energy scale during early universe is of the order ∼ 1016GeV (= 10−3MPl where recall that MPl is the Planck mass and MPl = 1/ √ 16πG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This indicates that the condition GH2 ≪ 1 holds during the early phase of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Owing to such condition, we can safely expand the Hypergeometric function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (31) as the Taylor series with respect to the argument containing GH2, and as a result, the above equation provides a constant Hubble parameter as the solution: H = 4πMPl �α+ β � (3 − β) (2 − β)(1 − β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (32) For α+ β ∼ 10−6 and β ≲ O(1), the constant Hubble parameter can be fixed at H ∼ 10−3MPl which can be identified with typical inflationary energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Therefore the entropic cosmology corresponding to the generalized entropy function Sg leads to a de-Sitter inflationary scenario during the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' However, a de-Sitter inflation has no exit mechanism, and moreover, the primordial curvature perturbation gets exactly scale invariant in the context of a de-Sitter inflation, which is not consistent with the recent Planck data [75] at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This indicates that the constant Hubble parameter obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (32) does not lead to a good inflationary scenario of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Thus in order to achieve a viable quasi de-Sitter inflation in the present context, we consider the parameters of Sg to be slowly varying functions with respect to the cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, we consider the parameter γ to vary and the other parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=', α+, α− and β) remain constant with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, γ(N) = � γ0 exp � − � Nf N σ(N) dN � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' N ≤ Nf γ0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' N ≥ Nf , (33) where γ0 is a constant and N denotes the inflationary e-folding number with Nf being the total e-folding number of the inflationary era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The function σ(N) has the following form, σ(N) = σ0 + e−(Nf −N) , (34) where σ0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The second term in the expression of σ(N) becomes effective only when N ≈ Nf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=', near the end of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The term e−(Nf −N) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (34) is actually considered to ensure an exit from inflation era and thus proves to be an useful one to make the inflationary scenario viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In such scenario where γ varies with N, the Friedmann equation turns out to be, − �2π G � � �� α+ � 1 + α+ β S �β−1 + α− � 1 + α− β S �−β−1 � 1 + α+ β S �β − � 1 + α− β S �−β � �� H′(N) H3 = σ(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (35) By using S = π/(GH2), or equivalently, 2HdH = − π GS2 dS, one can integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (35) to get H(N) as, H(N) = 4πMPl �α+ β � ���� 21/(2β) exp � − 1 2β � N 0 σ(N)dN � � 1 + � 1 + 4 (α+/α−)β exp � −2 � N 0 σ(N)dN ��1/(2β) � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (36) 8 The above solution of H(N) allows an exit from inflation at finite e-fold number which can be fixed at Nf = 58 for suitable choices of the entropic parameters [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Moreover we determine the spectral index for curvature perturbation (ns) and the tensor-to-scalar ratio (r) in the present context of entropic cosmology, and they are given by [51]: ns = 1 − 2σ0 � 1 + 4 (α+/α−)β exp [−2 (1 + σ0Nf)] (1 + σ0) � 1 + 4 (α+/α−)β − 8σ0 (α+/α−)β 1 + 4 (α+/α−)β , (37) and r = 16σ0 � 1 + 4 (α+/α−)β exp [−2 (1 + σ0Nf)] (1 + σ0) � 1 + 4 (α+/α−)β (38) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' It turns out that the theoretical expectations of ns and r get simultaneously compatible with the Planck data for the following ranges of the parameters: σ0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='013, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='017] , (α+/α−)β ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 , β = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='4] and (α+/β) ≈ 10−6 , (39) for Nf = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The consideration of α+ β ∼ 10−6 leads to the energy scale at the onset of inflation as H ∼ 10−3MPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Dark energy era from the 4-parameter generalized entropy In this section we will concentrate on late time cosmological implications of the generalized entropy function (Sg), where the cosmological constant Λ is considered to be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' During the late time, the parameter γ becomes constant, in particular γ = γ0, as we demonstrated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' As a result, the entropy function at the late time takes the following form, Sg = 1 γ0 �� 1 + α+ β S �β − � 1 + α− β S �−β� , (40) with S = π/(GH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Consequently, the energy density and pressure corresponding to the Sg are given by, ρg = 3H2 8πG � 1 − α+ γ0(2 − β) �GH2β πα+ �1−β� , pg = − ˙H 4πG � 1 − α+ γ0 �GH2β πα+ �1−β − �α+ γ0 � �α+ α− �β �GH2β πα+ �1+β� − ρg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (41) Therefore the dark energy density (ρD) is contributed from the entropic energy density (ρg) as well as from the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular ρD = ρg + 3 8πG �Λ 3 � , ρD + pD = ρg + pg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (42) Consequently, the dark energy EoS parameter comes with the following expression: ωD = pD/ρD = −1 − � 2 ˙H 3H2 � � �� 1 − α+ γ0 � GH2β πα+ �1−β − � α+ γ0 � � α+ α− �β � GH2β πα+ �1+β 1 − α+ γ0(2−β) � GH2β πα+ �1−β + Λ 3H2 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (43) In presence of the cosmological constant, the Friedmann equations are written as, H2 = 8πG 3 (ρm + ρD) = 8πG 3 (ρm + ρg) + Λ 3 , 9 ˙H = −4πG [ρm + (ρD + pD)] = −4πG [ρm + (ρg + pg)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (44) As usual, the fractional energy density of the pressureless matter and the dark energy satisfy Ωm + ΩD = 1 which along with ρm = ρm0 � a0 a �3 (with ρm0 being the present matter energy density) result to the Hubble parameter in terms of the red shift factor (z) as follows, H(z) = H0 � Ωm0(1 + z)3 √1 − ΩD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (45) Plugging the expression of ρg from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (41) into ΩD = � 8πG 3H2 � ρg + Λ 3 , and using the above form of H(z), we obtain, ΩD(z) = 1 − � α+ γ0(2−β) � 1 2−β � GH2 0β πα+ Ωm0(1 + z)3� 1−β 2−β � 1 + Λ 3H2 0Ωm0(1+z)3 �1/(2−β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (46) By using the above expressions, we determine the DE EoS parameter from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (43) as follows (see [51]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' ωD(z) = −1 + 1 (2 − β) � 1 + Λ 3H2 0Ωm0(1+z)3 � �N D � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (47) where N (the numerator) and D (the denominator) have the following forms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' N = 1 − Ωm0(2 − β) (1 + z) 3(1−β) (2−β) � � 1 + Λ 3H2 0Ωm0 [f(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Ωm0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' z)]1−β � + �α+ α− �β [Ωm0(2 − β)γ0/α+] 2β 1−β (1 + z) 6β 2−β × � � � � 1 + Λ 3H2 0Ωm0 �(1+β)/(1−β) [f(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Ωm0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' z)]1+β � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' and D = 1 − Ωm0 (1 + z) 3(1−β) (2−β) � 1 + Λ 3H2 0Ωm0 [f(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Ωm0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' z)]1−β � + Λ 3H2 0 �f(Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Ωm0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' z) (1 + z)3/(2−β) � (48) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Therefore ωD depends on the parameters: β, (α+/α−)β, γ0 and α+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Recall that the inflationary quantities are found to be simultaneously compatible with the Planck data if some of the parameters like α+, α− and β get constrained according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (39), while the parameter γ0 remains free from the inflationary requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With the aforementioned ranges of α+, α− and β, ωD(0) becomes compatible with the Planck observational data, provided γ0 lies within a small window as follows, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 × 10−4 ≤ γ0 (8πGH2 0)1−β ≤ 2 × 10−4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (49) Furthermore the deceleration parameter (symbolized by q) at present universe is obtained as, q = −1 + 3 2(2 − β) � 1 + Λ 3H2 0Ωm0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (50) Therefore for γm = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5×10−4, 2×10−4], the theoretical expression of q lies within q = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='56, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='42] which certainly contains the observational value of q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='535 from the Planck data [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='535 occurs for γm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='8 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Considering this value of γm and by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (47), we give the plot of ωD(z) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' z, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The figure reveals that that the theoretical expectation of the DE EoS parameter at present time acquires the value: ωD(0) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='950 which is well consistent with the Planck observational data [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' As a whole, we may argue that the entropic cosmology from the generalized entropy function Sg can unify the early inflation to the late dark energy era of the universe, for suitable ranges of the parameters given by: σ0 = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='013, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='017] , (α+/α−)β ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 , 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0 z ωD(z) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 1: ωD(z) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' z for a particular set of values of the parameters from their viable ranges as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (39) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (49), say β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='35, (α+/α−)β = 10, α+/β = 10−6 and γm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='8 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' β = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='4] and γm = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='5 × 10−4, 2 × 10−4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (51) Despite these successes, here it deserves mentioning that the entropy function Sg seems to be plagued with singularity for certain cosmological evolution of the universe, in particular, in the context of bounce cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Due to the reason that the Bekenstein-Hawking entropy can be expressed as S = π/ � GH2� , the generalized entropy Sg contains factor that is proportional to 1/H2 which diverges at H = 0, for instance at the instant of bounce in the context of bounce cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Therefore in a bounce scenario, the generalized entropy function shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) is not physical, and thus, we need to search for a different generalized entropy function which can lead to various known entropy functions for suitable choices of the parameters, and at the same time, proves to be non-singular for the entire cosmological evolution of the universe even at H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' SEARCH FOR A SINGULAR-FREE GENERALIZED ENTROPY With this spirit, we propose a new singular-free entropy function given by [52], Sns [α±, β, γ, ϵ] = 1 γ � � 1 + 1 ϵ tanh �ϵα+ β S ��β − � 1 + 1 ϵ tanh �ϵα− β S ��−β � , (52) where α±, β, γ and ϵ are the parameters which are considered to be positive, S symbolizes the Bekenstein-Hawking entropy and the suffix ’ns’ stands for ’non-singular’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In regard to the number of parameters, we propose a conjecture at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' First we demonstrate that the above entropy function remains finite, and thus is non-singular, during the whole cosmological evolution of a bouncing universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, the Sg takes the following form at the instant of bounce: Sns [α±, β, γ, ϵ] = 1 γ � � 1 + 1 ϵ �β − � 1 + 1 ϵ �−β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (53) Having demonstrated the non-singular behaviour of the entropy function, we now show that Sns of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (52), for suitable choices of the parameters, reduces to various known entropies proposed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For ϵ → 0, α+ → ∞ and α− = 0 along with the identification γ = (α+/β)β, Sns converges to the Tsallis entropy or to the Barrow entropy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The limit ϵ → 0, α− = 0, β → 0 and α+ β → finite results to the similarity between the non-singular generalized entropy Sg and the R´enyi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 11 For ϵ → 0 and α− → 0, the non-singular generalized entropy converges to the following form, Sns = 1 γ �� 1 + α+ β S �β − 1 � (54) Therefore with γ = R, α+ = R and β = R/δ, the above form of Sns becomes similar to the Sharma-Mittal entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For ϵ → 0, β → ∞, α+ = α− = γ 2 = K – the generalized entropy converges to the form of Kaniadakis entropy, Finally, ϵ → 0, α− → 0, β → ∞ and γ = α+ = (1 − q), the generalized entropy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (52) gets resemble with the Loop Quantum Gravity entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Furthermore, the generalized entropy function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (52) shares the following properties: (1) the non-singular generalized entropy satisfies the generalized third law of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (2) Sns [α±, β, γ, ϵ] turns out to be a monotonically increasing function of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (3) Sns [α±, β, γ, ϵ] proves to converge to the Bekenstein-Hawking entropy at certain limit of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' At this stage it deserves mentioning that we have proposed two different generalized entropy functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (9) and in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (52) respectively – the former entropy function contains four independent parameters while the latter one has five parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Furthermore both the entropies are able to generalize the known entropies for suitable choices of the respective parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' However as mentioned earlier that the entropy with four parameters becomes singular at H = 0 (for instance, in a bounce scenario when the Hubble parameter vanishes at the instant of bounce), while the entropy function having five parameters proves to be singular-free during the whole cosmological evolution of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Based on these findings, we give a second conjecture regarding the number of parameters in the non-singular generalized entropy function: Conjecture - II: “The minimum number of parameters required in a generalized entropy function that can gen- eralize all the known entropies, and at the same time, is also singular-free during the universe’s evolution – is equal to five”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' COSMOLOGY WITH THE NON-SINGULAR GENERALIZED ENTROPY Applying the thermodynamic laws to the non-singular generalized entropy function Sns and by following the same procedure as of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' [IV], one gets the cosmological field equations corresponding to the Sgns [52]: 1 γ � α+ sech2 � ϵπα+ βGH2 � � 1 + 1 ϵ tanh � ϵπα+ βGH2 ��β−1 + α− sech2 � ϵπα− βGH2 � � 1 + 1 ϵ tanh � ϵπα− βGH2 ��−β−1 � ˙H = −4πG (ρ + p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (55) Owing to the conservation equation of matter fields, in particular ˙ρ + 3H (ρ + p) = 0, the above expression can be integrated to get f (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' α±, β, γ, ϵ) = 8πGρ 3 + Λ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (56) Here the integration constant is symbolized by Λ and the function f has the following form: f (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' α±, β, γ, ϵ) = 2 γ � � α+ sech2 � ϵπα+ βGH2 � � 1 + 1 ϵ tanh � ϵπα+ βGH2 ��β−1 + α− sech2 � ϵπα− βGH2 � � 1 + 1 ϵ tanh � ϵπα− βGH2 ��−β−1 � H dH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (57) In regard to the functional form of f (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' α±, β, γ, ϵ), we would like to mention that the integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (57) may not be performed in a closed form, unless certain conditions are imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For example, we consider GH2 ≪ 1 which is, in 12 fact, valid during the universe’s evolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e the Hubble parameter is less than the Planck scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With GH2 ≪ 1, the functional form of f turns out to be, f (H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' α±, β, γ, ϵ) = 4 γ H2 � α+ � 1 + 1 ϵ �β−1 � exp � −2ϵπα+ βGH2 � + �2ϵπα+ βGH2 � Ei � −2ϵπα+ βGH2 �� + α− � 1 + 1 ϵ �−β−1 � exp � −2ϵπα− βGH2 � + �2ϵπα− βGH2 � Ei � −2ϵπα− βGH2 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (58) Therefore as a whole, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (55) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (56) are the cosmological field equations corresponding to the generalized entropy Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Non-singular entropy on bounce cosmology In this section, we will address the implications of the generalized entropy Sns on non-singular bounce cosmology, in particular, we will investigate whether the entropic energy density can trigger a viable bounce during the early stage of the universe that is consistent with the observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For this purpose, we take the matter field and the cosmological constant to be absent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=', ρ = p = Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In effect, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (55) becomes, 1 γ � α+ sech2 � ϵπα+ βGH2 � � 1 + 1 ϵ tanh � ϵπα+ βGH2 ��β−1 + α− sech2 � ϵπα− βGH2 � � 1 + 1 ϵ tanh � ϵπα− βGH2 ��−β−1 � ˙H = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (59) The parameters (α±, β, γ, ϵ) are positive, and thus the solution of the above equation is given by: ˙H = 0 or equivalently H = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Clearly H = constant does not lead to the correct evolution of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Thus similar to the previous case, we consider the parameters of Sns[α±, β, γ, ϵ] vary with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, we consider the parameter γ to vary with time, and all the other parameters remain fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ = γ(N) , (60) with N being the e-fold number of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In such scenario where γ(N) varies with time, the Friedmann equation corresponds to Sns[α±, β, γ, ϵ] gets modified compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (59), and is given by: � �� α+ sech2 � ϵα+ β S � � 1 + 1 ϵ tanh � ϵα+ β S ��β−1 + α− sech2 � ϵα− β S � � 1 + 1 ϵ tanh � ϵα− β S ��−β−1 � 1 + 1 ϵ tanh � ϵα+ β S ��β − � 1 + 1 ϵ tanh � ϵα− β S ��−β � �� dS = γ′(N) γ(N) dN (61) where an overprime denotes d dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (61) can be integrated to get, tanh � ϵπα βGH2 � = � γ(N) + � γ2(N) + 4 2 �1/β − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (62) where we take α+ = α− = α (say, without losing any generality) in order to extract an explicit solution of H(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Due to the appearance of quadratic power of H, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (62) allows a positive branch as well as a negative branch of the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This leads to a natural possibility of symmetric bounce in the present context of singular free generalized entropic cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Moreover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (62) also demonstrates that the explicit evolution of H(N) does depend on the form of γ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In the following, we will consider two cases where we will determine the form of γ(N) such that it gives two different symmetric bounce scenarios respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The exponential bounce described by the scale factor, a(t) = exp � a0t2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (63) This results to a symmetric bounce at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here a0 is a constant having mass dimension [+2] – this constant is related with the entropic parameters of Sns and thus, without losing any generality, we take a0 = ϵπα 4Gβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Such an exponential bounce can be achieved from singular free entropic cosmology provided the γ(N) is given by, γ(N) = � 1 + 1 ϵ tanh � 1 N ��β − � 1 + 1 ϵ tanh � 1 N ��−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (64) 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The quasi-matter bounce is described by, In this case, the scale factor is, a(t) = � 1 + a0 � t t0 �2�n (65) which is symmetric about t = 0 when the bounce happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The n, a0 and t0 considered in the scale factor are related to the entropic parameters, and we take it as follows: n = √α , a0 = π 4β and t0 = � G/ϵ , (66) with G being the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The relation between (n, a0, t0) with the entropic parameters can be considered in a different way compared to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (66), however for a simplified expression of γ(N) we consider the relations as of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='(66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Consequently the γ(N) which leads to such quasi-matter bounce, comes as, γ(N) = � 1 + 1 ϵ tanh � e−N/√α � eN/√α − 1 � 1 2 ��β − � 1 + 1 ϵ tanh � e−N/√α � eN/√α − 1 � 1 2 ��−β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (67) Here it deserves mentioning that in the case of exponential bounce, the comoving Hubble radius asymptotically goes to zero and thus the perturbation modes remain at the super-Hubble regime at the distant past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This may results to the “horizon problem” in the exponential bounce scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' On contrary, the comoving Hubble radius in the case of quasi-matter bounce asymptotically diverges to infinity at both sides of the bounce, and thus the perturbation modes lie within the deep sub-Hubble regime at the distant past – this resolves the horizon issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Based on this arguments, we will concentrate on the quasi-matter bounce to perform the perturbation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In regard to the perturbation analysis, we represent the present entropic cosmology with the ghost free Gauss- Bonnet (GB) theory of gravity proposed in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The motivation of such representation is due to the rich structure of the Gauss-Bonnet theory in various directions of cosmology [68–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The action for f(G) gravity is given by [67], S = � d4x√−g � 1 2κ2 R + λ �1 2∂µχ∂µχ + µ4 2 � − 1 2∂µχ∂µχ + h (χ) G − V (χ) � , (68) where µ is a constant having mass dimension [+1], λ represents the Lagrange multiplier, χ is a scalar field and V (χ) is its potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Moreover G = R2 − 4RµνRµν + RµναβRµναβ is the Gauss-Bonnet scalar and h(χ) symbolizes the Gauss-Bonnet coupling with the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Moreover we consider such class of Gauss-Bonnet coupling functions that satisfy ¨h = ˙hH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This condition actually leads to the speed of the gravitational wave as unity in the context of GB theory and makes the model compatible with the GW170817 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For a certain γ(N) in the context of entropic cosmology, there exists an equivalent set of GB parameters in the side of Gauss-Bonnet cosmology that results to the same cosmological evolution as of the generalized entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' the equivalent forms of ˜V (χ) and λ(t) for a certain γ(N) turn out to be,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' ˜V (χ) = −8πG F1 [γ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ′(N)] � 1 κ2 + 8h0a(t)H(t) � ���� t=χ/µ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (69) µ4λ(t) = −8πG F2 [γ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ′(N)] � 1 κ2 − 8h0a(t)H(t) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (70) where the functions F1 [γ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ′(N)] and F2 [γ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ′(N)] are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' F1 [γ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ′(N)] = − � 3ϵα 4βG2 � � ����ln � � � � � � � � � 1 2 � 2 γ(N)+√ γ2(N)+4 �1/β − 1 � � � � � � � � � � ���� −1 + H4 � γ′(N) 8π2γ(N) � × � � 1 + 1 ϵ tanh � ϵπα βGH2 ��β − � 1 + 1 ϵ tanh � ϵπα βGH2 ��−β α sech2 � ϵπα βGH2 � �� 1 + 1 ϵ tanh � ϵπα βGH2 ��β−1 + � 1 + 1 ϵ tanh � ϵπα βGH2 ��−β−1� � 14 and F2 [γ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' γ′(N)] = H4 � γ′(N) 8π2γ(N) � � ��� � 1 + 1 ϵ tanh � ϵπα βGH2 ��β − � 1 + 1 ϵ tanh � ϵπα βGH2 ��−β α sech2 � ϵπα βGH2 � �� 1 + 1 ϵ tanh � ϵπα βGH2 ��β−1 + � 1 + 1 ϵ tanh � ϵπα βGH2 ��−β−1� � ��� respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (69) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (70), we may argue that the entropic cosmology of Sns can be equivalently represented by Gauss-Bonnet cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' As mentioned earlier that we consider the quasi-matter bounce scenario described by the scale factor (65) to analyze the perturbation, where the perturbation modes generate during the contracting phase deep in the sub-Hubble regime, which in turn ensures the resolution of the horizon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' The important quantities that we will need are, Qa = −8˙hH2 = −4n2(1 + 2n) � �R πG � �R R0 � 1 2 −n , Qb = −16˙hH = 4n(1 + 2n) πG � �R R0 � 1 2 −n , Qc = Qd = 0 , Qe = −32˙h ˙H = 8n(1 + 2n) � �R πG � �R R0 � 1 2 −n , Qf = 16 � ¨h − ˙hH � = 0 , (71) respectively, where R0 = 1 t2 0 and �R(t) = R(t) 12n(1−4n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In regard to curvature perturbation, the Mukhanov-Sasaki (MS) equation in Fourier mode comes as, d2vk(η) dη2 + � k2 − σ η2 � vk(η) = 0 , (72) here η symbolizes the conformal time coordinate and v(k, η) is the scalar MS variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Moreover σ is given by, σ = ξ(ξ − 1) � �1 + 24 � 1 − 4n2� � �R R0 � 1 2 −n� � , (73) which is approximately a constant during the generation era of the perturbation modes in the sub-Hubble regime during the contracting phase, due to the condition n < 1/2 (required to solve the horizon problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In effect of which and considering the Bunch-Davies initial condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' the scalar power spectrum PΨ(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' η) in the super-horizon scale becomes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' PΨ(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' η) = �� 1 2π � 1 z |η| Γ(ν) Γ(3/2) �2 �k|η| 2 �3−2ν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (74) In regard to the tensor perturbation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' the Mukhanov-Sasaki equation takes the following form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' d2vT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' η) dη2 + � k2 − σT η2 � vT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' η) = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (75) where vT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' η) being the Fourier mode for the tensor MS variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' and σT has the following form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' σT = ξ(ξ − 1) � �1 − 16(1 − 4n2) � �R R0 � 1 2 −n� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (76) Due to n < 1/2, the quantity σT can be safely considered to be a constant during the generation era of the perturbation modes at the contracting phase of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here it may be mentioned that both the tensor polarization modes (+ and × polarization modes) obey the same evolution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (75) – this means that the two polarization modes equally contribute to the energy density of the tensor perturbation variable, and thus we will multiply by the factor ’2’ in the final expression of the tensor power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Similar to the curvature perturbation variable, the tensor perturbation initiates from the Bunch-Davies vacuum at the distant past, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' vT (k, η), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='e limk|η|≫1 vT (k, η) = 1 √ 2ke−ikη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With such initial condition, we obtain the tensor power spectrum for kth mode in the super-Hubble regime as, PT (k, τ) = 2 � 1 2π 1 zT |η| Γ(θ) Γ(3/2) �2 �k|η| 2 �3−2θ , (77) 15 where θ = � σT + 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Having obtained the scalar and tensor power spectra, we determine ns and r, and they are given by (the suffix ’h’ with a quantity represents the quantity at the instant of horizon crossing), ns = 4 − √ 1 + 4σh , r = 2 � z(ηh) zT (ηh) Γ(θ) Γ(ν) �2 (k |ηh|)2(ν−θ) , (78) where the quantities have the following forms, ν = � σh + 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' σh = ξ(ξ − 1) � �1 + 24 � 1 − 4n2� � �Rh R0 � 1 2 −n� � , θ = � σT,h + 1 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' σT,h = ξ(ξ − 1) � �1 − 16(1 − 4n2) � �Rh R0 � 1 2 −n� � , z(ηh) = − 1 √n � an 0 κ �Rn h � � �1 − 24n(1 + 2n) � �Rh R0 � 1 2 −n� � , zT (ηh) = 1 √ 2 � an 0 κ �Rn h � � �1 + 16n(1 + 2n) � �Rh R0 � 1 2 −n� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (79) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='01182 ns r FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 2: Parametric plot of ns (along x-axis) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' r (along y-axis) with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here we take α = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0938, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0939] and β = π 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here �Rh represents the Ricci scalar at the horizon crossing, and using the horizon crossing condition kηh = 2n 1−2n, it comes as, �Rh = � 1 26nan 0 �2/(1−2n) By−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' (80) Therefore it is clear that ns and r in the present context depends on the parameters n and a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Here we need to recall that n and a0 are related to the entropic parameters as n = √α and a0 = π/ (4β) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' It turns out that the theoretical predictions for ns and r get simultaneously compatible with the recent Planck data for a small range of the entropic parameters given by: α = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0938, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='0939] and β = π 16, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' 16 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' CONCLUSION In this short review article, we have proposed generalized entropic function(s) and have addressed their implications on black hole thermodynamics as well as on cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In the first half of the paper, a 4-parameter and a 3-parameter generalized entropy functions are shown, which are able to generalize the known entropies proposed so far, like the Tsallis, R´enyi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable choices of the respective entropic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' However the 4-parameter entropy functions proves to be more general compared to the 3-parameter entropy function, in particular, the 3-parameter entropy does not converge to the Kaniadakis entropy for any choices of the parameters, unlike to the entropy having 4 parameters which generalizes all the known entropies including the Kaniadakis one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Thus regarding to the number of parameters in a generalized entropy function, we have provided a conjecture – “The minimum number of parameters required in a generalized entropy function that can generalize all the known entropies mentioned above is equal to four”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Consequently the interesting implications of 3-parameter entropy on black hole thermodynamics and the 4-parameter entropy on cosmology have been addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' It turns out that the entropic cosmology corresponding to the 4-parameter generalized entropy results to an unified cosmological scenario of early inflation and the late dark energy era of the universe, where the observable quantities are found to be compatible with the recent Planck data for certain viable ranges of the entropic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Despite these successes, here it deserves mentioning that the 4-parameter entropy function (Sg) seems to be plagued with singularity for certain cosmological evolution of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' In particular, Sg diverges at the instant when the Hubble parameter vanishes, for instance at the instant of bounce in the context of bounce cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With this spirit, we have proposed a singular-free 5-parameter entropy function (Sns) which converges to all the known entropy functions for particular limits of the entropic parameters, and at the same time, also proves to be non-singular for the entire cosmological evolution of the universe even at H = 0 (where H represents the Hubble parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Regarding to the non-singular entropy, a second conjecture has been given : “The minimum number of parameters required in a generalized entropy function that can generalize all the known entropies, and at the same time, is also singular-free during the universe’s evolution – is equal to five”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Such non-singular behaviour of Sns proves to be useful in describing the bounce cosmology, in particular, the entropic cosmology corresponding to Sns naturally allows symmetric bounce universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With the perturbation analysis in the context of entropic bounce, it has been shown that the observable quantities like the spectral tilt and the tensor-to-scalar ratio are simultaneously compatible with the Planck data in the background of symmetric quasi-matter bounce scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Finally we would like to mention that the proposals of generalized entropy functions (Sg or Sns) opens a new directions in theoretical physics, and its vast consequences may hint some unexplored directions of black hole thermo- dynamics as well as of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' For example, it will be of utmost interest to study the aspects of the generalized entropy functions on primordial black hole formation or primordial gravitational wave or the recently found astro- physical black holes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' With the recent and future advancements of different detectors (like the GW detectors or regarding the black hole detection), we hope that these study can indirectly quantify the viable ranges of entropic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Acknowledgments This work was supported by MINECO (Spain), project PID2019-104397GB-I00 and also partially supported by the program Unidad de Excelencia Maria de Maeztu CEX2020-001058-M, Spain (SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' This research was also supported in part by the International Centre for Theoretical Sciences (ICTS) for the online program - Physics of the Early Universe (code: ICTS/peu2022/1) (TP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' [1] J.' metadata={'source': 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no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='4, 147 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='1140/epjp/i2019-12504-7 [arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='02417 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfFfrV/content/2301.01013v1.pdf'} +page_content=' Nojiri and S.' metadata={'source': 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Grizzle +Abstract—This paper presents a reactive planning system that +allows a Cassie-series bipedal robot to avoid multiple non- +overlapping obstacles via a single, continuously differentiable +control barrier function (CBF). The overall system detects an +individual obstacle via a height map derived from a LiDAR +point cloud and computes an elliptical outer approximation, +which is then turned into a CBF. The QP-CLF-CBF formalism +developed by Ames et al. is applied to ensure that safe trajectories +are generated. Liveness is ensured by an analysis of induced +equilibrium points that are distinct from the goal state. Safe +planning in environments with multiple obstacles is demonstrated +both in simulation and experimentally on the Cassie biped. +THIS IS AN INITIAL DRAFT +While the paper is not yet polished, it allows the co- +first authors to highlight their research skills while they +are seeking a PhD position. The full autonomy videos are +upload to our YouTube channel and the video for this +particular paper can be viewed here. This draft has been +approved by Huang and Grizzle. +I. INTRODUCTION AND CONTRIBUTIONS +Bipedal robots are typically conceived to achieve agile- +legged locomotion over irregular terrains, and maneuver in +cluttered environments [1]–[3]. To explore safely in such +environments, it is critical for robots to generate quick, yet +smooth responses to any changes in the obstacles, map, and +environment. In this paper, we propose a means to design and +compose control barrier functions (CBFs) for multiple non- +overlapping obstacles and evaluate the system on a 20-degree- +of-freedom (DoF) bipedal robot. +In an autonomous system, the task of avoiding obstacles +is usually handled by a planning algorithm because it has +access to the map of an entire environment. Given the map, +the planning algorithm is then able to design a collision-free +path from the robot’s current position to a goal. If the map +is updated due to a change in the environment, the planner +then needs to update the planned path, so-called replanning, +to accommodate the new environment. Such maps are typically +large and contain rich information such as semantics, terrain +characteristics, and uncertainty, and thus are slow to update. +This raises a concern when obstacles either move into the +planned path but the map has not been updated or a robot’s +new pose allows the detection of previously unseen obstacles. +The slow update rate of the map leads to either collision or +∗ equal contribution. +Jinze Liu, Minzhe Li, Jiunn-Kai Huang, and Jessy W. Grizzle are with the +Robotics Institute, University of Michigan, Ann Arbor, MI 48109, USA. { +minzlee, jzliu, bjhuang, grizzle}@umich.edu. +Fig. 1: In the top figure, Cassie Blue autonomously avoids multiple obstacles +via the developed CLF-CBF-QP obstacle avoidance system, comprised of +an intermediate goal selector, obstacle detection, and a CLF-CBF quadratic +programming solver. The bottom figure is the elevation map built in real time. +The blue and cyan blobs are obstacles that Cassie detects and avoids in real +time. A gantry is used in the experiments because the lab-built perception +system that has been added to the robot is unprotected in case of falls. +abrupt maneuvers to avoid collisions. The non-smooth aspects +arising from the map updates or changes in the perceived +environment can be detrimental to the stability of the overall +system. +Research on obstacle avoidance has been studied for sev- +eral decades as pioneered in classic probabilistic roadmap +approaches (PRM) [4] and cell decomposition [5, Chapter +arXiv:2301.01906v1 [cs.RO] 5 Jan 2023 + +6]. However, the omission of robot dynamics and the extra +computation for map discretization make these methods hard +to use in real-time applications. Artificial potential fields [6]– +[15] stand out for their simplicity, extendability, and efficiency, +leading to their wide adoption for real-time obstacle avoidance +planning problems. A drawback of potential field methods +is that they require the entire map of an environment to be +available when designing a potential function that will render +attractive one or more goal points in the map. Moreover, un- +wanted local minima and oscillations in the potential field have +limited their deployment in the field. A control barrier function +(CBF) [16], on the other hand, enables real-time controller +synthesis with provable safety for mobile robots operating in a +continuous (non-discretized) space and can work with a partial +(or incomplete) map. A control Lyapunov function (CLF) is a +(candidate) positive definite function for a closed-loop system +where at any given time instance there exists a control input +that renders the derivative of the Lyapunov function along +the system dynamics negative definite. The CLF is typically +designed to vanish at a desired goal state or pose. +The main theme of [16] is that a real-time quadratic program +(QP) can be used to combine a CLF and a CBF in such a way +that closed-loop trajectories induced by the CLF are minimally +modified to provide provable safety, that is, non-collision with +obstacles. This design philosophy has been explored in [16]– +[18]. +One means of avoiding obstacles is to come to a complete +stop, though it is at the cost of not reaching the goal state. +The papers [19]–[22] showed that such behavior can be an +unintended outcome of the CLF-CBF-QP design approach +of [16]. Specifically, the inequality constraints (of the QP) +associated with the derivatives of the control Lyapunov and +control barrier functions can induce equilibria in the closed- +loop system that are distinct from the equilibrium of the +CLF. Reference [19] characterizes these equilibria via the +KKT conditions associated with the QP, while reference [20] +emphasizes that if an induced equilibrium is unstable, then +“natural noise” in the environment will avoid the robot being +deadlocked at an unstable equilibrium. +Inspired by the above-cited works on CLF-CBF-QPs for +planning and control, we incorporate high-bandwidth obstacle +avoidance into the CLF-RRT* reactive planner of [1]. The +CLF in [1] takes into account features specific to bipeds, such +as the limited lateral leg motion that renders lateral walking +more laborious than sagittal plane walking. This paper seeks +to utilize the CLF designed specifically for bipedal robots +in tandem with a CBF to avoid multiple, non-overlapping +obstacles in a smooth fashion, while ensuring progress to a +goal state. +The main contributions of the new proposed CLF-CBF +system are the following: +1) We propose a novel CLF-CBF-QP obstacle avoidance +system specifically adapted for bipedal robots locomoting +in the presence of multiple non-overlapping obstacles. +The full system provides for real-time obstacle detection, +CBF design, and safe control input generation through a +QP. +2) We mathematically prove the validity of the proposed +CBF for both single and multiple obstacles. We also +analytically analyze the existence of spurious equilibrium +points induced by the CLF-CBF constraints on the QP. +3) We provide simulations that support the mathematical +analysis for obstacle avoidance while reaching a goal. +4) The overall reactive planning system is demonstrated +experimentally on a 20-degree-of-freedom Cassie-series +bipedal robot. +5) We +open-source +the +implementations +of +the +system +in +C++ +with +Robot +Operating +System +(ROS) +[23] +and +associated +videos +of +the +experiments; +see +https://github.com/UMich- +BipedLab/multi_object_avoidance_via_clf_cbf. +The rest of the paper is organized as follows. Section II +overviews related work. The design and validation of the +proposed CBF is presented in Sec. III. We analyze equilibrium +points of the proposed CBF in Appendix A. Section IV +proposes a novel and simple method to combine CBFs for non- +overlapping obstacles. Simulation and experimental results are +given in Sec. V. +II. RELATED WORK ON CONTROL WITH SAFETY +A continuously differentiable, proper, positive definite func- +tion V (x) that vanishes at a single point is called a candidate +Lyapunov function [24]. If the derivative of V (x) along the +trajectories of a control system can be rendered negative +definite by proper choice of the control input, it is called a +control Lyapunov Function, or CLF for short [25]–[27]. CLFs +are widely used in the design of controllers to asymptotically +drive a system to a goal state. Safety involves steering a control +system to a goal state while avoiding self-collisions, obstacles, +or other undesirable states, collectively referred to as unsafe +states. The set complement of the unsafe states is the set of +safe states. +A. Artificial Potential Fields and Navigation Functions +The first systematic method for real-time control and ob- +stacle avoidance was introduced by Khatib in [28]. Called +the method of Artificial Potential Functions, it revolutionized +feedback control for manipulators in that hard constraints +could be enforced in both the robot’s task space and joint space +in real time. Prior to this seminal work, obstacle avoidance, +or more generally the generation of safe paths, was relegated +to a path planner operating at a much slower time scale. A +survey of the method of potential functions can be found in +[29]. +Potential functions seek to construct “repulsive fields” +around obstacles that are active throughout the entire state +space of the robot’s dynamical system, without destroying +the presence of an attractive field steering the system to a +goal state. It has been recognized that superimposed attracting +and repelling fields can create undesired spurious equilibria, +which prevent a robot from reaching its goal state [30]. In +addition, potential fields have been observed to introduce tra- +jectory oscillations as a robot passes near obstacles. Heuristic +2 + +modifications have been proposed to avoid local minima [11]– +[13], while potential fields have been combined with other +gradient-based functions to reduce oscillations [14], [15]. +The method of Navigation Functions by Koditschek and +Rimon [31] sought to design a single function whose gradient +produces trajectories avoiding multiple obstacles while asymp- +totically converging to a single goal state from almost all +initial conditions [32]–[35]; specifically, all equilibria except +the goal state should be unstable. Because the design of a +navigation function takes into account the global topology +of the method of navigation functions is not appropriate for +problems requiring the online identification and avoidance of +obstacles; in addition, there are topological restrictions to the +existence of navigation functions. +B. Control Barrier Functions and Control Lyapunov Func- +tions +Barrier Functions provide Lyapunov-like conditions for +proving a given set of safe states is forward invariant, meaning +that trajectories starting in the safe set remain in the safe +set. The natural extension of a barrier function to a system +with control inputs is a Control Barrier Function or CBF +for short, first proposed by [36]. CBFs parallel the extension +of Lyapunov functions to CLFs, in that the key point is to +impose inequality constraints on the derivative of a candidate +CBF (resp., CLF) to establish entire classes of controllers +that render a given set forward invariant (resp., asymptotically +stable). +Importantly, barrier functions and CBFs focus solely on +safety and do not seek to simultaneously steer a system +to any particular point in the safe set. This allows CBFs +to be combined with other “goal-oriented” control methods +as a (maximally permissive) supervisor that only modifies +a trajectory when it is in conflict with the safety criteria +established by the CBF. The papers [37], [38] introduced the +notion of using a real-time quadratic program (QP) to combine +a CBF with a CLF to achieve convergence to a goal state +while avoiding unsafe states. The overall method goes by the +acronym CLF-CBF-QP. +For control systems that are affine in the control variable, +CLF-CBF-QPs have proven to be enormously popular in and +out of robotics applications [16]–[18], [39]–[43]. To highlight +just a few example, reference [17] uses a CLF-CBF-QP to +achieve stable walking for bipedal robots, while trajectory +planning under spatiotemporal and control input constraints +is presented in [18], [39], [40]. Applications to obstacle +avoidance are addressed in [41]–[43]. +The recent paper [44] shows that CBFs are a strict general- +ization of artificial potential functions and that in a practical +example, a CLF-CBF-QP has reduced issues with oscillations +as a robot passes near obstacles and improved liveness, mean- +ing the ability to reach the goal state. Hence, we use the +method of CLF-CBF-QPs in this paper. +C. Combining Multiple CBFs +Usually, a control barrier function is designed for a single +obstacle. When there are multiple obstacles in the control +system, the barrier functions for each obstacle must be com- +bined in some manner to provide safety guarantees. Reference +[45] shows that if the intersection of the set of “allowable +controls” of individual CBFs is non-empty, then the CLF- +CBF-QP method can be extended to several obstacles; the +reference does not show how to check this condition online (in +real time). Multiple CBF functions have also been combined +to obtain a single CBF so that existing methods can be +applied. Reference [46] combines several CBFs into an overall +CBF using max-min operations. The resulting CBF is non- +differentiable and hence this technique is not used here. Ref- +erence [47] combines multiple CBFS for disjoint unsafe sets +with a single CLF to produce a new CLF that simultaneously +provides asymptotic stability and obstacle avoidance. This +work is therefore related to the method navigation functions +reviewed above and suffers from the same drawbacks; how- +ever, a key technique used in this reference to combine the +CBFs before merging them with a CLF will be exploited in the +current paper, namely a continuously differentiable saturation +function. +D. CLF-CBF-QPs and Unwanted Equilibrium Points +The presence of multiple stable equilibrium points intro- +duces “deadlock” in a control system. Reference [19] shows +that the use of real-time QPs to combine safety and goal- +reaching in navigation problems can lead to unwanted equi- +librium points. With this awareness, the authors of [21] modify +the cost function in the quadratic program to remove the +unwanted equilibria. The modification induces a rotational +motion in the closed-loop system that steers it around the +obstacle, something a bipedal robot can do naturally. Hence, +here we only exploit their analysis method for finding the +unwanted equilibria and show that our method introduces +at most one undesired equilibrium point when obstacles are +disjoint. Moreover, we do not need to remove the unwanted +equilibrium using the methods in [22], [48] by transforming +the system’s state space into a convex manifold, or by increas- +ing the complexity of the system’s state space. +E. Summary +The presence of multiple obstacles is common in practice. +While existing works can treat disjoint obstacles, they are not +appropriate for use where obstacles are identified in real-time +via an onboard perception system. In this work, for a biped- +appropriate planning model, we propose a simple means to +combine CBFs for disjoint obstacles so that the complexity +of the real-time CLF-CBF-QP remains constant and induced +equilibrium points are easy to characterize and avoid. +III. CONSTRUCTION OF CONTROL LYAPUNOV FUNCTION +AND CONTROL BARRIER FUNCTION +This section introduces the CLF proposed in [1] and ana- +lyzes its trajectories when combined with a quadratic CBF +through a real-time QP. The goal is to ensure the closed- +loop system is able to reach a goal state while smoothly +avoiding a single obstacle. This section lays the foundation +for considering multiple obstacles in the next section. +3 + +Fig. 2: The red line is the distance between the obstacle and the robot. +A. State Representation +Denote P = (xr, yr, θ) the robot pose, G = (xt, yt) the +goal position in the world frame. We simplify an obstacle O +as a circle (and hence convex) described as its center (xo, yo) +and its radius ro. We define the robot state as +x = +� +� +r +δ +θ +� +� , +(1) +where r = +� +(xt − xr)2 + (yt − yr)2, θ is the heading angle +of the robot, and δ is the angle between θ and the line of sight +from the robot to the goal, as shown in Fig. 2. +The dynamics of the control system is defined as +˙x = f(x) + g(x)u += +� +� +0 +0 +0 +� +� + +� +� +− cos(δ) +− sin(δ) +0 +sin(δ) +r +− cos(δ) +r +1 +0 +0 +−1 +� +� +� +� +vx +vy +ω +� +� , +(2) +where we view u = +�vx, +vy, +ω�T as the control variables +in the robot frame, as shown in Fig. 2. +B. Design of CLF and CBF for Bipedal Robots +The control Lyapunov function leveraged in the reactive +planner proposed in [1] takes into account features specific +to bipeds, such as the limited lateral leg motion that renders +lateral walking more laborious than sagittal plane walking. +Therefore, we also define the CLF as +V (x) = r2 + γ2 sin2(βδ) +2 +, +(3) +where γ is the weight on the orientation, and β controls the +size of the field of view (FoV). Given P and G, we have a +closed-form solution for control u in (2), +ωref = r cos(δ) [rvδ cos(δ) − vr sin(δ)] +α + r2 cos2(δ) +vref +y += α(vr sin(δ) − rvδ cos(δ)) +r2cos(δ)2 + α +vref +x += vr cos(δ)r2 + αvδ sin(δ)r + αvr cos(δ) +r2cos(δ)2 + α +; +(4) +where vr and vδ are defined as: +vr = kr1 +r +kr2 + r +vδ = − 2 +β kδ1 +r +kδ2 + r sin(2βδ). +(5) +In (4) and (5), α, β, kr1, kr2, kδ1, kδ2 are positive constants. +See [1] for more details. +Next, we introduce a candidate CBF as +B(x) = +� xr − xo +yr − yo +�⊤ +Q +� xr − xo +yr − yo +� +− r2 +o, +(6) +where (xo, yo) gives the center of the obstacle, ro specifies the +“radius” of the obstacle, and Q is positive definite. We next +verify that (6) is a valid CBF. +C. Proof of CBF Validity +Following [49], we define the sets +D := {x ∈ R3 | B(x) ̸= −r2 +o, and r ̸= 0} +C := {x ∈ D | B(x) ≥ 0} +(7) +associated with the candidate CBF (6) and note that Int(C) ̸= +∅ and Int(C) = C. From [49], for (6) to be a valid CBF +function of (2), there must exist some η > 0, such that, +∀x ∈ D, ∃u ∈ R3, ˙B(x, u) + ηB(x) ≥ 0, +(8) +where ˙B(x, u) := LfB(x) + LgB(x)u is the time derivative +of B(x) along the dynamics of (2), η > 0 sets the repulsive +effort of the CBF, and +LfB(x) := ∂B(x) +∂x +f(x) +(9) +LgB(x) := ∂B(x) +∂x +g(x). +(10) +Because the drift term f(x) in (2) is identically zero, the +zero control u ≡ 0 satisfies (8) for x ∈ C. Hence, we need to +show that (8) can be met for x ∈∼ C, the set complement of +C. Direct application of the chain rule gives that +LgB(x) = a(x)b(x)g(x), +where +a(x) := 2 [ xt − r cos(δ + θ) − xo, +yt − r sin(δ + θ) − yo ] Q += 2 +� xr − xo, +yr − yo +� +Q +b(x) := +� − cos(δ + θ) +r sin(δ + θ) +r sin(δ + θ) +− sin(δ + θ) +−r cos(δ + θ) +−r cos(δ + θ) +� +g(x) = +� +��� +− cos(δ) +− sin(δ) +0 +sin(δ) +r +− cos(δ) +r +1 +0 +0 +−1 +� +��� . +(11) +Moreover, a(x) only vanishes at the center of an obstacle, +the rows of b(x) are linearly independent for all r > 0, and +det (g(x)) = − 1 +r ̸= 0 for all 0 < r < ∞. It follows that for +all x ∈ D, LgB(x) ̸= 0 and hence (8) is satisfied, proving +that (6) is a valid CBF. +4 + +V +W +O =(x +X +X +JD. Quadratic Program of the Proposed CLF-CBF System +A quadratic program (QP) is set up to optimize the control +u with the slack variable s while enforcing both the CLF and +CBF constraints. Let L(x, u, s) be the CLF constraints +L(x, u, s) := LfV (x) + LgV (x)u + µV (x) − s ≤ 0, +(12) +where Lpq(x) := ∇q(x) · p(x) is the Lie derivative, µ serves +as a decay rate of the upper bound of V (x). Next, we denote +B(x, u) the CBF constraints +B(x, u) := −LfB(x) − LgB(x)u − ηB(x) ≤ 0, +(13) +where η serves as a decay rate of the lower bound of B(x). +Finally, the QP for the control values is formulated as +u∗, s∗ = arg min +L(x,u,s)≤0 +B(x,u)≤0 +J(u, s), +(14) +where the cost function J(u, s) is defined as +J(u, s) := 1 +2(u − uref)T H(u − uref) + 1 +2ps2, +(15) +the positive definite, diagonal matrix H := diag([h1, h2, h3]) +weights the control variables, uref := +�vref +x +vref +y +ωref�T is +the control vector from the CLF (3) without obstacles, and +p ≥ 0 is the weight of the slack variable, s. +In the proposed CLF-CBF-QP system, uref is the closed- +form solution obtained from the CLF without obstacles, and H +assigns weights for different control variables. The proposed +CLF-CBF-QP cost function captures inherent features of a +Cassie-series robot, such as the low-cost of longitudinal move- +ment and high-cost of lateral movement, while guaranteeing +safety. We next look at liveness, that is, the ability of the +system to reach the desired goal. +E. Analysis for Unwanted Equilibria +Paper [19] points out very clearly that the CLF-CBF-QP +formulation of Sec. III-D can introduce unwanted equilibria +that may prevent the robot from reaching a goal state. The +paper [20] also considered this problem and noted that if the +equilibria are unstable, then liveness is preserved for almost +all initial conditions. In Appendix A, we follow the KKT- +analysis of the CLF-CBF-QP presented in [19] and show that +Fig. 3: Illustration of a case when the robot directly faces the obstacle and the +target creates an equilibrium in the continuous-time system. In a simulation +with discrete-time control updates, the robot walks back and forth at the +obstacle boundary. +Fig. +4: +Illustration +of +breaking +the +equilibrium +by +using +uref +2 : +�vref +x +vref +y +ωref + ϵ�T when δ = 0. The robot successfully reaches the +target position without colliding with the obstacle. +only one equilibrium point is created by the QP. Moreover, the +equilibrium occurs at an obstacle boundary for δ = 0, dy = +0, dx > 0, in other words, when the robot’s heading faces +directly to the obstacle and the target, as shown in Fig. 3. The +robot will move directly toward the obstacle and stop at the +obstacle boundary. +Remark 1. When the robot encounters the above equilibrium +state, we can add a constant ϵ > 0 to uref in (14) such that +uref = +�vref +x +vref +y +ωref + ϵ�T. As is shown in Fig. 4, the +robot breaks its equilibrium state, avoids the obstacle, and +reaches the target position. This is related to, but distinct +from, the method presented in [19] for resolving unwanted +equilibria. +IV. COMBINING CBFS FOR MULTIPLE OBSTACLES +So far, we have assumed there is only one obstacle perceived +by the robot. In this section, we will discuss how to handle +multiple obstacles in the environment when each obstacle is a +positive distance apart from the others [47]. Specifically, for +i ∈ {1, 2, . . . , M}, suppose that +Bi(x) := +� xr − xo,i +yr − yo,i +�⊤ +Qi +� xr − xo,i +yr − yo,i +� +− r2 +o,i +Di := {x ∈ R3 | Bi(x) ̸= −r2 +o,i, and r ̸= 0} +Ci := {x ∈ Di | Bi(x) ≥ 0} +(16) +are valid CBF functions for the dynamics (2). For i ̸= j, the +obstacles corresponding to Bi : R3 → R and Bj : R3 → R are +a positive distance apart if +∆ij := +inf +x ∈∼ Ci +y ∈∼ Cj +||x − y|| > 0. +(17) +A key innovation with respect to [46] is that we will +compose the associated CBFs in a smooth (C1) manner. +A potential drawback with respect to [46] is that we will +assume the obstacles giving rise to the CBFs are a positive +distance apart. Similar to [47], we saturate standard quadratic +CBFs before seeking to combine them. Distinct from [47], we +multiply the saturated CBFs instead of creating a weighted +5 + +-2 +-4 +-6 +-8 +-10E +-6 +-4 +-2 +0 +2 +4 +6 +X +m-2 +-4 +目 +-6 +-8 +-10 +-6 +-4 +-2 +0 +2 +4 +6 +X +msum. This greatly simplifies the analysis of the composite CBF +with respect to all previous works. +A. Smooth Saturation Function +We introduce a continuously differentiable saturation func- +tion that will allow us to compose in a simple manner CBFs +corresponding to obstacles that are a positive distance apart. +Consider σ : R → R by +σ(s) := +� +� +� +� +� +s +s ≤ 0 +s(1 + s − s2) +0 < s < 1 +1 +s ≥ 1. +(18) +Then straightforward calculations show that for all s ∈ R, +dσ(s) +ds +exists and satisfies +dσ(s) +ds +:= +� +� +� +� +� +1 +s ≤ 0 +1 + 2s − 3s2 +0 < s < 1 +0 +s ≥ 1. +(19) +Upon noting that for all 0 < s < 1, 0 < dσ(s) +ds +< 1, it follows +that σ : R → R is continuously differentiable and monotonic. +Remark 2. For 0 ≤ s ≤ 1, σ is constructed from a degree- +three Bézier polynomial p : [0, 1] → R such that p(0) = 0, +dp(0) +ds += 1, p(1) = 1, dp(1) +ds += 0. Moreover, for 0 < s < 1, +dp(s) +ds +> 0. +Definition 1. For κ > 0, we define σκ : R → R by +σκ(s) := σ( s +κ). +(20) +Proposition 1. Suppose that κ > 0 and B : D → R is a +candidate CBF with D and C defined as in (7). Then σκ ◦ B : +D → R is a valid CBF for the system (2) if, and only if, +B : D → R is a valid CBF. +Proof. For x ∈ C, σκ ◦ B(x) > 0 and hence satisfies (8) for +u = 0. For x ∈∼ C, by the chain rule and the construction of +σ : R → R, +∂σκ ◦ B(x) +∂x += dσ(s) +ds +���� +s= B(x) +κ +∂B(x) +∂x += 1 +κ +∂B(x) +∂x +. +(21) +Hence, the proof of Sect. III-C applies. +■ +Proposition 2. Suppose for 1 ≤ i ≤ M, the CBFs Bi(x) : +R3 → R are a positive distance apart. Then there exist κ1 > 0, +κ2 > 0, . . ., κM > 0, such that for all i ̸= j, +{x ∈ R3 | σκi ◦ Bi(x) < 1} ∩ {x ∈ R3 | Bj(x) < 0} = ∅. +(22) +Proof. By the disjointness property, ∆i := min +j̸=i +∆ij > 0. +For S ⊂ R3 and x ∈ R3, define the distance from x to S +by +d(x, S) := inf +y∈S ||x − y||. +(23) +Then, because (i) Bi is continuous, (ii) the set complement of +Ci is bounded, and (iii) d(x, ∼ Ci) > 0 =⇒ Bi(x) > 0, it +follows that +m∗ +i := +sup +d(x,∼Ci)≤∆i +Bi(x) +(24) +is a finite positive number. Therefore, for all 0 < κi < m∗ +i , +{x ∈ R3 | σκi ◦ Bi(x) < 1} ⊂ {x ∈ R3 | d(x, ∼ Ci) ≤ ∆i}, +(25) +and hence (22) holds. +■ +B. Multiplication Property of Smooth Saturated CBFs +For M ≥ 2 CBFs corresponding to disjoint obstacles, define +the sets +DM := +M +� +i=1 +Di +CM := +M +� +i=1 +{x ∈ DM | Bi(x) ≥ 0} += +M +� +i=1 +Ci. +(26) +Theorem 1. Under the assumed disjointness property, the +product of smoothly saturated valid CBFs, +BM(x) := +M +� +i=1 +σκi ◦ Bi(x), +(27) +is a valid CBF for DM, CM, and the dynamic system (2). +Proof. For x ∈ CM, the zero control u ≡ 0 satisfies (8) +because the drift term f(x) is zero. We show that for x ̸∈ CM, +(8) can be satisfied. +By the disjoint property of the assumed CBF functions, +when BM(x) < 0, we have ∃i, such that σκi ◦ Bi(x) = +Bi(x) < 0, and σκj ◦ Bj(x) = 1 for j ̸= i. Hence, +BM(x) = Bi(x). Because Bi(x) is assumed to be a valid +CBF function, and both DM ⊂ Di and CM ⊂ Ci hold, the +CBF property holds for BM(x). +■ +Remark 3. Due to the way we have constructed the multi- +obstacle CBF, the equilibrium analysis for a single obstacle +carries over here without changes. This is because, when +the robot is at a boundary of an obstacle, the values of the +saturated CBFs for the other obstacles will all be one. +V. SIMULATION RESULTS WITH SINGLE AND MULTIPLE +OBSTACLES +In this section, we first use simulation to study the behavior +and liveness of the proposed CLF-CBF system with a single +obstacle. Next, we run the system on several synthetic envi- +ronments with 20 obstacles in Robot Operating System (ROS) +[23] with C++. +Remark 4. For the CBF in (6), we take Q = I and in +Prop. 1, we take κ1 = · · · = κM = min{∆2 +i }M +i=1, which +is the minimum of the square of the distance between any of +the obstacles. +6 + +A. Robot Model in Simulation +In MATLAB and ROS, the bipedal robot is represented +by the Angular momentum Linear Inverted Pendulum (ALIP) +model [2]. The ALIP robot takes piece-wise constant inputs +from the CLF-CBF-QP system. Let g, H, τ be the gravitational +constant, the robot’s center of mass height, and the time +interval of a swing phase, respectively. The motion of an ALIP +model on the x-axis satisfies +�xk+1 +˙xk+1 +� += +� cosh(ξ) +1 +ρ sinh(ξ) +ρ sinh(ξ) +cosh(ξ) +� �xk +˙xk +� ++ +�1 − cosh(ξ) +−ρ sinh(ξ) +� +px, +(28) +where xk and ˙xk are the contact position and velocity of the +swing foot on the x-axis, px is the center of mass (CoM) +position on the x-axis of the robot, ξ = ρτ and ρ = +� +g/H. +The motion of the robot on the y-axis can be similarly defined. +Fig. 5: Illustration of how the trajectories vary as a function of differ- +ent obstacle positions. The target (marked in black) and the robot pose +(−15, −15, −15◦) are fixed throughout all of the simulations. The different +colors denote different simulations with only one obstacle present at a time. +The red trajectory is generated without any obstacle present from the QP only +containing CLF constraint. +B. Behavior Study with Single Obstacle in MATLAB +The optimal control command of the robot is the solution of +the CLF-CBF-QP problem defined in (14). The time interval +of a swing phase is set to τ = 0.3s. The robot updates its pose +based on the ALIP model and the optimal control command. +The updated pose is then fed back to the CLF-CBF-QP system +to compute the optimal control for the next iteration. This +process continues until the robot reaches the target or collides +with an obstacle. +Figure 5 shows how the trajectories vary as a function of +a single obstacle’s position with a fixed initial robot pose +of (−15, −15, −15◦), marked as the magenta arrow. The +red trajectory is the nominal trajectory without any obstacles +present. Each colored trajectory and matching circle represent +a distinct simulation result. The robot successfully avoids the +obstacle in all cases. In Fig. 6, we show how the trajectories +vary as a function of different robot orientations with a fixed +obstacle location. +Remark 5. When the robot is within an obstacle, there is also +a valid solution that pushes the robot outside of the obstacle. +Fig. 6: Illustration of how the trajectories vary as a function of different robot +orientations with a fixed obstacle location. The target (marked in cyan) and +obstacle at (−4, −4) are fixed through out all the simulations. A different +color stands for a different robot orientation. +Consider the CBF constraint (13), +LfB(x) + LgB(x)u + ηB(x) ≥ 0. +(29) +When the robot is withing an obstacle, B(x) < 0 and the QP +selects u such that LfB(x) + LgB(x)u ≥ −ηB(x), causing +the robot to leave the obstacle. +Fig. 7: Liveness analysis for the CLF-CBF system. The initial pose is +(−15, −15, −15◦), and the target is located at (0, 0). Each dot in the figure +represents the center of an object with radius (r = 1). The interval between +each center dots are 0.2 meter in both x and y direction. Note that all the red +points either originally collide with the robot or the target. +C. Liveness Analysis in MATLAB +We analyze the liveness by placing an obstacle with a +fixed radius (r = 1) at different locations. The robot starts +at (−15, −15, −15◦) and the target is located at (0, 0). The +obstacle is placed at every 0.2 meter. If the robot successfully +reaches the target without collision, the obstacle location is +marked in green otherwise in red, as shown in Fig. 7. All the +red points either originally collide with the robot or the target. +D. Multi-Obstacle Simulation with ROS in C++ +In this simulation, we implement a local map centering at +robot position with a fixed size and a sub-goal selector to place +a target within the local map to achieve long-term planning +as not all the obstacles are perceived by the robot at the +beginning in practice. Even though the global map is available +in simulation but it is not available in practice, therefore, only +7 + +0 +-2 +-4 +-6 +-10 +-12 +-14 +-16 +-15 +-10 +-5 +00 +-1 +-2 +-3 +-4 +-5 +-6 +-7 +-8 +-9 +-10 +-10 +-8 +9- +-4 +-2 +0 +x m0 +success +hit +-2 +-4 +-6 +-10 +-12 +-14 +-16 +-16 +-14 +-12 +-10 +-8 +9- +-4 +-2 +0 +2 +x [m](a) +(b) +(c) +(d) +Fig. 8: Trajectories of 39 obstacles in noise-free (top two) and 20 obstacles in noisy (bottom two) synthetic maps with the size of 50 × 30 meters. The +highlighted areas are the local map at that specific timestamp. The dark blue circles are the obstacles. Different colors represent different runs in the map. +the information within the local map at the specific timestamp +is provided to the robot. The robot model is the same ALIP +model in Sec. V-A. +In Fig. 8, we generate two noise-free and two noisy syn- +thetic maps with the size of 50×30 meters. Each map contains +20 obstacles marked as blue circles. We run six different initial +poses and final goals for each map. Different colors represent +different runs in the map. The highlighted area is the local +map at that specific timestamp. An intermediate goal is chosen +at the intersection between the boundary of the local map +and the line connecting the robot and the final goal at the +current timestamp. If the intermediate goal collides with an +obstacle, it is moved back along the line. The intermediate +goal is updated when it is reached or becomes inside of an +obstacle due to the update of the local map. The robot with +ALIP model successfully reaches the goals in all 6 × 4 = 24 +runs. +VI. EXPERIMENTAL RESULTS ON A BIPEDAL ROBOT +We perform several experiments of the proposed CLF- +CBF-QP system on Cassie Blue, a bipedal robot with 20 +degrees of freedom. The entire system integrates elevation +mapping, intermediate goal selection, and the low-level CLF- +CBF obstacle avoidance system. +(a) +(b) +Fig. 9: The left shows the sensor suite with different sensors, and the right +shows the sensor suite mounted on Cassie Blue. +A. Autonomy System Integration +The following is summarized from [1] for the completeness +of the paper. To allow the robot to perceive its surroundings +under different lighting conditions and environments, we de- +signed a perception suite that consists of an RGB-D camera +8 + +(a) +(b) +(c) +(d) +Fig. 10: Autonomy experiments with Cassie Blue on the first floor of FRB. The green arrow is Cassie’s pose and the green lines are the resulting trajectories. +The blue sphere is the selected target position. The map is colored by height and the highlighted area is the local map. +(Intel RealSense™ D435) and a 32-Beam Velodyne ULTRA +Puck LiDAR, as shown in Fig. 9. The sensor calibrations are +performed via [50]–[53]. The invariant extended Kalman filter +(InEKF) [54] estimates the pose of Cassie at 2k Hz. The raw +point cloud is motion compensated by the InEKF and then +used to build an elevation map. +B. Autonomy Experiment on Cassie Blue +We conducted several indoor experiments with Cassie Blue +on the first floor of the Ford Robotics Building (FRB) where +tables and chairs are considered obstacles. To detect obstacles +in the environment, an occupancy grid map is updated in real- +time using the timestamped elevation map. Grids with heights +greater than 0.2 meters are considered occupied. An occupied +grid is defined as the boundary of obstacles if there is an +unoccupied grid in its neighborhood. The Breadth First Search +(BFS) algorithm [55] is utilized to find the separated obstacles +in the map. Next, we apply the Gift Wrapping Algorithm [56] +to the boundary grids of obstacles to find the convex hulls of +the obstacles. Finally, the minimum bounding ball algorithm +[57] is applied to the convex hulls to find the minimum +bounding circles of the obstacles. The circles are used to +represent obstacles in the CBF function (6). The target position +is selected by clicking a point in the global map. If the final +target is not within the current local map, an intermediate goal +will be selected within the local map. When an intermediate +goal is reached by Cassie or becomes invalid because of the +update of the local map, it is updated. In the experiments, +Cassie successfully avoids all the obstacles and reaches the +target position, as shown in Fig. 10. +VII. CONCLUSION +This paper presented a reactive planning system that al- +lows a Cassie-series bipedal robot to avoid multiple non- +overlapping obstacles via a single, continuously differentiable +control barrier function (CBF). The overall system detects an +individual obstacle via a height map derived from a LiDAR +point cloud and computes an elliptical outer approximation, +which is then turned into a quadratic CBF. A continuously +differentiable saturation function is presented that preserves +the CBF property of a quadratic CBF while allowing the +9 + +saturated CBFs for individual obstacles to be turned into a +single CBF. The CLF-CBF-QP formalism developed by Ames +et al. can then be applied to ensure that safe trajectories are +generated in the presence of multiple obstacles. Liveness is +ensured by an analysis of induced equilibrium points that are +distinct from the goal state. Safe planning in environments +with multiple obstacles is demonstrated both in simulation and +experimentally on the Cassie bipedal robot. +ACKNOWLEDGMENT +Toyota Research Institute provided funds to support this work. +Funding for J. Grizzle was in part provided by NSF Award +No. 1808051. This article solely reflects the opinions and conclusions +of its authors and not the funding entities. +REFERENCES +[1] J.-K. Huang and J. W. Grizzle, “Efficient anytime clf reactive plan- +ning system for a bipedal robot on undulating terrain,” arXiv preprint +arXiv:2108.06699, 2021. +[2] Y. Gong and J. Grizzle, “Angular momentum about the contact point +for control of bipedal locomotion: Validation in a lip-based controller,” +arXiv preprint arXiv:2008.10763, 2020. +[3] G. Gibson, O. Dosunmu-Ogunbi, Y. Gong, and J. Grizzle, “Terrain- +adaptive, alip-based bipedal locomotion controller via model predictive +control and virtual constraints,” 2021. [Online]. 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Harris, “Encyclopedia of operations research and +management science,” Journal of the Operational Research Society, +vol. 48, no. 7, pp. 759–760, 1997. +APPENDIX A +EQUILIBRIUM ANALYSIS OF MULTI-OBSTACLE SYSTEMS +We give the complete analysis for a single obstacle, following +the work of [19]. Because the drift term of our model is zero, any +equilibrium points are where the optimal control is the zero vector. +To avoid this undesirable situation, we seek to find all equilibrium +points E = {x|u∗ = [0, 0, 0]T, r > 0, and δ, θ ∈ (−π, π]}, where +u∗ is the optimal control variable. Recall thatf = +�0 +0 +0�T in (2), +which leads to LfV (x) = LfB(x) = 0. We denote the following to +re-write the CLF and the CBF constraints: +�dx +dy +0� += −LgB(x) +�ax +ay +aω +� += LgV (x), +(30) +where +ax = −r cos(δ) + βγ2 sin(2βδ) sin(δ) +2r +ay = −r sin(δ) − βγ2 sin(2βδ) cos(δ) +2r +aω = βγ sin(2βδ) +2 +. +(31) +The constraints become +L(x, u, s) = axvx + ayvy + aωω − s + µV (x), +(32) +B(x, u) = dxvx + dyvy − ηB(x), +(33) +and the cost function (15) of the QP can then be re-written as: +J(u, s) = 1 +2h1(vx − vref +x )2 + 1 +2h2(vy − vref +y )2 ++ 1 +2h3(ω − ωref)2 + 1 +2ps2, +(34) +where {hi}3 +i=1 are the diagonal elements of H in (15), the weights +of control variables [vx, vy, ω] with hi > 0. +The KKT conditions [58] of this quadratic program are: +∂L +∂u = Hu∗ − Huref + λ1LgV T − λ2LgBT = 0 +(35) +∂L +∂s = ps − λ1 = 0 +(36) +0 = λ1(LfV + LgV u∗ + µV − s) +(37) +0 = λ2(−LfB − LgBu∗ − ηB) +(38) +0 ≥ LfV + LgV u∗ + µV − s +(39) +0 ≥ −LfB − LgBu∗ − ηB +(40) +0 ≤ λ1, λ2, +(41) +where λ1, λ2 ∈ R, and L is the Lagrangian function and defined as +L(u, s, λ1, λ2) = J(u, s) + λ1L(x, u, s) + λ2B(x, u). +(42) +Next, we analyze equilibrium points (if any) via four cases +depending on whether each CLF or CBF constraint is active or +inactive following [19]. +A. Both CLF and CBF are inactive +When both constraints are inactive, we have +λ1 = 0 +λ2 = 0 +0 > LfV + LgV u∗ + µV − s +0 > −LfB − LgBu∗ − ηB. +(43) +With (35) and (36), u∗ and s∗ in this case are +u∗ = uref +s∗ = 0. +(44) +From (4), as long as the goal is not reached, uref is not a zero vector. +Hence, there is no equilibrium point in this case. +B. CLF constraint inactive and CBF constraint active +We prove that there is no equilibrium point in this case by +contradiction. When the CLF constraint is inactive and the CBF +constraint is active, we have +λ1 = 0 +λ2 ≥ 0 +0 > LfV + LgV u∗ + µV − s +0 = −LfB − LgBu∗ − ηB. +(45) +With (35) and (36), u∗, s∗ and λ2 in this case are +u∗ = uref + λ2H−1LgBT +s∗ = 0 +λ2 = −ηB + LfB + LgBuref +LgBH−1LgBT +. +(46) +11 + +If there is an equilibrium point, then u∗ is the zero vector. Hence, at +the equilibrium point, by LfV (x) = 0, u∗ = 0 and s∗ = 0, we have +LfV + LgV u∗ + µV − s∗ = µV > 0, +(47) +which conflicts with (45). Therefore, there is no equilibrium point in +this case. +C. CLF constraint active and CBF constraint inactive +When the CLF constraint is active and the CBF constraint is +inactive, we have +λ1 ≥ 0 +λ2 = 0 +0 = LfV + LgV u∗ + µV − s +0 > −LfB − LgBu∗ − ηB. +(48) +With (35) and (36), u∗, s∗ and λ1 in this case are +u∗ = uref − λ1H−1LgV T +s∗ = λ1 +p +λ1 = pµV + pLfV + pLgV uref +pLgV H−1LgV T + 1 +. +(49) +Using the variables defined in (30), u∗ can be rewritten as: +u∗ = +� +� +v∗ +x +v∗ +y +ω∗ +� +� = +� +�� +vref +x +− λ1ax +h1 +vref +y +− λ1ay +h2 +ωref − λ1aω +h3 +� +�� . +(50) +We know from (31) that +(ay = 0 & aω = 0) ⇐⇒ δ = 0. +(51) +In addition, we know from (4) that +(vref +y += 0 & ωref = 0) ⇐⇒ δ = 0. +(52) +Therefore, we split this case into three cases based on the value of +δ. +1) δ = 0 (Case I): Substituting δ = 0 to (30), we have ax = +−r < 0, ay = 0, aω = 0, and to (4), we have vref +x +> 0, vref +y += +0, ωref = 0. Finally, with (41), the optimal control command (50) +can be simplified as: +u∗ = +� +� +v∗ +x +v∗ +y +ω∗ +� +� = +� +� +vref +x ++ λ1r +h1 > 0 +0 +0 +� +� . +(53) +The optimal control command is not a zero vector, and hence there +is no equilibrium point in this case. +2) δ > 0 (Case II): When δ > 0, by the definitions in (31), we +have ay < 0, aω > 0, and by (4), we have vref +y +> 0, ωref < 0. With +(41) and (50), we have +v∗ +y = vref +y +− λ1ay +h2 +> 0 +ω∗ = ωref − λ1aω +h3 +< 0. +(54) +The optimal control command is not a zero vector in this case. +Therefore, there is no equilibrium points in this case either. +3) δ < 0 (Case III): Similarly, by (30) and (4), we have ay > +0, aω < 0 and vref +y +< 0, ωref > 0. With (41) and (50), we have +v∗ +y = vref +y +− λ1ay +h2 +< 0 +ω∗ = ωref − λ1aω +h3 +> 0 +(55) +The optimal control command is not a zero vector in this case; there +is, thus, no equilibrium point in this case. +In summary, there is no equilibrium point when the CLF constraint +is active and the CBF constraint is inactive. +D. Both CLF and CBF constraint are active +When the CLF constraint is active and the CBF constraint is active, +we have +λ1 ≥ 0 +λ2 ≥ 0 +0 = LfV + LgV u∗ + µV − s +0 = −LfB − LgBu∗ − ηB. +(56) +We can rewrite (35) and (36) as: +u∗ = uref − λ1H−1LgV T + λ2H−1LgBT +s∗ = λ1 +p +(57) +Using the variables defined in (30), u∗ can be rewritten as: +u∗ = +� +� +v∗ +x +v∗ +y +ω∗ +� +� = +� +�� +vref +x +− λ1ax +h1 +− λ2dx +h1 +vref +y +− λ1ay +h2 +− λ2dy +h2 +ωref − λ1aω +h3 +� +�� . +(58) +When the robot is at an equilibrium point, u∗ is the zero vector. +By (56) and LfB = 0, u∗ = 0, we have B = 0, which implies that +the robot is at the boundary of an obstacle. In the following proof of +Sec. A-D, we will assume the robot is at the boundary of obstacles. +The property of B = 0 leads to an immediate proposition which +is helpful in finding the equilibrium point in the system when one of +the components of the optimal control is 0. +Proposition 3. dy = 0 =⇒ v∗ +x = 0. +Proof. By the proof in III-C, we have LgB(x) = ∇B(x) · g(x) ̸= 0 +for x ∈ D. Therefore, when dy = 0, we have dx ̸= 0. Then, we can +further have LgB(x)u∗ = 0 =⇒ v∗ +x = 0. +■ +In addition, with the properties (51) and (52), we split this case +into four cases based on whether δ and dy are zero. +1) dy = δ = 0 (Case I): Substituting to (30), we have ax = +−r < 0, ay = 0, aω = 0, and to (4), we have vref +x +> 0, vref +y += +0, ωref = 0. Finally, with Proposition 3, in this case the optimal +control command (58) can be written as: +u∗ = +� +� +v∗ +x +v∗ +y +ω∗ +� +� = +� +� +vref +x +− λ1ax +h1 +− λ2dx +h1 +0 +0 +� +� = +� +� +0 +0 +0 +� +� . +(59) +λ1 and λ2 can be obtained by (59), (56) and (57): +λ1 = pµV > 0 +λ2 = h1vref +x +− pµV ax +dx +(60) +By (41) and (60), we have +∵ vref +x +> 0, ax < 0, h1vref +x +− pµV ax +dx +≥ 0 −→ dx > 0. +(61) +Hence, there is an equilibrium point when B = 0, dy = δ = 0 and +dx > 0. +2) dy ̸= 0, δ = 0 (Case II): When δ = 0, by (30) and (4), +we have ax = −r < 0, ay = 0, aω = 0 and vref +x +> 0, vref +y += +0, ωref = 0. Finally, with (41), the optimal control command (58) +can be simplified as: +u∗ = +� +� +v∗ +x +v∗ +y +ω∗ +� +� = +� +� +vref +x +− λ1ax +h1 +− λ2dx +h1 +− λ2dy +h2 +̸= 0 +0 +� +� . +(62) +Because v∗ +y ̸= 0, the optimal command is not a zero vector in this +case. Equilibrium points don’t exist when dy ̸= and δ = 0. +12 + +3) δ > 0 (Case III): When δ > 0, by (54), ω∗ < 0. Hence, the +optimal command is not a zero vector and there are no equilibrium +points in this case. +4) δ < 0 (Case IV): When δ < 0, by (55), ω∗ > 0. Hence, the +optimal command is not a zero vector and there are no equilibrium +points in this case. +13 + diff --git a/0dAzT4oBgHgl3EQf8f7E/content/tmp_files/load_file.txt b/0dAzT4oBgHgl3EQf8f7E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..12920405507c1cb0bcbd6fabb0b09d6309b029b8 --- /dev/null +++ b/0dAzT4oBgHgl3EQf8f7E/content/tmp_files/load_file.txt @@ -0,0 +1,809 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf,len=808 +page_content='1 Realtime Safety Control for Bipedal Robots to Avoid Multiple Obstacles via CLF-CBF Constraints Jinze Liu∗, Minzhe Li∗, Jiunn-Kai Huang, and Jessy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Grizzle Abstract—This paper presents a reactive planning system that allows a Cassie-series bipedal robot to avoid multiple non- overlapping obstacles via a single, continuously differentiable control barrier function (CBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The overall system detects an individual obstacle via a height map derived from a LiDAR point cloud and computes an elliptical outer approximation, which is then turned into a CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The QP-CLF-CBF formalism developed by Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' is applied to ensure that safe trajectories are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Liveness is ensured by an analysis of induced equilibrium points that are distinct from the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Safe planning in environments with multiple obstacles is demonstrated both in simulation and experimentally on the Cassie biped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' THIS IS AN INITIAL DRAFT While the paper is not yet polished, it allows the co- first authors to highlight their research skills while they are seeking a PhD position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The full autonomy videos are upload to our YouTube channel and the video for this particular paper can be viewed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This draft has been approved by Huang and Grizzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' INTRODUCTION AND CONTRIBUTIONS Bipedal robots are typically conceived to achieve agile- legged locomotion over irregular terrains, and maneuver in cluttered environments [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' To explore safely in such environments, it is critical for robots to generate quick, yet smooth responses to any changes in the obstacles, map, and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In this paper, we propose a means to design and compose control barrier functions (CBFs) for multiple non- overlapping obstacles and evaluate the system on a 20-degree- of-freedom (DoF) bipedal robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In an autonomous system, the task of avoiding obstacles is usually handled by a planning algorithm because it has access to the map of an entire environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Given the map, the planning algorithm is then able to design a collision-free path from the robot’s current position to a goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' If the map is updated due to a change in the environment, the planner then needs to update the planned path, so-called replanning, to accommodate the new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Such maps are typically large and contain rich information such as semantics, terrain characteristics, and uncertainty, and thus are slow to update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This raises a concern when obstacles either move into the planned path but the map has not been updated or a robot’s new pose allows the detection of previously unseen obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The slow update rate of the map leads to either collision or ∗ equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Jinze Liu, Minzhe Li, Jiunn-Kai Huang, and Jessy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Grizzle are with the Robotics Institute, University of Michigan, Ann Arbor, MI 48109, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' { minzlee, jzliu, bjhuang, grizzle}@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 1: In the top figure, Cassie Blue autonomously avoids multiple obstacles via the developed CLF-CBF-QP obstacle avoidance system, comprised of an intermediate goal selector, obstacle detection, and a CLF-CBF quadratic programming solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The bottom figure is the elevation map built in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The blue and cyan blobs are obstacles that Cassie detects and avoids in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A gantry is used in the experiments because the lab-built perception system that has been added to the robot is unprotected in case of falls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' abrupt maneuvers to avoid collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The non-smooth aspects arising from the map updates or changes in the perceived environment can be detrimental to the stability of the overall system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Research on obstacle avoidance has been studied for sev- eral decades as pioneered in classic probabilistic roadmap approaches (PRM) [4] and cell decomposition [5, Chapter arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='01906v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='RO] 5 Jan 2023 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' However, the omission of robot dynamics and the extra computation for map discretization make these methods hard to use in real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Artificial potential fields [6]– [15] stand out for their simplicity, extendability, and efficiency, leading to their wide adoption for real-time obstacle avoidance planning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A drawback of potential field methods is that they require the entire map of an environment to be available when designing a potential function that will render attractive one or more goal points in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Moreover, un- wanted local minima and oscillations in the potential field have limited their deployment in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A control barrier function (CBF) [16], on the other hand, enables real-time controller synthesis with provable safety for mobile robots operating in a continuous (non-discretized) space and can work with a partial (or incomplete) map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A control Lyapunov function (CLF) is a (candidate) positive definite function for a closed-loop system where at any given time instance there exists a control input that renders the derivative of the Lyapunov function along the system dynamics negative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The CLF is typically designed to vanish at a desired goal state or pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The main theme of [16] is that a real-time quadratic program (QP) can be used to combine a CLF and a CBF in such a way that closed-loop trajectories induced by the CLF are minimally modified to provide provable safety, that is, non-collision with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This design philosophy has been explored in [16]– [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' One means of avoiding obstacles is to come to a complete stop, though it is at the cost of not reaching the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The papers [19]–[22] showed that such behavior can be an unintended outcome of the CLF-CBF-QP design approach of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Specifically, the inequality constraints (of the QP) associated with the derivatives of the control Lyapunov and control barrier functions can induce equilibria in the closed- loop system that are distinct from the equilibrium of the CLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Reference [19] characterizes these equilibria via the KKT conditions associated with the QP, while reference [20] emphasizes that if an induced equilibrium is unstable, then “natural noise” in the environment will avoid the robot being deadlocked at an unstable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Inspired by the above-cited works on CLF-CBF-QPs for planning and control, we incorporate high-bandwidth obstacle avoidance into the CLF-RRT* reactive planner of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The CLF in [1] takes into account features specific to bipeds, such as the limited lateral leg motion that renders lateral walking more laborious than sagittal plane walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This paper seeks to utilize the CLF designed specifically for bipedal robots in tandem with a CBF to avoid multiple, non-overlapping obstacles in a smooth fashion, while ensuring progress to a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The main contributions of the new proposed CLF-CBF system are the following: 1) We propose a novel CLF-CBF-QP obstacle avoidance system specifically adapted for bipedal robots locomoting in the presence of multiple non-overlapping obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The full system provides for real-time obstacle detection, CBF design, and safe control input generation through a QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 2) We mathematically prove the validity of the proposed CBF for both single and multiple obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We also analytically analyze the existence of spurious equilibrium points induced by the CLF-CBF constraints on the QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 3) We provide simulations that support the mathematical analysis for obstacle avoidance while reaching a goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 4) The overall reactive planning system is demonstrated experimentally on a 20-degree-of-freedom Cassie-series bipedal robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 5) We open-source the implementations of the system in C++ with Robot Operating System (ROS) [23] and associated videos of the experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' see https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='com/UMich- BipedLab/multi_object_avoidance_via_clf_cbf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Section II overviews related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The design and validation of the proposed CBF is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We analyze equilibrium points of the proposed CBF in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Section IV proposes a novel and simple method to combine CBFs for non- overlapping obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Simulation and experimental results are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' RELATED WORK ON CONTROL WITH SAFETY A continuously differentiable, proper, positive definite func- tion V (x) that vanishes at a single point is called a candidate Lyapunov function [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' If the derivative of V (x) along the trajectories of a control system can be rendered negative definite by proper choice of the control input, it is called a control Lyapunov Function, or CLF for short [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CLFs are widely used in the design of controllers to asymptotically drive a system to a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Safety involves steering a control system to a goal state while avoiding self-collisions, obstacles, or other undesirable states, collectively referred to as unsafe states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The set complement of the unsafe states is the set of safe states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Artificial Potential Fields and Navigation Functions The first systematic method for real-time control and ob- stacle avoidance was introduced by Khatib in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Called the method of Artificial Potential Functions, it revolutionized feedback control for manipulators in that hard constraints could be enforced in both the robot’s task space and joint space in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Prior to this seminal work, obstacle avoidance, or more generally the generation of safe paths, was relegated to a path planner operating at a much slower time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A survey of the method of potential functions can be found in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Potential functions seek to construct “repulsive fields” around obstacles that are active throughout the entire state space of the robot’s dynamical system, without destroying the presence of an attractive field steering the system to a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' It has been recognized that superimposed attracting and repelling fields can create undesired spurious equilibria, which prevent a robot from reaching its goal state [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In addition, potential fields have been observed to introduce tra- jectory oscillations as a robot passes near obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Heuristic 2 modifications have been proposed to avoid local minima [11]– [13], while potential fields have been combined with other gradient-based functions to reduce oscillations [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The method of Navigation Functions by Koditschek and Rimon [31] sought to design a single function whose gradient produces trajectories avoiding multiple obstacles while asymp- totically converging to a single goal state from almost all initial conditions [32]–[35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' specifically, all equilibria except the goal state should be unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Because the design of a navigation function takes into account the global topology of the method of navigation functions is not appropriate for problems requiring the online identification and avoidance of obstacles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' in addition, there are topological restrictions to the existence of navigation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Control Barrier Functions and Control Lyapunov Func- tions Barrier Functions provide Lyapunov-like conditions for proving a given set of safe states is forward invariant, meaning that trajectories starting in the safe set remain in the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The natural extension of a barrier function to a system with control inputs is a Control Barrier Function or CBF for short, first proposed by [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CBFs parallel the extension of Lyapunov functions to CLFs, in that the key point is to impose inequality constraints on the derivative of a candidate CBF (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=', CLF) to establish entire classes of controllers that render a given set forward invariant (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=', asymptotically stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Importantly, barrier functions and CBFs focus solely on safety and do not seek to simultaneously steer a system to any particular point in the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This allows CBFs to be combined with other “goal-oriented” control methods as a (maximally permissive) supervisor that only modifies a trajectory when it is in conflict with the safety criteria established by the CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The papers [37], [38] introduced the notion of using a real-time quadratic program (QP) to combine a CBF with a CLF to achieve convergence to a goal state while avoiding unsafe states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The overall method goes by the acronym CLF-CBF-QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For control systems that are affine in the control variable, CLF-CBF-QPs have proven to be enormously popular in and out of robotics applications [16]–[18], [39]–[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' To highlight just a few example, reference [17] uses a CLF-CBF-QP to achieve stable walking for bipedal robots, while trajectory planning under spatiotemporal and control input constraints is presented in [18], [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Applications to obstacle avoidance are addressed in [41]–[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The recent paper [44] shows that CBFs are a strict general- ization of artificial potential functions and that in a practical example, a CLF-CBF-QP has reduced issues with oscillations as a robot passes near obstacles and improved liveness, mean- ing the ability to reach the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, we use the method of CLF-CBF-QPs in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Combining Multiple CBFs Usually, a control barrier function is designed for a single obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' When there are multiple obstacles in the control system, the barrier functions for each obstacle must be com- bined in some manner to provide safety guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Reference [45] shows that if the intersection of the set of “allowable controls” of individual CBFs is non-empty, then the CLF- CBF-QP method can be extended to several obstacles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' the reference does not show how to check this condition online (in real time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Multiple CBF functions have also been combined to obtain a single CBF so that existing methods can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Reference [46] combines several CBFs into an overall CBF using max-min operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The resulting CBF is non- differentiable and hence this technique is not used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Ref- erence [47] combines multiple CBFS for disjoint unsafe sets with a single CLF to produce a new CLF that simultaneously provides asymptotic stability and obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This work is therefore related to the method navigation functions reviewed above and suffers from the same drawbacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' how- ever, a key technique used in this reference to combine the CBFs before merging them with a CLF will be exploited in the current paper, namely a continuously differentiable saturation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CLF-CBF-QPs and Unwanted Equilibrium Points The presence of multiple stable equilibrium points intro- duces “deadlock” in a control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Reference [19] shows that the use of real-time QPs to combine safety and goal- reaching in navigation problems can lead to unwanted equi- librium points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' With this awareness, the authors of [21] modify the cost function in the quadratic program to remove the unwanted equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The modification induces a rotational motion in the closed-loop system that steers it around the obstacle, something a bipedal robot can do naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, here we only exploit their analysis method for finding the unwanted equilibria and show that our method introduces at most one undesired equilibrium point when obstacles are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Moreover, we do not need to remove the unwanted equilibrium using the methods in [22], [48] by transforming the system’s state space into a convex manifold, or by increas- ing the complexity of the system’s state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Summary The presence of multiple obstacles is common in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' While existing works can treat disjoint obstacles, they are not appropriate for use where obstacles are identified in real-time via an onboard perception system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In this work, for a biped- appropriate planning model, we propose a simple means to combine CBFs for disjoint obstacles so that the complexity of the real-time CLF-CBF-QP remains constant and induced equilibrium points are easy to characterize and avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CONSTRUCTION OF CONTROL LYAPUNOV FUNCTION AND CONTROL BARRIER FUNCTION This section introduces the CLF proposed in [1] and ana- lyzes its trajectories when combined with a quadratic CBF through a real-time QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The goal is to ensure the closed- loop system is able to reach a goal state while smoothly avoiding a single obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This section lays the foundation for considering multiple obstacles in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 2: The red line is the distance between the obstacle and the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' State Representation Denote P = (xr, yr, θ) the robot pose, G = (xt, yt) the goal position in the world frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We simplify an obstacle O as a circle (and hence convex) described as its center (xo, yo) and its radius ro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We define the robot state as x = � � r δ θ � � , (1) where r = � (xt − xr)2 + (yt − yr)2, θ is the heading angle of the robot, and δ is the angle between θ and the line of sight from the robot to the goal, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The dynamics of the control system is defined as ˙x = f(x) + g(x)u = � � 0 0 0 � � + � � − cos(δ) − sin(δ) 0 sin(δ) r − cos(δ) r 1 0 0 −1 � � � � vx vy ω � � , (2) where we view u = �vx, vy, ω�T as the control variables in the robot frame, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Design of CLF and CBF for Bipedal Robots The control Lyapunov function leveraged in the reactive planner proposed in [1] takes into account features specific to bipeds, such as the limited lateral leg motion that renders lateral walking more laborious than sagittal plane walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Therefore, we also define the CLF as V (x) = r2 + γ2 sin2(βδ) 2 , (3) where γ is the weight on the orientation, and β controls the size of the field of view (FoV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Given P and G, we have a closed-form solution for control u in (2), ωref = r cos(δ) [rvδ cos(δ) − vr sin(δ)] α + r2 cos2(δ) vref y = α(vr sin(δ) − rvδ cos(δ)) r2cos(δ)2 + α vref x = vr cos(δ)r2 + αvδ sin(δ)r + αvr cos(δ) r2cos(δ)2 + α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (4) where vr and vδ are defined as: vr = kr1 r kr2 + r vδ = − 2 β kδ1 r kδ2 + r sin(2βδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (5) In (4) and (5), α, β, kr1, kr2, kδ1, kδ2 are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' See [1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Next, we introduce a candidate CBF as B(x) = � xr − xo yr − yo �⊤ Q � xr − xo yr − yo � − r2 o, (6) where (xo, yo) gives the center of the obstacle, ro specifies the “radius” of the obstacle, and Q is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We next verify that (6) is a valid CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Proof of CBF Validity Following [49], we define the sets D := {x ∈ R3 | B(x) ̸= −r2 o, and r ̸= 0} C := {x ∈ D | B(x) ≥ 0} (7) associated with the candidate CBF (6) and note that Int(C) ̸= ∅ and Int(C) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' From [49], for (6) to be a valid CBF function of (2), there must exist some η > 0, such that, ∀x ∈ D, ∃u ∈ R3, ˙B(x, u) + ηB(x) ≥ 0, (8) where ˙B(x, u) := LfB(x) + LgB(x)u is the time derivative of B(x) along the dynamics of (2), η > 0 sets the repulsive effort of the CBF, and LfB(x) := ∂B(x) ∂x f(x) (9) LgB(x) := ∂B(x) ∂x g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (10) Because the drift term f(x) in (2) is identically zero, the zero control u ≡ 0 satisfies (8) for x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, we need to show that (8) can be met for x ∈∼ C, the set complement of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Direct application of the chain rule gives that LgB(x) = a(x)b(x)g(x), where a(x) := 2 [ xt − r cos(δ + θ) − xo, yt − r sin(δ + θ) − yo ] Q = 2 � xr − xo, yr − yo � Q b(x) := � − cos(δ + θ) r sin(δ + θ) r sin(δ + θ) − sin(δ + θ) −r cos(δ + θ) −r cos(δ + θ) � g(x) = � ��� − cos(δ) − sin(δ) 0 sin(δ) r − cos(δ) r 1 0 0 −1 � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (11) Moreover, a(x) only vanishes at the center of an obstacle, the rows of b(x) are linearly independent for all r > 0, and det (g(x)) = − 1 r ̸= 0 for all 0 < r < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' It follows that for all x ∈ D, LgB(x) ̸= 0 and hence (8) is satisfied, proving that (6) is a valid CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 4 V W O =(x X X JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Quadratic Program of the Proposed CLF-CBF System A quadratic program (QP) is set up to optimize the control u with the slack variable s while enforcing both the CLF and CBF constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Let L(x, u, s) be the CLF constraints L(x, u, s) := LfV (x) + LgV (x)u + µV (x) − s ≤ 0, (12) where Lpq(x) := ∇q(x) · p(x) is the Lie derivative, µ serves as a decay rate of the upper bound of V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Next, we denote B(x, u) the CBF constraints B(x, u) := −LfB(x) − LgB(x)u − ηB(x) ≤ 0, (13) where η serves as a decay rate of the lower bound of B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Finally, the QP for the control values is formulated as u∗, s∗ = arg min L(x,u,s)≤0 B(x,u)≤0 J(u, s), (14) where the cost function J(u, s) is defined as J(u, s) := 1 2(u − uref)T H(u − uref) + 1 2ps2, (15) the positive definite, diagonal matrix H := diag([h1, h2, h3]) weights the control variables, uref := �vref x vref y ωref�T is the control vector from the CLF (3) without obstacles, and p ≥ 0 is the weight of the slack variable, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In the proposed CLF-CBF-QP system, uref is the closed- form solution obtained from the CLF without obstacles, and H assigns weights for different control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The proposed CLF-CBF-QP cost function captures inherent features of a Cassie-series robot, such as the low-cost of longitudinal move- ment and high-cost of lateral movement, while guaranteeing safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We next look at liveness, that is, the ability of the system to reach the desired goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Analysis for Unwanted Equilibria Paper [19] points out very clearly that the CLF-CBF-QP formulation of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' III-D can introduce unwanted equilibria that may prevent the robot from reaching a goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The paper [20] also considered this problem and noted that if the equilibria are unstable, then liveness is preserved for almost all initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In Appendix A, we follow the KKT- analysis of the CLF-CBF-QP presented in [19] and show that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 3: Illustration of a case when the robot directly faces the obstacle and the target creates an equilibrium in the continuous-time system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In a simulation with discrete-time control updates, the robot walks back and forth at the obstacle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 4: Illustration of breaking the equilibrium by using uref 2 : �vref x vref y ωref + ϵ�T when δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot successfully reaches the target position without colliding with the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' only one equilibrium point is created by the QP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Moreover, the equilibrium occurs at an obstacle boundary for δ = 0, dy = 0, dx > 0, in other words, when the robot’s heading faces directly to the obstacle and the target, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot will move directly toward the obstacle and stop at the obstacle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' When the robot encounters the above equilibrium state, we can add a constant ϵ > 0 to uref in (14) such that uref = �vref x vref y ωref + ϵ�T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' As is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 4, the robot breaks its equilibrium state, avoids the obstacle, and reaches the target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This is related to, but distinct from, the method presented in [19] for resolving unwanted equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' COMBINING CBFS FOR MULTIPLE OBSTACLES So far, we have assumed there is only one obstacle perceived by the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In this section, we will discuss how to handle multiple obstacles in the environment when each obstacle is a positive distance apart from the others [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Specifically, for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' , M}, suppose that Bi(x) := � xr − xo,i yr − yo,i �⊤ Qi � xr − xo,i yr − yo,i � − r2 o,i Di := {x ∈ R3 | Bi(x) ̸= −r2 o,i, and r ̸= 0} Ci := {x ∈ Di | Bi(x) ≥ 0} (16) are valid CBF functions for the dynamics (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For i ̸= j, the obstacles corresponding to Bi : R3 → R and Bj : R3 → R are a positive distance apart if ∆ij := inf x ∈∼ Ci y ∈∼ Cj ||x − y|| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (17) A key innovation with respect to [46] is that we will compose the associated CBFs in a smooth (C1) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A potential drawback with respect to [46] is that we will assume the obstacles giving rise to the CBFs are a positive distance apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Similar to [47], we saturate standard quadratic CBFs before seeking to combine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Distinct from [47], we multiply the saturated CBFs instead of creating a weighted 5 2 4 6 8 10E 6 4 2 0 2 4 6 X m-2 4 目 6 8 10 6 4 2 0 2 4 6 X msum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This greatly simplifies the analysis of the composite CBF with respect to all previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Smooth Saturation Function We introduce a continuously differentiable saturation func- tion that will allow us to compose in a simple manner CBFs corresponding to obstacles that are a positive distance apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Consider σ : R → R by σ(s) := � � � � � s s ≤ 0 s(1 + s − s2) 0 < s < 1 1 s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (18) Then straightforward calculations show that for all s ∈ R, dσ(s) ds exists and satisfies dσ(s) ds := � � � � � 1 s ≤ 0 1 + 2s − 3s2 0 < s < 1 0 s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (19) Upon noting that for all 0 < s < 1, 0 < dσ(s) ds < 1, it follows that σ : R → R is continuously differentiable and monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For 0 ≤ s ≤ 1, σ is constructed from a degree- three Bézier polynomial p : [0, 1] → R such that p(0) = 0, dp(0) ds = 1, p(1) = 1, dp(1) ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Moreover, for 0 < s < 1, dp(s) ds > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For κ > 0, we define σκ : R → R by σκ(s) := σ( s κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (20) Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Suppose that κ > 0 and B : D → R is a candidate CBF with D and C defined as in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Then σκ ◦ B : D → R is a valid CBF for the system (2) if, and only if, B : D → R is a valid CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For x ∈ C, σκ ◦ B(x) > 0 and hence satisfies (8) for u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For x ∈∼ C, by the chain rule and the construction of σ : R → R, ∂σκ ◦ B(x) ∂x = dσ(s) ds ���� s= B(x) κ ∂B(x) ∂x = 1 κ ∂B(x) ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (21) Hence, the proof of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' III-C applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' ■ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Suppose for 1 ≤ i ≤ M, the CBFs Bi(x) : R3 → R are a positive distance apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Then there exist κ1 > 0, κ2 > 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=', κM > 0, such that for all i ̸= j, {x ∈ R3 | σκi ◦ Bi(x) < 1} ∩ {x ∈ R3 | Bj(x) < 0} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' By the disjointness property, ∆i := min j̸=i ∆ij > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For S ⊂ R3 and x ∈ R3, define the distance from x to S by d(x, S) := inf y∈S ||x − y||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (23) Then, because (i) Bi is continuous, (ii) the set complement of Ci is bounded, and (iii) d(x, ∼ Ci) > 0 =⇒ Bi(x) > 0, it follows that m∗ i := sup d(x,∼Ci)≤∆i Bi(x) (24) is a finite positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Therefore, for all 0 < κi < m∗ i , {x ∈ R3 | σκi ◦ Bi(x) < 1} ⊂ {x ∈ R3 | d(x, ∼ Ci) ≤ ∆i}, (25) and hence (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' ■ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Multiplication Property of Smooth Saturated CBFs For M ≥ 2 CBFs corresponding to disjoint obstacles, define the sets DM := M � i=1 Di CM := M � i=1 {x ∈ DM | Bi(x) ≥ 0} = M � i=1 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (26) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Under the assumed disjointness property, the product of smoothly saturated valid CBFs, BM(x) := M � i=1 σκi ◦ Bi(x), (27) is a valid CBF for DM, CM, and the dynamic system (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For x ∈ CM, the zero control u ≡ 0 satisfies (8) because the drift term f(x) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We show that for x ̸∈ CM, (8) can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' By the disjoint property of the assumed CBF functions, when BM(x) < 0, we have ∃i, such that σκi ◦ Bi(x) = Bi(x) < 0, and σκj ◦ Bj(x) = 1 for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, BM(x) = Bi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Because Bi(x) is assumed to be a valid CBF function, and both DM ⊂ Di and CM ⊂ Ci hold, the CBF property holds for BM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' ■ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Due to the way we have constructed the multi- obstacle CBF, the equilibrium analysis for a single obstacle carries over here without changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This is because, when the robot is at a boundary of an obstacle, the values of the saturated CBFs for the other obstacles will all be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' SIMULATION RESULTS WITH SINGLE AND MULTIPLE OBSTACLES In this section, we first use simulation to study the behavior and liveness of the proposed CLF-CBF system with a single obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Next, we run the system on several synthetic envi- ronments with 20 obstacles in Robot Operating System (ROS) [23] with C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' For the CBF in (6), we take Q = I and in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 1, we take κ1 = · · · = κM = min{∆2 i }M i=1, which is the minimum of the square of the distance between any of the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Robot Model in Simulation In MATLAB and ROS, the bipedal robot is represented by the Angular momentum Linear Inverted Pendulum (ALIP) model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The ALIP robot takes piece-wise constant inputs from the CLF-CBF-QP system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Let g, H, τ be the gravitational constant, the robot’s center of mass height, and the time interval of a swing phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The motion of an ALIP model on the x-axis satisfies �xk+1 ˙xk+1 � = � cosh(ξ) 1 ρ sinh(ξ) ρ sinh(ξ) cosh(ξ) � �xk ˙xk � + �1 − cosh(ξ) −ρ sinh(ξ) � px, (28) where xk and ˙xk are the contact position and velocity of the swing foot on the x-axis, px is the center of mass (CoM) position on the x-axis of the robot, ξ = ρτ and ρ = � g/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The motion of the robot on the y-axis can be similarly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 5: Illustration of how the trajectories vary as a function of differ- ent obstacle positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The target (marked in black) and the robot pose (−15, −15, −15◦) are fixed throughout all of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The different colors denote different simulations with only one obstacle present at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The red trajectory is generated without any obstacle present from the QP only containing CLF constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Behavior Study with Single Obstacle in MATLAB The optimal control command of the robot is the solution of the CLF-CBF-QP problem defined in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The time interval of a swing phase is set to τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='3s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot updates its pose based on the ALIP model and the optimal control command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The updated pose is then fed back to the CLF-CBF-QP system to compute the optimal control for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This process continues until the robot reaches the target or collides with an obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Figure 5 shows how the trajectories vary as a function of a single obstacle’s position with a fixed initial robot pose of (−15, −15, −15◦), marked as the magenta arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The red trajectory is the nominal trajectory without any obstacles present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Each colored trajectory and matching circle represent a distinct simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot successfully avoids the obstacle in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 6, we show how the trajectories vary as a function of different robot orientations with a fixed obstacle location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' When the robot is within an obstacle, there is also a valid solution that pushes the robot outside of the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 6: Illustration of how the trajectories vary as a function of different robot orientations with a fixed obstacle location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The target (marked in cyan) and obstacle at (−4, −4) are fixed through out all the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A different color stands for a different robot orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Consider the CBF constraint (13), LfB(x) + LgB(x)u + ηB(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (29) When the robot is withing an obstacle, B(x) < 0 and the QP selects u such that LfB(x) + LgB(x)u ≥ −ηB(x), causing the robot to leave the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 7: Liveness analysis for the CLF-CBF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The initial pose is (−15, −15, −15◦), and the target is located at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Each dot in the figure represents the center of an object with radius (r = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The interval between each center dots are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='2 meter in both x and y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Note that all the red points either originally collide with the robot or the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Liveness Analysis in MATLAB We analyze the liveness by placing an obstacle with a fixed radius (r = 1) at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot starts at (−15, −15, −15◦) and the target is located at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The obstacle is placed at every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='2 meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' If the robot successfully reaches the target without collision, the obstacle location is marked in green otherwise in red, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' All the red points either originally collide with the robot or the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Multi-Obstacle Simulation with ROS in C++ In this simulation, we implement a local map centering at robot position with a fixed size and a sub-goal selector to place a target within the local map to achieve long-term planning as not all the obstacles are perceived by the robot at the beginning in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Even though the global map is available in simulation but it is not available in practice, therefore, only 7 0 2 4 6 10 12 14 16 15 10 5 00 1 2 3 4 5 6 7 8 9 10 10 8 9- 4 2 0 x m0 success hit 2 4 6 10 12 14 16 16 14 12 10 8 9- 4 2 0 2 x [m](a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 8: Trajectories of 39 obstacles in noise-free (top two) and 20 obstacles in noisy (bottom two) synthetic maps with the size of 50 × 30 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The highlighted areas are the local map at that specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The dark blue circles are the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Different colors represent different runs in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' the information within the local map at the specific timestamp is provided to the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot model is the same ALIP model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 8, we generate two noise-free and two noisy syn- thetic maps with the size of 50×30 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Each map contains 20 obstacles marked as blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We run six different initial poses and final goals for each map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Different colors represent different runs in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The highlighted area is the local map at that specific timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' An intermediate goal is chosen at the intersection between the boundary of the local map and the line connecting the robot and the final goal at the current timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' If the intermediate goal collides with an obstacle, it is moved back along the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The intermediate goal is updated when it is reached or becomes inside of an obstacle due to the update of the local map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The robot with ALIP model successfully reaches the goals in all 6 × 4 = 24 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' EXPERIMENTAL RESULTS ON A BIPEDAL ROBOT We perform several experiments of the proposed CLF- CBF-QP system on Cassie Blue, a bipedal robot with 20 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The entire system integrates elevation mapping, intermediate goal selection, and the low-level CLF- CBF obstacle avoidance system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 9: The left shows the sensor suite with different sensors, and the right shows the sensor suite mounted on Cassie Blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Autonomy System Integration The following is summarized from [1] for the completeness of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' To allow the robot to perceive its surroundings under different lighting conditions and environments, we de- signed a perception suite that consists of an RGB-D camera 8 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 10: Autonomy experiments with Cassie Blue on the first floor of FRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The green arrow is Cassie’s pose and the green lines are the resulting trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The blue sphere is the selected target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The map is colored by height and the highlighted area is the local map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (Intel RealSense™ D435) and a 32-Beam Velodyne ULTRA Puck LiDAR, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The sensor calibrations are performed via [50]–[53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The invariant extended Kalman filter (InEKF) [54] estimates the pose of Cassie at 2k Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The raw point cloud is motion compensated by the InEKF and then used to build an elevation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Autonomy Experiment on Cassie Blue We conducted several indoor experiments with Cassie Blue on the first floor of the Ford Robotics Building (FRB) where tables and chairs are considered obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' To detect obstacles in the environment, an occupancy grid map is updated in real- time using the timestamped elevation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Grids with heights greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content='2 meters are considered occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' An occupied grid is defined as the boundary of obstacles if there is an unoccupied grid in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The Breadth First Search (BFS) algorithm [55] is utilized to find the separated obstacles in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Next, we apply the Gift Wrapping Algorithm [56] to the boundary grids of obstacles to find the convex hulls of the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Finally, the minimum bounding ball algorithm [57] is applied to the convex hulls to find the minimum bounding circles of the obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The circles are used to represent obstacles in the CBF function (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The target position is selected by clicking a point in the global map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' If the final target is not within the current local map, an intermediate goal will be selected within the local map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' When an intermediate goal is reached by Cassie or becomes invalid because of the update of the local map, it is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In the experiments, Cassie successfully avoids all the obstacles and reaches the target position, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CONCLUSION This paper presented a reactive planning system that al- lows a Cassie-series bipedal robot to avoid multiple non- overlapping obstacles via a single, continuously differentiable control barrier function (CBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The overall system detects an individual obstacle via a height map derived from a LiDAR point cloud and computes an elliptical outer approximation, which is then turned into a quadratic CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A continuously differentiable saturation function is presented that preserves the CBF property of a quadratic CBF while allowing the 9 saturated CBFs for individual obstacles to be turned into a single CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The CLF-CBF-QP formalism developed by Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' can then be applied to ensure that safe trajectories are generated in the presence of multiple obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Liveness is ensured by an analysis of induced equilibrium points that are distinct from the goal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Safe planning in environments with multiple obstacles is demonstrated both in simulation and experimentally on the Cassie bipedal robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' ACKNOWLEDGMENT Toyota Research Institute provided funds to support this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Funding for J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Grizzle was in part provided by NSF Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 1808051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' This article solely reflects the opinions and conclusions of its authors and not the funding 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Harris, “Encyclopedia of operations research and management science,” Journal of the Operational Research Society, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 759–760, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' APPENDIX A EQUILIBRIUM ANALYSIS OF MULTI-OBSTACLE SYSTEMS We give the complete analysis for a single obstacle, following the work of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Because the drift term of our model is zero, any equilibrium points are where the optimal control is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' To avoid this undesirable situation, we seek to find all equilibrium points E = {x|u∗ = [0, 0, 0]T, r > 0, and δ, θ ∈ (−π, π]}, where u∗ is the optimal control variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Recall thatf = �0 0 0�T in (2), which leads to LfV (x) = LfB(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' We denote the following to re-write the CLF and the CBF constraints: �dx dy 0� = −LgB(x) �ax ay aω � = LgV (x), (30) where ax = −r cos(δ) + βγ2 sin(2βδ) sin(δ) 2r ay = −r sin(δ) − βγ2 sin(2βδ) cos(δ) 2r aω = βγ sin(2βδ) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (31) The constraints become L(x, u, s) = axvx + ayvy + aωω − s + µV (x), (32) B(x, u) = dxvx + dyvy − ηB(x), (33) and the cost function (15) of the QP can then be re-written as: J(u, s) = 1 2h1(vx − vref x )2 + 1 2h2(vy − vref y )2 + 1 2h3(ω − ωref)2 + 1 2ps2, (34) where {hi}3 i=1 are the diagonal elements of H in (15), the weights of control variables [vx, vy, ω] with hi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The KKT conditions [58] of this quadratic program are: ∂L ∂u = Hu∗ − Huref + λ1LgV T − λ2LgBT = 0 (35) ∂L ∂s = ps − λ1 = 0 (36) 0 = λ1(LfV + LgV u∗ + µV − s) (37) 0 = λ2(−LfB − LgBu∗ − ηB) (38) 0 ≥ LfV + LgV u∗ + µV − s (39) 0 ≥ −LfB − LgBu∗ − ηB (40) 0 ≤ λ1, λ2, (41) where λ1, λ2 ∈ R, and L is the Lagrangian function and defined as L(u, s, λ1, λ2) = J(u, s) + λ1L(x, u, s) + λ2B(x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (42) Next, we analyze equilibrium points (if any) via four cases depending on whether each CLF or CBF constraint is active or inactive following [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Both CLF and CBF are inactive When both constraints are inactive, we have λ1 = 0 λ2 = 0 0 > LfV + LgV u∗ + µV − s 0 > −LfB − LgBu∗ − ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (43) With (35) and (36), u∗ and s∗ in this case are u∗ = uref s∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (44) From (4), as long as the goal is not reached, uref is not a zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, there is no equilibrium point in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CLF constraint inactive and CBF constraint active We prove that there is no equilibrium point in this case by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' When the CLF constraint is inactive and the CBF constraint is active, we have λ1 = 0 λ2 ≥ 0 0 > LfV + LgV u∗ + µV − s 0 = −LfB − LgBu∗ − ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (45) With (35) and (36), u∗, s∗ and λ2 in this case are u∗ = uref + λ2H−1LgBT s∗ = 0 λ2 = −ηB + LfB + LgBuref LgBH−1LgBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (46) 11 If there is an equilibrium point, then u∗ is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, at the equilibrium point, by LfV (x) = 0, u∗ = 0 and s∗ = 0, we have LfV + LgV u∗ + µV − s∗ = µV > 0, (47) which conflicts with (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Therefore, there is no equilibrium point in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' CLF constraint active and CBF constraint inactive When the CLF constraint is active and the CBF constraint is inactive, we have λ1 ≥ 0 λ2 = 0 0 = LfV + LgV u∗ + µV − s 0 > −LfB − LgBu∗ − ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (48) With (35) and (36), u∗, s∗ and λ1 in this case are u∗ = uref − λ1H−1LgV T s∗ = λ1 p λ1 = pµV + pLfV + pLgV uref pLgV H−1LgV T + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (49) Using the variables defined in (30), u∗ can be rewritten as: u∗ = � � v∗ x v∗ y ω∗ � � = � �� vref x − λ1ax h1 vref y − λ1ay h2 ωref − λ1aω h3 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (50) We know from (31) that (ay = 0 & aω = 0) ⇐⇒ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (51) In addition, we know from (4) that (vref y = 0 & ωref = 0) ⇐⇒ δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (52) Therefore, we split this case into three cases based on the value of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 1) δ = 0 (Case I): Substituting δ = 0 to (30), we have ax = −r < 0, ay = 0, aω = 0, and to (4), we have vref x > 0, vref y = 0, ωref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Finally, with (41), the optimal control command (50) can be simplified as: u∗ = � � v∗ x v∗ y ω∗ � � = � � vref x + λ1r h1 > 0 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (53) The optimal control command is not a zero vector, and hence there is no equilibrium point in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 2) δ > 0 (Case II): When δ > 0, by the definitions in (31), we have ay < 0, aω > 0, and by (4), we have vref y > 0, ωref < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' With (41) and (50), we have v∗ y = vref y − λ1ay h2 > 0 ω∗ = ωref − λ1aω h3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (54) The optimal control command is not a zero vector in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Therefore, there is no equilibrium points in this case either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 3) δ < 0 (Case III): Similarly, by (30) and (4), we have ay > 0, aω < 0 and vref y < 0, ωref > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' With (41) and (50), we have v∗ y = vref y − λ1ay h2 < 0 ω∗ = ωref − λ1aω h3 > 0 (55) The optimal control command is not a zero vector in this case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' there is, thus, no equilibrium point in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In summary, there is no equilibrium point when the CLF constraint is active and the CBF constraint is inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Both CLF and CBF constraint are active When the CLF constraint is active and the CBF constraint is active, we have λ1 ≥ 0 λ2 ≥ 0 0 = LfV + LgV u∗ + µV − s 0 = −LfB − LgBu∗ − ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (56) We can rewrite (35) and (36) as: u∗ = uref − λ1H−1LgV T + λ2H−1LgBT s∗ = λ1 p (57) Using the variables defined in (30), u∗ can be rewritten as: u∗ = � � v∗ x v∗ y ω∗ � � = � �� vref x − λ1ax h1 − λ2dx h1 vref y − λ1ay h2 − λ2dy h2 ωref − λ1aω h3 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (58) When the robot is at an equilibrium point, u∗ is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' By (56) and LfB = 0, u∗ = 0, we have B = 0, which implies that the robot is at the boundary of an obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' In the following proof of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' A-D, we will assume the robot is at the boundary of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' The property of B = 0 leads to an immediate proposition which is helpful in finding the equilibrium point in the system when one of the components of the optimal control is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' dy = 0 =⇒ v∗ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' By the proof in III-C, we have LgB(x) = ∇B(x) · g(x) ̸= 0 for x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Therefore, when dy = 0, we have dx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Then, we can further have LgB(x)u∗ = 0 =⇒ v∗ x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' ■ In addition, with the properties (51) and (52), we split this case into four cases based on whether δ and dy are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 1) dy = δ = 0 (Case I): Substituting to (30), we have ax = −r < 0, ay = 0, aω = 0, and to (4), we have vref x > 0, vref y = 0, ωref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Finally, with Proposition 3, in this case the optimal control command (58) can be written as: u∗ = � � v∗ x v∗ y ω∗ � � = � � vref x − λ1ax h1 − λ2dx h1 0 0 � � = � � 0 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (59) λ1 and λ2 can be obtained by (59), (56) and (57): λ1 = pµV > 0 λ2 = h1vref x − pµV ax dx (60) By (41) and (60), we have ∵ vref x > 0, ax < 0, h1vref x − pµV ax dx ≥ 0 −→ dx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (61) Hence, there is an equilibrium point when B = 0, dy = δ = 0 and dx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 2) dy ̸= 0, δ = 0 (Case II): When δ = 0, by (30) and (4), we have ax = −r < 0, ay = 0, aω = 0 and vref x > 0, vref y = 0, ωref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Finally, with (41), the optimal control command (58) can be simplified as: u∗ = � � v∗ x v∗ y ω∗ � � = � � vref x − λ1ax h1 − λ2dx h1 − λ2dy h2 ̸= 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' (62) Because v∗ y ̸= 0, the optimal command is not a zero vector in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Equilibrium points don’t exist when dy ̸= and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 12 3) δ > 0 (Case III): When δ > 0, by (54), ω∗ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, the optimal command is not a zero vector and there are no equilibrium points in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 4) δ < 0 (Case IV): When δ < 0, by (55), ω∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' Hence, the optimal command is not a zero vector and there are no equilibrium points in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQf8f7E/content/2301.01906v1.pdf'} diff --git a/0dFRT4oBgHgl3EQfkjfS/content/tmp_files/2301.13596v1.pdf.txt b/0dFRT4oBgHgl3EQfkjfS/content/tmp_files/2301.13596v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..18fa167f406392b3376a5fd92e26b36c2a78df06 --- /dev/null +++ b/0dFRT4oBgHgl3EQfkjfS/content/tmp_files/2301.13596v1.pdf.txt @@ -0,0 +1,704 @@ +Confinement of fractional excitations in a triangular lattice antiferromagnet +L. Facheris,1, ∗ S. D. Nabi,1 A. Glezer Moshe,2 U. Nagel,2 T. R˜o˜om,2 +K. Yu. Povarov,1, 3 J. R. Stewart,4 Z. Yan,1 and A. Zheludev1, † +1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland +2National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia +3Present address: Dresden High Magnetic Field Laboratory +(HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany +4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11 0QX, United Kingdom +(Dated: February 1, 2023) +High-resolution neutron and THz spectroscopies are used to study the magnetic excitation spec- +trum of Cs2CoBr4, a distorted-triangular-lattice antiferromagnet with nearly XY-type anisotropy. +What was previously thought of as a broad excitation continuum [Phys. Rev. Lett. 129, 087201 +(2022)] is shown to be a series of dispersive bound states reminiscent of “Zeeman ladders” in quasi- +one-dimensional Ising systems. At wave vectors where inter-chain interactions cancel at the Mean +Field level, they can indeed be interpreted as bound finite-width kinks in individual chains. Else- +where in the Brillouin zone their true two-dimensional structure and propagation are revealed. +In conventional magnetic insulators the dynamic re- +sponse is typically dominated by coherent single-particle +S = 1 excitations, aka magnons or spin waves. In many +low-dimensional and highly frustrated quantum spin sys- +tems elementary excitations carry fractional quantum +numbers, be they spinons in Heisenberg spin chains [1–4] +or Majorana fermions in the now-famous Kitaev model +[5–7]. +The physical excitation spectrum, such as that +measured by neutron spectroscopy, is then dominated by +broad multi-particle continua [8–11]. In addition to the +continuum, fractional excitations may also form bound +states due to attractive interactions between them. +A +spectacular new phenomenon emerges when interactions +are confining, i.e. do not fall off with distance, much like +strong forces that bind quarks in hadrons [12]. This pro- +duces an entire series of bound states inside the resulting +potential well. An example is the sequence of domain wall +(kink) bound states in quasi-one-dimensional Ising spin +chains [13–15]. +The confining potential for this model +is linear and results from 3-dimensional couplings, which +generate an effective field acting on individual chains [14]. +The binding energies are, in supreme mathematical ele- +gance, spaced according to the negative zeros of the Airy +function [13, 15]. +The best-known experimental examples of such “Zee- +man ladder” spectra are the quasi-one-dimensional Ising +ferromagnet CoNb2O6 [16] and antiferromagnet (AF) +BaCo2V2O8 +[17], as well as the isostructural com- +pound SrCo2V2O8 [18], where as many as 8 consecutive +bound states are observed. Shorter sequences have been +found in another prototypical Ising spin chain material, +RbCoCl3 [19]. +In the present work we report the ob- +servation of a somewhat similar phenomenon in an en- +tirely different type of system, namely in a quasi-two- +dimensional distorted-triangular-lattice AF where the +effective magnetic anisotropy is predominantly of XY, +rather than Ising character. +That the quintessentially +one-dimensional physics of bound kinks survives in two +dimensions is remarkable. We argue that it is “rescued” +at certain special wave vectors by the intrinsic frustration +in triangular lattice geometry. Elsewhere in the Brillouin +zone the bound states are no longer restricted to sin- +gle chains and are to be viewed as 2-dimensional objects +propagating on the entire triangular plane. +The material in question, Cs2CoBr4 (space group +Pnma, a = 10.19, b = 7.73, c = 13.51 ˚A), is a very +interesting J − J′ model distorted-triangular-lattice AF +[20, 21]. +Despite a prominent triangular motif in its +structure, it demonstrates certain one-dimensional fea- +tures such as a field-induced incommensurate spin den- +sity wave with Tomonaga-Luttinger spin liquid type dy- +namics and a propagation vector controlled by a one- +dimensional nesting in the spinon Fermi sea. +Its true +2-dimensional nature is manifest in the presence of a ro- +bust m = 1/3 magnetization plateau, typical of a trian- +gular AF. The model magnetic Hamiltonian is described +in detail in Refs. [20, 21]. The key structural features +are chains of Co2+ ions that run along the crystallo- +graphic b axis of the orthorhombic lattice (see Fig. +1 +in Ref. [20]). The chains are coupled in the (bc) plane in +a zigzag fashion to form a distorted triangular network +(inset of Fig. 1(d)). Easy-plane single-ion anisotropy en- +sures that the low-energy physics of the spin-3/2 Co2+ +ions can be described in terms of effective S = 1/2 +pseudo-spins. The components of the effective exchange +coupling constants are subject to restrictions imposed +by the pseudo-spin projection. +A simplistic spin-wave +analysis of previous inelastic neutron data provided a +rough estimate for the nearest-neighbor in-chain AF ex- +change tensor components: +JXX ∼ J, JY Y +∼ 1.1J, +JZZ ∼ 0.25J, J = 0.8 meV [21]. Here Y is chosen along +the b crystallographic direction, and X and Z alternate +between adjacent chains, where anisotropy planes are al- +most orthogonal. Note that this is practically a planar +arXiv:2301.13596v1 [cond-mat.str-el] 31 Jan 2023 + +2 +(b) +(c) +(d) +(a) +ground state +J +J' +bound state m3 +FIG. 1. +(a)-(b) Neutron scattering intensity (solid sym- +bols) measured at T = 40 mK versus energy transfer at +the one-dimensional AF zone-centers q = (0, 0.5, 0.5) and +q = (0, 1, 0.5), respectively. +The data are integrated fully +along h direction and in ±0.025 r.l.u. and ±0.25 r.l.u. along +k and l, respectively. Solid lines are fits to a series of Gaus- +sian peaks. +Dashed Gaussians represent the calculated ex- +perimental energy resolution. Black dotted lines indicate the +fitted flat background. +(c) Measured terahertz absorption +(solid line) versus absorbed photon energy for light propa- +gating along the c axis at 0.2 K. Dashed areas highlight the +individual components that find counterparts in the neutron +spectra. (d) Measured excitation energy plotted versus the +value of negative roots of the Airy function. The solid line is +a linear fit as described in the text. The blue area highlights +the points used for the fit. Inset: cartoons of the magnetic +ground state and a representative m = 3 2-kink bound state. +exchange anisotropy, with only a tiny in-plane Ising com- +ponent to account for the ∆ ∼ 0.4 meV spectral gap +found in this system. +The frustrated inter-chain cou- +pling J′ is significant, of the order of 0.45J, and is of +predominantly Ising (Y Y ) character. Inter-plane inter- +actions J′′ are not frustrated. The material orders mag- +netically in a colinear stripe-type structure, with an or- +dering wavevector (0, 1/2, 1/2) (see inset in Fig. 1(d)). +The N´eel temperature TN = 1.3 K allows us to esti- +mate J′′. If this were the only coupling between chains +with no additional frustration due to J′, we could expect +kBTN ∼ 2∆/ ln(∆/J′′) [22]. The actual value of J′′ must +be larger than thus obtained, as the in-plane frustration +interferes with the emerging magnetic structure. A cer- +tain upper estimate is given by the mean field picture +where kBTN ∼ 2J′′S(S + 1). This leads us to conclude +that 3 · 10−4 meV ≲ J′′ ≲ 0.075 meV ≪ J, confirming +the quasi-2-dimensional character of the material. +Our previous inelastic neutron scattering experiments +indicated that the excitation spectrum in zero applied +field is a gapped continuum of states, with intensity con- +centrated on its lower bound, and a strong dispersion +along the chain axis [21]. +The central finding of the +present work is that this “continuum” is actually a se- +quence of at least 9 sharp bound states that previously +could not be observed due to poor experimental energy +resolution. New neutron data were collected at the LET +time-of-flight spectrometer at ISIS (UK), using 2.35 meV +incident energy neutrons in repetition-rate-multiplication +mode [23]. +We used the same 1.16 g single crystal as +in [21] mounted on a 3He-4He dilution refrigerator. All +measurements were performed at a base temperature of +40 mK. In the experiment the sample was rotated 180◦ +around the a axis in steps of 1◦. The spectra were mea- +sured for ∼ 10 minute counting time at each sample po- +sition. +We first focus on the one-dimensional AF zone-centers +(qb = 0, π), where inter-chain interactions within the tri- +angular planes cancel out at the Mean Field-RPA level, +and where spin wave theory predicts no transverse disper- +sion or intensity modulation of excitations. Fig. 1(a),(b) +show constant-q cuts through the data at wave vectors +q = (0, 0.5, 0.5) and q = (0, 1, 0.5), respectively. A se- +quence of sharp peaks is clearly apparent in both cases. +A fit to the data using empirical Gaussian profiles yields +an accurate measure of the peak positions and shows +that their widths are essentially resolution-limited. +In +Fig. 1(a),(b) this is emphasized by the shaded Gaussians +representing the computed experimental resolution [24]. +Corroborative evidence is also obtained by THz spec- +troscopy. The experiment was performed with a Martin- +Puplett-type interferometer and a 3He-4He dilution re- +frigerator with base temperature of 150 mK using a 3He- +cooled Si bolometer at 0.3 K. The sample was a circu- +lar plate approximately 1 mm thick in c direction and +4 mm in diameter. THz radiation propagating along the + +3 +crystal c axis was unpolarized and the apodized instru- +mental resolution was 0.025 meV. The THz absorption +spectrum is shown in Fig. 1(c). It is calculated as a dif- +ference of spectra measured at 0.2 K and 2 K, i.e. in +the magnetically ordered phase and above TN. The THz +spectrum appears to have some features absent in the +neutron spectrum, but all peaks found in the latter are +also present here. The positions of these peaks were de- +termined in Gaussian fits (shaded peaks) in a narrow +range ±0.025 meV near each peak value. +The spacing between the excitation peaks present in +both measurements corresponds to confinement in an ap- +proximately linear one-dimensional potential. To demon- +strate this, we plot the excitation energies deduced from +neutron spectra at several wave vectors, as well as the +positions of corresponding THz peaks, versus the neg- +ative roots zi of the Airy function in Fig. 1(d). For a +precise linear attractive potential λ|x| between the dis- +persive particles, near the minimum ϵ(k) = m0+ℏ2k2/2µ +we expect the excitation energies to be [15, 16] +mi = 2m0 + (ℏλ)2/3µ−1/3zi with i = 1, 2, . . . . +(1) +In the actual data, the linear dependence is appar- +ent for all but the first few points. +As will be ad- +dressed in more detail below, this slight deviation in- +dicates that the confining force increases somewhat at +short distances. From a linear fit to the higher-energy +peaks we can immediately extract the slope 0.072(3) meV +and the energy of a single particle m0 = 0.18(1) meV +(half-intercept). +Using the single-particle kinetic mass +ℏ2/µ = 0.39 meV×b2 [24], we estimate the confining force +constant λ = 0.031(2) meV/b [25]. +The next point that we make is that the observed +bound states at the one-dimensional AF zone-center are +essentially one-dimensional objects. This is concluded by +analyzing the neutron spectra shown in Figs. 2(a),(b). +The bound states do not propagate in either transverse +direction and thus have an essentially flat dispersion. +Moreover, their intensity shows no modulation trans- +verse to the chains, as shown for the first two modes in +Figs. 2(e),(f). The measured transverse wave vector de- +pendencies are entirely accounted for (solid lines) by the +combined effects of i) the magnetic form factor of Co2+ +and ii) a neutron polarization factor for spin components +perpendicular to the chain axis (to the direction of or- +dered moments in the ground state). This implies that +these excitations do not involve cross-chain correlations +and are confined to a single chain. +This consideration prompts a simple interpretation +of the observed behavior. +Similarly to the situation +in CoNb2O6 and BaCo2V2O8, the observed modes are +bound states of two kinks (domain walls) in individual +chains. +Such an excitation is illustrated by the car- +toon in the inset of Fig. 1(d). +Since the ordered mo- +ments are along the b crystallographic axis, they are po- +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +FIG. 2. (a)-(d) False color plot of neutron scattering inten- +sity measured at T = 40 mK plotted versus energy transfer +and momentum transfer transverse to the crystallographic b- +axis. Gray areas mask regions of elastic-incoherent scattering. +Background subtraction has been performed as described in +[24]. +The orange regions represent energy-integration win- +dows used to extract the cuts in panels below. +(e)-(h) +Intensity-momentum cuts (solid symbols) for the first two +modes in the Zeeman ladder. The blue line shows the product +of calculated neutron polarization factor for excitations polar- +ized perpendicular to the direction of the ordered moment and +the magnetic form-factor-squared for Co2+. +larized transverse to that direction [21], in agreement +with the measurement. +The energy m0 is to be asso- +ciated with that of a single domain wall. As a consis- +tency check, we can compare that to the computed en- +ergy of a domain wall in a classical spin chain. Using +JY Y /JXX ∼ 1.1 as estimated for Cs2CoBr4, with a triv- +ial numerical classical-energy minimization procedure we +get m0 ∼ 0.9JS2 = 0.18 meV, in excellent agreement +with the measured value. +Geometric frustration ensures that at the magnetic +zone-center these strings of flipped spins within a single +chain incur no energy cost due to interactions with adja- +cent chains within the triangular lattice. Moreover, any +transverse dispersion is suppressed. At the same time, +the interaction energy due to unfrustrated inter-layer +coupling is proportional to the string length, resulting +in confinement. In this simplistic picture, the confining +force is λ = 2J′′S2/b. This yields an inter-layer coupling +constant J′′ = 0.062(4) meV, inside the possible range +deduced from TN. The first lowest-energy bound state +with energy m1 corresponds to a single spin flip in the +chain, in other words to a single-magnon excitation. The +i-th higher-energy states are two domain walls separated +by a length-i string of spins that are aligned opposite to +the ground state AF spin configuration. + +4 +The deviation from linear-potential behavior at low en- +ergies is also readily explained by this picture. Since the +material is almost planar, the domain walls are not con- +fined to a single bond as in the ideal Ising case, but have +a characteristic size l [26]. We can estimate that quan- +tity in a classical spin chain using the above-mentioned +anisotropy parameters: l ∼ 2b. The energy of the first +few bound states is thus modified due to a physical over- +lap of the two bounding domain walls. Experimentally, +the bound state energy is reduced, which corresponds to +an additional attractive interaction between kinks. Once +the kinks are separated by a distance of more than ∼ l, +this interaction becomes negligible and the confinement +potential becomes linear, originating only from inter- +layer interactions. +Away from the one-dimensional AF zone-centers, the +excitations are considerably more complex. This is very +clear in the longitudinal dispersion of the bound states +shown in Fig. 3(a),(b). Other than at qb = 0, π (k = +0, 1/2) the m1 mode splits into two branches, each with +an asymmetric dispersion relation. In fact, the m1 state +at qb = π seems to be continuously connected to the m2 +excitations at qb = 2π (k = 1) and vice versa. Fitting +the dispersion of the strongest low-energy mode in the +vicinity of qb = π to a Lorentz-invariant relativistic form +(ℏωq)2 = ℏ2∆ (qb)2 /µ + ∆2, +(2) +yields the value of kinetic mass quoted above. +A look at the intensities reveals that other than at the +special wave vectors, the bound states can no longer be +seen as strings in a single chain, but are “dressed” with +correlations extending to several neighboring chains in +the triangular plane. +This conclusion is reached from +Fig. 2(c),(d), that show a transverse cut of the spectrum +at qb = 5π/4 and qb = 3π/2, respectively. +As plot- +ted in Fig. 2(g),(h), the measured intensity of the first +two modes now shows a much steeper transverse wave +vector dependence than computed from just the polar- +ization and form factors (solid line). The second mode +even seems to show signs of intensity oscillations. +Our data reveal that away from the special wave vec- +tors the bound states also propagate in two dimensions, +albeit with a small bandwidth. Indeed, in Fig. 2(d) one +can see that at qb = 3π/2 the bound states develop a +non-zero dispersion along the c∗ direction, in contrast to +what is seen at qb = 0, π. Although the bandwidth of +transverse dispersion, 0.08 meV, is at the limit of our +experimental resolution, qualitatively one can say that +qc = 0, 4π are dispersion minima for the m1 mode, while +the maximum is at qc = 2π. That periodicity is consis- +tent with having two chains per unit cell along the c-axis +direction in the crystal structure. +Overall, the differences between our results and spectra +of Ising spin chains [16, 17] are striking. In the latter, +all bound states, including the first one, are much less +dispersive than the lower edge of the entire spectrum, +(a) +(b) +FIG. 3. (a)-(b) False color plot of neutron scattering inten- +sity measured at T = 40 mK plotted versus energy transfer +and momentum transfer along q = (0, k, 0.5) and q = (0, k, 1) +respectively. The data were fully integrated along h, and in +the range ±0.25 r.l.u. along l around the central value. The +gray areas mask regions where the incoherent scattering domi- +nates the signal. Background subtraction has been performed +as described in [24]. +which approximately corresponds to the lower edge of the +two-kink continuum in the absence of long-range order. +As a result, each bound state persists only in a restricted +area in the Brillouin zone. In contrast, in Cs2CoBr4 a +few of the lower-energy bound states are highly dispersive +and span across the entire zone. +In summary, +we demonstrate that “Zeeman lad- +ders” of confined fractional excitations can exist in a +bona fide quasi-two-dimensional system. +These states +are inherently related to those in the one-dimensional +model, as revealed at special wave vectors where two- +dimensional interactions are canceled by geometric frus- +tration. +However, elsewhere in reciprocal space their +true 2-dimensional character is manifest. +Once again, +the distorted triangular lattice model provides a link be- +tween one- and two-dimensional quantum magnetism. +This work was supported by a MINT grant of the Swiss +National Science Foundation. We acknowledge support +by the Estonian Research Council grants PRG736 and +MOBJD1103, and by European Regional Development +Fund Project No. TK134. 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Bennington., LET, a cold +neutron multi-disk chopper spectrometer at ISIS, Nuclear +Instruments and Methods in Physics Research Section A: +Accelerators, Spectrometers, Detectors and Associated +Equipment 637, 128 (2011). +[24] See Supplemental Material for detailed discussion of the +resolution calculations, additional inelastic neutron scat- +tering data, background subtraction procedure, and esti- +mate of the kinetic mass for a kink. +[25] This force of ∼ 6 fN corresponds to the gravity pull be- +tween two average humans at a separation of 8 km. +[26] We define the domain wall width in a spin-S chain as the +distance over which the z-axis spin component changes +from S/2 to −S/2 near its center. +[27] L. Facheris, et al.; +(2022): +Spin-density wave dy- +namics in a 2D distorted triangular lattice antifer- +romagnet, +STFC +ISIS +Neutron +and +Muon +Source, +https://doi.org/10.5286/ISIS.E.RB2210048 . + +Supplemental Material for “Confinement of fractional excitations in a triangular +lattice antiferromagnet” +L. Facheris,1, ∗ S. D. Nabi,1 A. Glezer Moshe,2 U. Nagel,2 T. R˜o˜om,2 +K. Yu. Povarov,1, 3 J. R. Stewart,4 Z. Yan,1 and A. Zheludev1, † +1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland +2National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia +3Present address: Dresden High Magnetic Field Laboratory +(HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat, +Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany +4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11 0QX, United Kingdom +(Dated: January 31, 2023) +This Supplemental Material provides further details supporting the main text that may be of +interest to the specialized reader. +In particular, the resolution calculations, additional inelastic +data, the background subtraction for the neutron spectroscopic measurements, and estimate for the +kink’s kinetic mass are presented. +CONTENTS +I. Determination of energy resolution for the LET +experiment +1 +II. Additional cuts used for Fig. 1(d) +1 +III. Background subtraction procedure for LET data +1 +IV. Estimating a kink’s kinetic mass µ. +2 +References +2 +I. +DETERMINATION OF ENERGY +RESOLUTION FOR THE LET EXPERIMENT +The neutron scattering data presented in the main text +were obtained on the direct-geometry time-of-flight LET +spectrometer at ISIS (UK) [1]. The instrument was op- +erated in the high-flux mode, with a chopper resolution +frequency of 210 Hz and a pulse remover frequency of +140 Hz. A phase delay time for chopper 2 of 87000 µs +was introduced to avoid contamination on the main in- +coming channel Ei = 2.35 meV by slower neutrons. The +resolution calculations were performed with the PyChop +interface of Mantid Workbench [2]. The obtained resolu- +tion profile is shown in SUPP. FIG. 1. +The widths of the shaded Gaussian profiles in Fig. +1(a),(b) of the main text were calculated based on the +fitted peak positions and the data in SUPP. FIG. 1. +∗ lfacheri@phys.ethz.ch +† zhelud@ethz.ch; http://www.neutron.ethz.ch/ +SUPP. FIG. 1. Calculated energy resolution (solid line) ver- +sus neutron energy transfer for the spectrometer settings +listed in the text. +Dotted lines mark the positions mi at +q = (0, 0.5, 0.5) as obtained from Fig. 1(a) of the main text. +II. +ADDITIONAL CUTS USED FOR FIG. 1(d) +The additional cuts at q = (0, 0.5, 1) and q = (0, 1, 1) +(not shown in the main text) are displayed in SUPP. FIG. +2. The fit is performed in full analogy to Fig. 1(a),(b) as +described in the main text. The extracted peak positions +from SUPP. FIG. 2 (a),(b) are plotted in Fig. 1(d) of the +main text. +III. +BACKGROUND SUBTRACTION +PROCEDURE FOR LET DATA +The inelastic neutron scattering data presented in Fig. +2 and Fig. 3 of the main text are background subtracted. +Although the dataset was rather clean, a background +subtraction similar to that in [3] was nonetheless per- +formed. In this section the model adopted to describe +the background is outlined. The analysis was performed +using the Horace software package [4]. +SUPP. FIG. 3 shows raw data corresponding to Fig. 3 +of the main text. Strong sharp lines at the edges of the + +2 +(a) +(b) +SUPP. FIG. 2. +(a)-(b) Neutron scattering intensity (solid +symbols) measured at T = 40 mK versus energy transfer at +q = (0, 0.5, 1) and q = (0, 1, 1), respectively. The data are +integrated fully along h direction and in ±0.025 r.l.u. and +±0.25 r.l.u. along k and l, respectively. Solid lines are fits to +a series of Gaussian peaks. Dashed Gaussians represent the +calculated experimental energy resolution. Black dotted lines +indicate the fitted flat background. +dataset below 0.4 meV are known spurious originating +from scattering from the sample environment employed. +The total background was modeled assuming no mag- +netic scattering below the gap and above the top of the +spectrum. Thus, the background dataset is identical to +original data for ℏω ≤ 0.34 meV and ℏω ≥ 1.28 meV +(see dashed horizontal lines in SUPP. FIG. 3 for the +background regions projected on these particular cuts). +In the intermediate energy region, momentum-dependent +boxes were constructed as shown in SUPP. FIG. 3 and +numerically interpolated over the total explored (q, ℏω)- +space. The so-obtained background was then point-to- +point subtracted from the original data. +IV. +ESTIMATING A KINK’S KINETIC MASS µ. +Near it’s minimum at a one-dimensional wave vector +k0 = π +b , the dispersion relation for a single kink can be +approximated as +ϵk = m0 + ℏ2 +2µ(k − k0)2. +(S.1) +The parameter µ is the kinetic “mass” of this quasiparti- +cle. We can access it from the experimentally measured +spectrum of two-kink excitations. For a two-kink state, +energy-momentum conservation dictates +ℏω(2−kink) +q += ϵk+ϵq−k = 2m0+ ℏ2 +2µ +� +(k − k0)2 + (q − k + k0)2� +. +(S.2) +Minimizing (S.2) with respect to the “hidden” quasi- +momentum k, we find that the lower boundary of the +two-particle continuum lies at k = q. Thus, the lowest +magnon-like dispersion is given by: +ℏωq = 2m0 + ℏ2 +2µ +� +(q − k0)2 + k2 +0 +� +(S.3) +Near the minimum wavevector q0 = k0 → π/b, we find +that the curvature of the parabola-like dispersion is ac- +tually the same for a single kink and the lowest bound +state. +(a) +(b) +SUPP. FIG. 3. (a)-(b) False color plot of raw neutron scat- +tering intensity measured at T = 40 mK plotted versus en- +ergy transfer and momentum transfer along q = (0, k, 0.5) +and q = (0, k, 1) respectively. The data were fully integrated +along h, and in the range ±0.25 r.l.u. +along l around the +central value. The gray areas mask regions where the inco- +herent scattering dominates the signal. Orange dashed lines +and boxes delimit the edges of the background dataset, as de- +scribed in the text. +[1] R. Bewley, J. Taylor, and S. Bennington., LET, a cold +neutron multi-disk chopper spectrometer at ISIS, Nuclear +Instruments and Methods in Physics Research Section + +AA3 +A: Accelerators, Spectrometers, Detectors and Associated +Equipment 637, 128 (2011). +[2] O. Arnold, J. C. Bilheux, J. M. Borreguero, A. Buts, S. I. +Campbell, L. Chapon, M. Doucet, N. Draper, R. Fer- +raz Leal, M. A. Gigg, V. E. Lynch, A. Markvardsen, +D. J. Mikkelson, R. L. Mikkelson, R. Miller, K. Palmen, +P. Parker, G. Passos, T. G. Perring, P. F. Peterson, S. Ren, +M. A. Reuter, A. T. Savici, J. W. Taylor, R. J. Taylor, +R. Tolchenov, W. Zhou, and J. Zikovsky, Mantid—Data +analysis and visualization package for neutron scattering +and µSR experiments, Nuclear Instruments and Meth- +ods in Physics Research Section A: Accelerators, Spec- +trometers, Detectors and Associated Equipment 764, 156 +(2014). +[3] L. Facheris, K. Y. Povarov, S. D. Nabi, D. G. Mazzone, +J. Lass, B. Roessli, E. Ressouche, Z. Yan, S. Gvasaliya, +and A. Zheludev, Spin Density Wave versus Fractional +Magnetization Plateau in a Triangular Antiferromagnet, +Phys. Rev. Lett. 129, 087201 (2022). +[4] R. A. Ewings, A. Buts, M. D. Le, J. van Duijn, I. Bustin- +duy, and T. G. Perring, Horace: Software for the anal- +ysis of data from single crystal spectroscopy experiments +at time-of-flight neutron instruments, Nuclear Instruments +and Methods in Physics Research Section A: Accelerators, +Spectrometers, Detectors and Associated Equipment 834, +132 (2016). + diff --git a/0dFRT4oBgHgl3EQfkjfS/content/tmp_files/load_file.txt b/0dFRT4oBgHgl3EQfkjfS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c8d299b8c51337d3e81f8cec6752b06b245bdb7 --- /dev/null +++ b/0dFRT4oBgHgl3EQfkjfS/content/tmp_files/load_file.txt @@ -0,0 +1,701 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf,len=700 +page_content='Confinement of fractional excitations in a triangular lattice antiferromagnet L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Facheris,1, ∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Nabi,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Glezer Moshe,2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Nagel,2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' R˜o˜om,2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Povarov,1, 3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Stewart,4 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Yan,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Zheludev1, † 1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland 2National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia 3Present address: Dresden High Magnetic Field Laboratory (HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany 4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11 0QX, United Kingdom (Dated: February 1, 2023) High-resolution neutron and THz spectroscopies are used to study the magnetic excitation spec- trum of Cs2CoBr4, a distorted-triangular-lattice antiferromagnet with nearly XY-type anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' What was previously thought of as a broad excitation continuum [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 129, 087201 (2022)] is shown to be a series of dispersive bound states reminiscent of “Zeeman ladders” in quasi- one-dimensional Ising systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' At wave vectors where inter-chain interactions cancel at the Mean Field level, they can indeed be interpreted as bound finite-width kinks in individual chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Else- where in the Brillouin zone their true two-dimensional structure and propagation are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In conventional magnetic insulators the dynamic re- sponse is typically dominated by coherent single-particle S = 1 excitations, aka magnons or spin waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In many low-dimensional and highly frustrated quantum spin sys- tems elementary excitations carry fractional quantum numbers, be they spinons in Heisenberg spin chains [1–4] or Majorana fermions in the now-famous Kitaev model [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The physical excitation spectrum, such as that measured by neutron spectroscopy, is then dominated by broad multi-particle continua [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In addition to the continuum, fractional excitations may also form bound states due to attractive interactions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A spectacular new phenomenon emerges when interactions are confining, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' do not fall off with distance, much like strong forces that bind quarks in hadrons [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This pro- duces an entire series of bound states inside the resulting potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' An example is the sequence of domain wall (kink) bound states in quasi-one-dimensional Ising spin chains [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The confining potential for this model is linear and results from 3-dimensional couplings, which generate an effective field acting on individual chains [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The binding energies are, in supreme mathematical ele- gance, spaced according to the negative zeros of the Airy function [13, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The best-known experimental examples of such “Zee- man ladder” spectra are the quasi-one-dimensional Ising ferromagnet CoNb2O6 [16] and antiferromagnet (AF) BaCo2V2O8 [17], as well as the isostructural com- pound SrCo2V2O8 [18], where as many as 8 consecutive bound states are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Shorter sequences have been found in another prototypical Ising spin chain material, RbCoCl3 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In the present work we report the ob- servation of a somewhat similar phenomenon in an en- tirely different type of system, namely in a quasi-two- dimensional distorted-triangular-lattice AF where the effective magnetic anisotropy is predominantly of XY, rather than Ising character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' That the quintessentially one-dimensional physics of bound kinks survives in two dimensions is remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' We argue that it is “rescued” at certain special wave vectors by the intrinsic frustration in triangular lattice geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Elsewhere in the Brillouin zone the bound states are no longer restricted to sin- gle chains and are to be viewed as 2-dimensional objects propagating on the entire triangular plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The material in question, Cs2CoBr4 (space group Pnma, a = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='19, b = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='73, c = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='51 ˚A), is a very interesting J − J′ model distorted-triangular-lattice AF [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Despite a prominent triangular motif in its structure, it demonstrates certain one-dimensional fea- tures such as a field-induced incommensurate spin den- sity wave with Tomonaga-Luttinger spin liquid type dy- namics and a propagation vector controlled by a one- dimensional nesting in the spinon Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Its true 2-dimensional nature is manifest in the presence of a ro- bust m = 1/3 magnetization plateau, typical of a trian- gular AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The model magnetic Hamiltonian is described in detail in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The key structural features are chains of Co2+ ions that run along the crystallo- graphic b axis of the orthorhombic lattice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The chains are coupled in the (bc) plane in a zigzag fashion to form a distorted triangular network (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Easy-plane single-ion anisotropy en- sures that the low-energy physics of the spin-3/2 Co2+ ions can be described in terms of effective S = 1/2 pseudo-spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The components of the effective exchange coupling constants are subject to restrictions imposed by the pseudo-spin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A simplistic spin-wave analysis of previous inelastic neutron data provided a rough estimate for the nearest-neighbor in-chain AF ex- change tensor components: JXX ∼ J, JY Y ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='1J, JZZ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='25J, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='8 meV [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Here Y is chosen along the b crystallographic direction, and X and Z alternate between adjacent chains, where anisotropy planes are al- most orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Note that this is practically a planar arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='13596v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content="str-el] 31 Jan 2023 2 (b) (c) (d) (a) ground state J J' bound state m3 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (a)-(b) Neutron scattering intensity (solid sym- bols) measured at T = 40 mK versus energy transfer at the one-dimensional AF zone-centers q = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5) and q = (0, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The data are integrated fully along h direction and in ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='025 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='25 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' along k and l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Solid lines are fits to a series of Gaus- sian peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Dashed Gaussians represent the calculated ex- perimental energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Black dotted lines indicate the fitted flat background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (c) Measured terahertz absorption (solid line) versus absorbed photon energy for light propa- gating along the c axis at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Dashed areas highlight the individual components that find counterparts in the neutron spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (d) Measured excitation energy plotted versus the value of negative roots of the Airy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The solid line is a linear fit as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The blue area highlights the points used for the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Inset: cartoons of the magnetic ground state and a representative m = 3 2-kink bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' exchange anisotropy, with only a tiny in-plane Ising com- ponent to account for the ∆ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='4 meV spectral gap found in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The frustrated inter-chain cou- pling J′ is significant, of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='45J, and is of predominantly Ising (Y Y ) character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Inter-plane inter- actions J′′ are not frustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The material orders mag- netically in a colinear stripe-type structure, with an or- dering wavevector (0, 1/2, 1/2) (see inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The N´eel temperature TN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='3 K allows us to esti- mate J′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' If this were the only coupling between chains with no additional frustration due to J′, we could expect kBTN ∼ 2∆/ ln(∆/J′′) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The actual value of J′′ must be larger than thus obtained, as the in-plane frustration interferes with the emerging magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A cer- tain upper estimate is given by the mean field picture where kBTN ∼ 2J′′S(S + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This leads us to conclude that 3 · 10−4 meV ≲ J′′ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='075 meV ≪ J, confirming the quasi-2-dimensional character of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Our previous inelastic neutron scattering experiments indicated that the excitation spectrum in zero applied field is a gapped continuum of states, with intensity con- centrated on its lower bound, and a strong dispersion along the chain axis [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The central finding of the present work is that this “continuum” is actually a se- quence of at least 9 sharp bound states that previously could not be observed due to poor experimental energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' New neutron data were collected at the LET time-of-flight spectrometer at ISIS (UK), using 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='35 meV incident energy neutrons in repetition-rate-multiplication mode [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' We used the same 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='16 g single crystal as in [21] mounted on a 3He-4He dilution refrigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' All measurements were performed at a base temperature of 40 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In the experiment the sample was rotated 180◦ around the a axis in steps of 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The spectra were mea- sured for ∼ 10 minute counting time at each sample po- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' We first focus on the one-dimensional AF zone-centers (qb = 0, π), where inter-chain interactions within the tri- angular planes cancel out at the Mean Field-RPA level, and where spin wave theory predicts no transverse disper- sion or intensity modulation of excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(a),(b) show constant-q cuts through the data at wave vectors q = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5) and q = (0, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A se- quence of sharp peaks is clearly apparent in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A fit to the data using empirical Gaussian profiles yields an accurate measure of the peak positions and shows that their widths are essentially resolution-limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(a),(b) this is emphasized by the shaded Gaussians representing the computed experimental resolution [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Corroborative evidence is also obtained by THz spec- troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The experiment was performed with a Martin- Puplett-type interferometer and a 3He-4He dilution re- frigerator with base temperature of 150 mK using a 3He- cooled Si bolometer at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The sample was a circu- lar plate approximately 1 mm thick in c direction and 4 mm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' THz radiation propagating along the 3 crystal c axis was unpolarized and the apodized instru- mental resolution was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='025 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The THz absorption spectrum is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' It is calculated as a dif- ference of spectra measured at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='2 K and 2 K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' in the magnetically ordered phase and above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The THz spectrum appears to have some features absent in the neutron spectrum, but all peaks found in the latter are also present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The positions of these peaks were de- termined in Gaussian fits (shaded peaks) in a narrow range ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='025 meV near each peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The spacing between the excitation peaks present in both measurements corresponds to confinement in an ap- proximately linear one-dimensional potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' To demon- strate this, we plot the excitation energies deduced from neutron spectra at several wave vectors, as well as the positions of corresponding THz peaks, versus the neg- ative roots zi of the Airy function in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' For a precise linear attractive potential λ|x| between the dis- persive particles, near the minimum ϵ(k) = m0+ℏ2k2/2µ we expect the excitation energies to be [15, 16] mi = 2m0 + (ℏλ)2/3µ−1/3zi with i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (1) In the actual data, the linear dependence is appar- ent for all but the first few points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' As will be ad- dressed in more detail below, this slight deviation in- dicates that the confining force increases somewhat at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' From a linear fit to the higher-energy peaks we can immediately extract the slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='072(3) meV and the energy of a single particle m0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='18(1) meV (half-intercept).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Using the single-particle kinetic mass ℏ2/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='39 meV×b2 [24], we estimate the confining force constant λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='031(2) meV/b [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The next point that we make is that the observed bound states at the one-dimensional AF zone-center are essentially one-dimensional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This is concluded by analyzing the neutron spectra shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2(a),(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The bound states do not propagate in either transverse direction and thus have an essentially flat dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Moreover, their intensity shows no modulation trans- verse to the chains, as shown for the first two modes in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2(e),(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The measured transverse wave vector de- pendencies are entirely accounted for (solid lines) by the combined effects of i) the magnetic form factor of Co2+ and ii) a neutron polarization factor for spin components perpendicular to the chain axis (to the direction of or- dered moments in the ground state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This implies that these excitations do not involve cross-chain correlations and are confined to a single chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This consideration prompts a simple interpretation of the observed behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Similarly to the situation in CoNb2O6 and BaCo2V2O8, the observed modes are bound states of two kinks (domain walls) in individual chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Such an excitation is illustrated by the car- toon in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Since the ordered mo- ments are along the b crystallographic axis, they are po- (a) (b) (c) (d) (e) (f) (g) (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (a)-(d) False color plot of neutron scattering inten- sity measured at T = 40 mK plotted versus energy transfer and momentum transfer transverse to the crystallographic b- axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Gray areas mask regions of elastic-incoherent scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Background subtraction has been performed as described in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The orange regions represent energy-integration win- dows used to extract the cuts in panels below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (e)-(h) Intensity-momentum cuts (solid symbols) for the first two modes in the Zeeman ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The blue line shows the product of calculated neutron polarization factor for excitations polar- ized perpendicular to the direction of the ordered moment and the magnetic form-factor-squared for Co2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' larized transverse to that direction [21], in agreement with the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The energy m0 is to be asso- ciated with that of a single domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' As a consis- tency check, we can compare that to the computed en- ergy of a domain wall in a classical spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Using JY Y /JXX ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='1 as estimated for Cs2CoBr4, with a triv- ial numerical classical-energy minimization procedure we get m0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='9JS2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='18 meV, in excellent agreement with the measured value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Geometric frustration ensures that at the magnetic zone-center these strings of flipped spins within a single chain incur no energy cost due to interactions with adja- cent chains within the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Moreover, any transverse dispersion is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' At the same time, the interaction energy due to unfrustrated inter-layer coupling is proportional to the string length, resulting in confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In this simplistic picture, the confining force is λ = 2J′′S2/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This yields an inter-layer coupling constant J′′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='062(4) meV, inside the possible range deduced from TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The first lowest-energy bound state with energy m1 corresponds to a single spin flip in the chain, in other words to a single-magnon excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The i-th higher-energy states are two domain walls separated by a length-i string of spins that are aligned opposite to the ground state AF spin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 4 The deviation from linear-potential behavior at low en- ergies is also readily explained by this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Since the material is almost planar, the domain walls are not con- fined to a single bond as in the ideal Ising case, but have a characteristic size l [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' We can estimate that quan- tity in a classical spin chain using the above-mentioned anisotropy parameters: l ∼ 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The energy of the first few bound states is thus modified due to a physical over- lap of the two bounding domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Experimentally, the bound state energy is reduced, which corresponds to an additional attractive interaction between kinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Once the kinks are separated by a distance of more than ∼ l, this interaction becomes negligible and the confinement potential becomes linear, originating only from inter- layer interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Away from the one-dimensional AF zone-centers, the excitations are considerably more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This is very clear in the longitudinal dispersion of the bound states shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3(a),(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Other than at qb = 0, π (k = 0, 1/2) the m1 mode splits into two branches, each with an asymmetric dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In fact, the m1 state at qb = π seems to be continuously connected to the m2 excitations at qb = 2π (k = 1) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Fitting the dispersion of the strongest low-energy mode in the vicinity of qb = π to a Lorentz-invariant relativistic form (ℏωq)2 = ℏ2∆ (qb)2 /µ + ∆2, (2) yields the value of kinetic mass quoted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A look at the intensities reveals that other than at the special wave vectors, the bound states can no longer be seen as strings in a single chain, but are “dressed” with correlations extending to several neighboring chains in the triangular plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This conclusion is reached from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2(c),(d), that show a transverse cut of the spectrum at qb = 5π/4 and qb = 3π/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' As plot- ted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2(g),(h), the measured intensity of the first two modes now shows a much steeper transverse wave vector dependence than computed from just the polar- ization and form factors (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The second mode even seems to show signs of intensity oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Our data reveal that away from the special wave vec- tors the bound states also propagate in two dimensions, albeit with a small bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Indeed, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2(d) one can see that at qb = 3π/2 the bound states develop a non-zero dispersion along the c∗ direction, in contrast to what is seen at qb = 0, π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Although the bandwidth of transverse dispersion, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='08 meV, is at the limit of our experimental resolution, qualitatively one can say that qc = 0, 4π are dispersion minima for the m1 mode, while the maximum is at qc = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' That periodicity is consis- tent with having two chains per unit cell along the c-axis direction in the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Overall, the differences between our results and spectra of Ising spin chains [16, 17] are striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In the latter, all bound states, including the first one, are much less dispersive than the lower edge of the entire spectrum, (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (a)-(b) False color plot of neutron scattering inten- sity measured at T = 40 mK plotted versus energy transfer and momentum transfer along q = (0, k, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5) and q = (0, k, 1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The data were fully integrated along h, and in the range ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='25 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' along l around the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The gray areas mask regions where the incoherent scattering domi- nates the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Background subtraction has been performed as described in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' which approximately corresponds to the lower edge of the two-kink continuum in the absence of long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' As a result, each bound state persists only in a restricted area in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In contrast, in Cs2CoBr4 a few of the lower-energy bound states are highly dispersive and span across the entire zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In summary, we demonstrate that “Zeeman lad- ders” of confined fractional excitations can exist in a bona fide quasi-two-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' These states are inherently related to those in the one-dimensional model, as revealed at special wave vectors where two- dimensional interactions are canceled by geometric frus- tration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' However, elsewhere in reciprocal space their true 2-dimensional character is manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Once again, the distorted triangular lattice model provides a link be- tween one- and two-dimensional quantum magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' This work was supported by a MINT grant of the Swiss National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' We acknowledge support by the Estonian Research Council grants PRG736 and MOBJD1103, and by European Regional Development Fund Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' TK134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Experiments at the ISIS Neu- tron and Muon Source were supported by beamtime allo- cation RB2210048 from the Science and Technology Fa- cilities Council [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' ∗ lfacheri@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ch AA5 † zhelud@ethz.' 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neutron multi-disk chopper spectrometer at ISIS, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 637, 128 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [24] See Supplemental Material for detailed discussion of the resolution calculations, additional inelastic neutron scat- tering data, background subtraction procedure, and esti- mate of the kinetic mass for a kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [25] This force of ∼ 6 fN corresponds to the gravity pull be- tween two average humans at a separation of 8 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [26] We define the domain wall width in a spin-S chain as the distance over which the z-axis spin component changes from S/2 to −S/2 near its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Facheris, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (2022): Spin-density wave dy- namics in a 2D distorted triangular lattice antifer- romagnet, STFC ISIS Neutron and Muon Source, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5286/ISIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='RB2210048 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Supplemental Material for “Confinement of fractional excitations in a triangular lattice antiferromagnet” L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Facheris,1, ∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Nabi,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Glezer Moshe,2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Nagel,2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' R˜o˜om,2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Povarov,1, 3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Stewart,4 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Yan,1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Zheludev1, † 1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland 2National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia 3Present address: Dresden High Magnetic Field Laboratory (HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany 4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11 0QX, United Kingdom (Dated: January 31, 2023) This Supplemental Material provides further details supporting the main text that may be of interest to the specialized reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In particular, the resolution calculations, additional inelastic data, the background subtraction for the neutron spectroscopic measurements, and estimate for the kink’s kinetic mass are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Determination of energy resolution for the LET experiment 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Additional cuts used for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d) 1 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Background subtraction procedure for LET data 1 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Estimating a kink’s kinetic mass µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2 References 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' DETERMINATION OF ENERGY RESOLUTION FOR THE LET EXPERIMENT The neutron scattering data presented in the main text were obtained on the direct-geometry time-of-flight LET spectrometer at ISIS (UK) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The instrument was op- erated in the high-flux mode, with a chopper resolution frequency of 210 Hz and a pulse remover frequency of 140 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' A phase delay time for chopper 2 of 87000 µs was introduced to avoid contamination on the main in- coming channel Ei = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='35 meV by slower neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The resolution calculations were performed with the PyChop interface of Mantid Workbench [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The obtained resolu- tion profile is shown in SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The widths of the shaded Gaussian profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(a),(b) of the main text were calculated based on the fitted peak positions and the data in SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' ∗ lfacheri@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ch † zhelud@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='ch/ SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Calculated energy resolution (solid line) ver- sus neutron energy transfer for the spectrometer settings listed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Dotted lines mark the positions mi at q = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5) as obtained from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(a) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' ADDITIONAL CUTS USED FOR FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d) The additional cuts at q = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5, 1) and q = (0, 1, 1) (not shown in the main text) are displayed in SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The fit is performed in full analogy to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(a),(b) as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The extracted peak positions from SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2 (a),(b) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 1(d) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' BACKGROUND SUBTRACTION PROCEDURE FOR LET DATA The inelastic neutron scattering data presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3 of the main text are background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Although the dataset was rather clean, a background subtraction similar to that in [3] was nonetheless per- formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In this section the model adopted to describe the background is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The analysis was performed using the Horace software package [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3 shows raw data corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Strong sharp lines at the edges of the 2 (a) (b) SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (a)-(b) Neutron scattering intensity (solid symbols) measured at T = 40 mK versus energy transfer at q = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5, 1) and q = (0, 1, 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The data are integrated fully along h direction and in ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='025 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' and ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='25 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' along k and l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Solid lines are fits to a series of Gaussian peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Dashed Gaussians represent the calculated experimental energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Black dotted lines indicate the fitted flat background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' dataset below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='4 meV are known spurious originating from scattering from the sample environment employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The total background was modeled assuming no mag- netic scattering below the gap and above the top of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Thus, the background dataset is identical to original data for ℏω ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='34 meV and ℏω ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='28 meV (see dashed horizontal lines in SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3 for the background regions projected on these particular cuts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' In the intermediate energy region, momentum-dependent boxes were constructed as shown in SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3 and numerically interpolated over the total explored (q, ℏω)- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The so-obtained background was then point-to- point subtracted from the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' ESTIMATING A KINK’S KINETIC MASS µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Near it’s minimum at a one-dimensional wave vector k0 = π b , the dispersion relation for a single kink can be approximated as ϵk = m0 + ℏ2 2µ(k − k0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='1) The parameter µ is the kinetic “mass” of this quasiparti- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' We can access it from the experimentally measured spectrum of two-kink excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' For a two-kink state, energy-momentum conservation dictates ℏω(2−kink) q = ϵk+ϵq−k = 2m0+ ℏ2 2µ � (k − k0)2 + (q − k + k0)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='2) Minimizing (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='2) with respect to the “hidden” quasi- momentum k, we find that the lower boundary of the two-particle continuum lies at k = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Thus, the lowest magnon-like dispersion is given by: ℏωq = 2m0 + ℏ2 2µ � (q − k0)2 + k2 0 � (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='3) Near the minimum wavevector q0 = k0 → π/b, we find that the curvature of the parabola-like dispersion is ac- tually the same for a single kink and the lowest bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (a) (b) SUPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' (a)-(b) False color plot of raw neutron scat- tering intensity measured at T = 40 mK plotted versus en- ergy transfer and momentum transfer along q = (0, k, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='5) and q = (0, k, 1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The data were fully integrated along h, and in the range ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='25 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' along l around the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' The gray areas mask regions where the inco- herent scattering dominates the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Orange dashed lines and boxes delimit the edges of the background dataset, as de- scribed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Bewley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} +page_content=' Taylor, and S.' metadata={'source': 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for the anal- ysis of data from single crystal spectroscopy experiments at time-of-flight neutron instruments, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 834, 132 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFRT4oBgHgl3EQfkjfS/content/2301.13596v1.pdf'} diff --git a/1tE2T4oBgHgl3EQf5Qgr/content/tmp_files/2301.04188v1.pdf.txt b/1tE2T4oBgHgl3EQf5Qgr/content/tmp_files/2301.04188v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd3b90641f5505a6cf8ce9e41d617374c976688b --- /dev/null +++ b/1tE2T4oBgHgl3EQf5Qgr/content/tmp_files/2301.04188v1.pdf.txt @@ -0,0 +1,2627 @@ +MNRAS 000, 1–19 (2023) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Trajectory Based RFI Subtraction and Calibration +for Radio Interferometry +Chris Finlay,1,2,3,5★ Bruce A. Bassett,2,3,4,5 Martin Kunz1 and Nadeem Oozeer3,5,6 +1Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland +2Department of Pure and Applied Mathematics, University of Cape Town, South Africa +3African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa +4South African Astronomical Observatory, Cape Town, South Africa +5South African Radio Astronomy Observatory (SARAO), 2 Fir Street, Observatory, Cape Town, 7925, South Africa +6Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University, P.O. Box 94, Makhanda 6140, South Africa +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Radio interferometry calibration and Radio Frequency Interference (RFI) removal are usually done separately. Here we show +that jointly modelling the antenna gains and RFI has significant benefits when the RFI follows precise trajectories, such as for +satellites. One surprising benefit is improved calibration solutions, by leveraging the RFI signal itself. We present tabascal +(TrAjectory BAsed RFI Subtraction and CALibration), a new algorithm that jointly models the RFI signal & trajectory as well as +the calibration parameters in post-correlation visibilities allowing for curved wavefronts. tabascal can use either optimisation +or fully Bayesian statistical methods to find calibration solutions in contaminated data that would otherwise be thrown away. We +test tabascal on simulated MeerKAT calibration observations contaminated by satellite-based RFI with amplitudes varying +between -20 dB and 15 dB relative to the 1 Jy calibrator source. We obtain gain estimates that are both unbiased and up to an +order of magnitude better constrained compared to the case of no RFI. tabascal can be further applied to an adjacent target +observation: using 5 minutes of calibration data results in a target image with about half the noise achieved when using purely +flagged data, and only 23% higher than a completely uncontaminated observation. The source detection threshold and recovered +flux distribution of tabascal-processed data was on par with uncontaminated data. In contrast, the standard method of RFI +flagging alone resulted in a higher detection threshold (2×) and led to consistent underestimation of source fluxes. For a mean +RFI amplitude of 17 Jy, using tabascal leads to less than 1% loss of data compared to ∼ 75% data loss from an ideal 3𝜎 +flagging algorithm, a very significant increase in data available for science analysis. Although we have examined the case of +satellite RFI, tabascal should work for any RFI moving on parameterizable trajectories, relative to the phase centre, such as +planes or objects fixed to the ground. +Key words: Bayesian Methods – Radio Frequency Interference – Radio Interferometry – Calibration +1 INTRODUCTION +A major problem plaguing radio astronomy observatories across the +world is the problem of Radio Frequency Interference (RFI). In the +context of radio astronomy, RFI is generally any unwanted radio +signal that can result from both man-made and natural sources. The +increasing sensitivity of radio telescopes coupled with more RFI +sources has led to an exponentially growing number of detected +sources of RFI. Several signal processing methods exist and are used +to handle RFI, however, in practice there is no universal fool-proof +technique for RFI mitigation. For reviews on RFI mitigation see +Kesteven (2010); Briggs & Kocz (2005); Fridman & Baan (2001); +Ekers & Bell (2000). +RFI flagging, the process of identifying data points contaminated +by RFI, is the most commonly used post-processing technique for +RFI mitigation in use across all observatories. Though there has +★ E-mail: christopher.finlay@unige.ch +been significant progress in applying advances in machine learning +to RFI flagging (Vafaei Sadr et al. 2020; Sun et al. 2022; Mesarcik +et al. 2022), these techniques come at the expense of data loss. In +this paper we instead explore statistical methods to filter out the RFI +since it is reasonable to expect it to produce advantages similar to that +which occurred in analysis of the Cosmic Microwave Background, +see e.g. Aghanim et al. (2020). In particular, Bayesian methods show +significant promise for radio astronomy, e.g. Lochner et al. (2015); +Arras et al. (2021). If successful this means losing less useful data. +To do so we exploit the key defining property of RFI: namely that +it is not stationary in the reference frame of the celestial sky. RFI is +always moving relative to the sky, either due to the earth’s rotation +or artificial orbits (planes and satellites). +Methods for the subtraction of RFI signal have been investigated +in the past with limited success. Perley & Cornwell (2003) proposed +a method for subtracting RFI visibility contributions. In Cornwell +et al. (2004) the proposed method was tested on carefully chosen real +data and managed to reduce the RFI contribution by up to a factor +© 2023 The Authors +arXiv:2301.04188v1 [astro-ph.IM] 10 Jan 2023 + +2 +Chris Finlay et al. +Figure 1. The fraction of time that is flagged by the MeerKAT RFI flagger +in the L-band. The blue and orange curves show the fraction of time flagged +for baselines shorter than 1 km and longer than 1 km respectively. The most +heavily affected sub-bands have a static mask on the shorter baselines, that +is why they are flagged 100 % of the time. We can see that the majority of +the RFI present in this band is from satellite-based RFI of which the Global +Navigation Satellite System (GNSS) are the main culprit. Source: SARAO +of 1000 in specific channels of a ground-based RFI source. This +method therefore showed real promise and appears to have resulted +in a task named UVRFI (Kogan & Owen 2010) that is available in +the Astronomical Image Processing System (AIPS) software (Wells +1985). The UVRFI task has two sub tasks: (1) CIRC, an extension of +RfiX (Athreya 2009), that is closely related to the original method, is +used for ground-based sources and fits a linearly varying amplitude +given the implied fringe-frequency, and (2) CEXP, that applies CLEAN +(Högbom 1974) in the Fourier domain of the visibility time series, +that can be used for any RFI source as long as its fringe frequency is +suitably different from that of the celestial signal. +Our proposed method, tabascal, has some minor similarities +with the CIRC method from UVRFI. However, it is more general as it +includes the ability to work on both stationary and moving sources +of RFI, given that the RFI source moves on a fixed trajectory we can +parameterize. This is the case for all stationary sources as well as most +moving sources such as planes and satellites. The key differences are +(1) the full decomposition of 𝑁(𝑁 − 1)/2 baseline signals into 𝑁 +antenna-based signals, (2) the modelling of time-smearing effects +on the signal, and (3) the joint estimation of antenna gains and +RFI parameters. In this paper we apply tabascal to simulations +of MeerKAT (Jonas & Team 2016) observations contaminated by +satellite-based RFI. +We have chosen this situation as a testbed as MeerKAT is one of the +most sensitive radio telescopes in the world, in its operational bands, +and is the precursor for the Square Kilometre Array (SKA) Mid tele- +scope. Additionally, its L-band (900 - 1670 MHz) is already severely +affected by satellite-based RFI, not to mention the increasing number +of satellites yet to be deployed, such as the SpaceX Starlink (10.7 - +12.7 GHz) and Amazon Kuiper constellations, which will affect radio +telescopes in a number of frequency bands. The MeerKAT L-band +data currently suffers around 36.6 % data loss due to RFI flagging +(Sihlangu et al. 2021) of which the majority is caused by satellite- +based sources. Furthermore, satellites are a particularly interesting +source of RFI from the perspective of RFI subtraction due to their +predictability. The positions of satellites can be predicted based on +previous measurements, such as from two-line element sets (TLEs), +and for many, the spectral and temporal signal profiles are known a +priori (Harper & Dickinson 2018). The identified satellite constella- +tions that currently affect MeerKAT L-band data are summarised in +Table 1. +Constellation +Frequency +Orbit +Orbit +Bands (MHz) +Elevation +Inclination +GPS +L1: 1565 - 1585 +L2: 1217 - 1237 +L3: 1375 - 1387 +L5: 1166 - 1186 +20,180 km +55◦ +GLONASS +L1: 1592 - 1610 +L2: 1242 - 1249 +L3: 1202.025 +19,140 km +64.8◦ +Galileo +E1: 1575.42 +E5a: 1176.45 +E5b: 1207.14 +E5 AltBOC: 1191.795 +E6: 1278.75 +23,222 km +56◦ +Inmarsat +1526 - 1554 +35,785 km +0◦ +Iridium +1616 - 1626 +780 km +86.4◦ +Table 1. A summary of the characteristics of the satellite-based RFI that +affects the MeerKAT L-band. All of the active satellites in these constellations +have eccentricities of less than 1 %. Sources: GLONASS ESA, GLONASS +Novatel, GPS ESA, Galileo ESA, SARAO RFI Report. Iridium ESA +This paper is organized as follows: Section 2 describes the data +simulations, the probabilistic inverse model and the methods used to +recover parameters of interest. In Section 3 we discuss the recovered +posterior parameter distributions and how these results are used to +improve science performance in a target observation. Finally, in Sec- +tion 4 we summarize the problem and our findings as well as further +directions for this work. +2 METHODS AND SIMULATIONS +This section is organised as follows: in Sections 2.1 & 2.2 we discuss +the principle of radio interferometry and how we model the response +of the telescope to the sky brightness distribution. In Sections 2.3 & +2.4 we discuss modifications to the standard model necessary for most +types of RFI and the specific implementation of our MeerKAT data +simulations. Sections 2.5, 2.6 & 2.7 start with a brief introduction to +Bayesian concepts used in this paper, and then go on to describe our +likelihood term and forward model along with the associated priors. +We finish off with Sections 2.8 & 2.9 where we describe the methods +used to recover the posterior distributions of our model parameters by +means of a Laplace approximation and Markov chain Monte Carlo. +We begin by describing the problem of transfer calibration, also +referred to as 1st Generation Calibration (1GC). 1GC is the first +step in the calibration of an interferometer that provides a starting +point for further calibration i.e. 2GC/selfcal. In 1GC observations +of a calibrator source, i.e. a source of known position and flux, is +used to estimate the antenna gains which can then be used do an +initial calibration on a following target observation where the sky +distribution is not known a priori. A suitable calibrator source is +a bright, unresolved (smaller then the synthesized beam) source, is +isolated in the sky (relative to the primary beam width) and should be +close to the target (within 10◦ on the sky) (Mauch et al. 2020). The +brighter the calibrator source, the greater the signal-to-noise ratio +(SNR) achieved leading to better constrained antenna gain estimates +for a given observation time. The source should be isolated in the sky +so that the apparent sky model can be estimated as only consisting +of the calibrator source. Finally, the requirement of the calibrator +MNRAS 000, 1–19 (2023) + +Fraction of time flagged forbaselines <1 km and >1km for 4hr track (Xx pol) +1.0 +ged +flagge +0.8 +Aircraft transponders +GSM down +GLONASS L2 +Fraction of time +0.6 +GSM up +Galileo 2 +Inmarsat +GPS L5 +Galileo +3 +GPS L1 +GLONASS +Iridium +GPS L +GPS +0.4 +0.2 +bl <1 km +MMUM +bl >1 km +0.0 +900 +1000 +1100 +1200 +1300 +1400 +1500 +1600 +1700 +Frequency (MHz)tabascal +3 +lying within 10◦ of the target field comes from the angular scale +of ionospheric fluctuations which affect the gain phases. A list of +suitable calibrators is available from the SARAO External Service +Desk1. +2.1 Radio Interferometry +A radio interferometer measures the sky brightness distribution +(spectral radiance) by taking samples in visibility space. The do- +main of visibility space is denoted by the coordinates (𝑢, 𝑣, 𝑤). To +sample a point in visibility space, the signal from two antennas with +some spatial separation, measured in wavelengths, is correlated. The +separation vector, given by the components (𝑢, 𝑣, 𝑤)𝑝𝑞, referred to +as a baseline, points from antenna 𝑝 to antenna 𝑞. This is done for +all antenna pair combinations in an interferometric array at multi- +ple time steps leading to samples of many locations in the visibility +space. One is then able to infer the sky brightness distribution from +the visibility samples using the formalism described in this section. +For an ideal fringe-stopping interferometer, the visibility distribu- +tion is defined by Equation (1). +𝑉(𝑢, 𝑣, 𝑤) = +∬ +𝑙𝑚 +𝐼(𝑙, 𝑚) exp +� +−2𝜋𝑖�𝑢𝑙 +𝑣𝑚+𝑤(𝑛−1)�� 𝑑𝑙𝑑𝑚 +𝑛 +(1) +In Equation (1) 𝑖 is the imaginary unit and 𝐼(𝑙, 𝑚) is the brightness +distribution on the sky. The sky coordinates, (𝑙, 𝑚, 𝑛), are unitless +direction cosines lying in the range [−1, 1]. Since the domain of +the sky brightness distribution, 𝐼, lies on a sphere of fixed arbitrary +radius, the celestial sphere, the third direction cosine, 𝑛, is fixed, +i.e. 𝑛 = +√ +1 − 𝑙2 − 𝑚2. The origin of the sky coordinates, (0, 0, 1), +is called the phase centre and is fixed to a location on the celestial +sphere. This is the defining property of a fringe-stopping interferom- +eter. When considering the field of view (FoV) of the telescope to +be very small, i.e. the sky signal is dominated by an area of the sky +where 𝑙2 + 𝑚2 ≪ 1, Equation (1) reduces to the van Cittert-Zernike +(vCZ) theorem. In this case, 𝑛 ≈ 1, producing an exact 2D Fourier +relation between the visibilities and the sky brightness distribution +(Thompson et al. 2017). +2.2 Telescope Response +The telescope response, Γ𝑝𝑞, to measuring some sky brightness +distribution includes the modulation of the primary beam of each +of the antennas 𝑝 and 𝑞 and their respective bandpass filters. When +the primary beam intensity pattern of antenna 𝑝 is |𝐸𝑝(𝑙, 𝑚, 𝜈, 𝑡)|2 +and its bandpass is |𝐺 𝑝(𝜈, 𝑡)|, the telescope response is as given in +Equation (2). This is very similar to the form given in Thompson +et al. (2017). +Γ𝑝𝑞 = 1 +Δ𝑡 +∬ +𝑡𝑗±Δ𝑡/2 +𝜈0±Δ𝜈/2 +𝐺 𝑝𝐺∗ +𝑞 +∬ +𝑙𝑚 +𝐸𝑝𝐸∗ +𝑞𝐼 exp [...] 𝑑𝑙𝑑𝑚 +𝑛 +𝑑𝑡𝑑𝜈 +(2) +In Equation (2), ∗ indicates the complex conjugate, Δ𝑡 ≫ Δ𝜈−1 +is the integration time in the correlator for a single sample at time +𝑡 𝑗 and Δ𝜈 is the bandwidth of a single frequency channel centred +on 𝜈0. We have emitted the arguments of 𝐺, 𝐸 & 𝐼 but in principle +all of these depend on 𝜈 and 𝑡 and are generally assumed constant +1 https://skaafrica.atlassian.net/wiki/spaces/ESDKB/overview +over the intervals Δ𝜈 and Δ𝑡. Only 𝐸 and 𝐼 are functions of 𝑙 and 𝑚. +The expression inside the exponential denoted by [...] is the same +as in Equation (1). Once the integrals are calculated we are left with +Equation (3) +Γ𝑝𝑞(𝑢, 𝑣, 𝑤) = |𝐺 𝑝||𝐺𝑞|𝑒𝑖(𝜑𝑝−𝜑𝑞)Δ𝜈 +√︃ +𝐴𝑝 𝐴𝑞𝑉𝑝𝑞(𝑢, 𝑣, 𝑤) +(3) +where 𝑉𝑝𝑞 is the true visibility at the point (𝑢, 𝑣, 𝑤)𝑝𝑞, 𝐴𝑝 is the +collecting area of the dish on antenna 𝑝 in the direction of the phase +centre, |𝐺 𝑝| is the magnitude of the bandpass/gain at antenna 𝑝 and +𝜑𝑝 − 𝜑𝑞 is the phase difference of the gains between antennas 𝑝 and +𝑞. +In Equation (3) we have assumed that the gains are constant over +the integration time and the channel bandwidth. This is a standard +assumption as telescopes are designed to have such stability. Addi- +tionally, due to the rotation of the Earth and frequency dependence of +the (𝑢, 𝑣, 𝑤) coordinates, the visibility phases (from the exponential +term in Equation (1)) vary over time and frequency. This variation +cause the phases to change slightly over the integration window lead- +ing to decorrelation of the signal. This results in a reduction in the +amplitude of the visibilities. This effect is known as time/frequency +smearing. The strength of this effect increases with baseline length. +Therefore, integration windows and frequency channels are chosen +to be small enough to minimize this effect for astronomical sources. +Typically, the integration windows are still too large for signals from +RFI sources to correlate fully. This is treated as a feature to reduce +the level of contamination. The telescope response Γ𝑝𝑞 is in units +of Watts when the sky brightness distribution, 𝐼(𝑙, 𝑚), is in units of +W.m−2.Hz−1. +When modelling the visibilities we discretize the sky brightness +distribution as a collection of point sources which are summed over, +as in Equation (4). +˜𝑉𝑝𝑞 = 𝐺 𝑝 +�∑︁ +𝑠 +𝐸𝑝𝑠𝐾∗ +𝑝𝑠𝐵𝑠𝐾∗ +𝑞𝑠𝐸𝑞𝑠 +� +𝐺∗ +𝑞 +(4) +where the measured visibility, ˜𝑉𝑝𝑞 = Γ𝑝𝑞, the 𝐸 terms are nor- +malized as 𝐸𝑝 → 𝐸𝑝/√︁𝐴𝑝 and the 𝐺 𝑝 → 𝐺 𝑝/ +√ +Δ𝜈 making them +dimensionless in the equation. This way we have both the point +source brightnesses, 𝐵𝑠, and the measured visibilities, ˜𝑉𝑝𝑞, in the +same units, Jansky (Jy). Those familiar with the Radio Interferom- +etry Measurement Equation (RIME) (Smirnov 2011) will recognize +Equation (4), however, here we use it in a scalar form and do not +consider polarization. In Equation (4) we use the subscript 𝑠 to label +a point source at position (𝑙𝑠, 𝑚𝑠, 𝑛𝑠) in the sky. +In Equation (4) the 𝐾 terms, as defined in Equation (5), +𝐾𝑝𝑠 = exp �−2𝜋𝑖 �𝑢𝑝𝑙𝑠 + 𝑣 𝑝𝑚𝑠 + 𝑤 𝑝(𝑛𝑠 − 1)�� +(5) +are the geometric delay between an antenna and an arbitrary refer- +ence position. Their combination 𝐾𝑝𝑠𝐾∗𝑞𝑠 produce the exponential +term in Equation (1) for a specific location 𝑠. The 𝐵𝑠 term is the spec- +tral flux density of the point source 𝑠 in units of Jy. Extended sources +are represented by a discretized/pixelized version where each pixel is +treated as a point source. The 𝐸 terms are the direction-dependent ef- +fects (DDEs), previously used for the primary beam in Equation (2). +The 𝐺 terms are the direction-independent effects (DIEs), previously +used for the gains in Equation (2). +In Equation (5) the (𝑙, 𝑚, 𝑛)𝑠 are the coordinates of the point source +𝑠 and (𝑢, 𝑣, 𝑤)𝑝 are the coordinates of antenna 𝑝 relative to a global +reference position, in units of wavelength. +MNRAS 000, 1–19 (2023) + +4 +Chris Finlay et al. +2.3 Modifications for RFI +A number of features make signals from RFI sources distinct from +astronomical sources. RFI sources are much closer to us than astro- +nomical sources and move relative to the celestial sphere on which +astronomical sources, outside our galaxy, remain stationary. We will +start by discussing the implications of receiving signal from a source +closer than expected. +Spherical waves, originating from a source, resemble plane waves +when the source is very far away from the receiver, relative to the +receiver dimensions and wavelength of the emission. This is known +as the far-field regime and is defined in Equation (6): +𝑑𝐹 = 2𝐷2 +𝜆 , +given +𝑑𝐹 ≫ 𝐷 +& +𝑑𝐹 ≫ 𝜆 +(6) +For a single MeerKAT receptor observing in the L-band (𝜆 ≈ 21 +cm), 𝑑𝐹 is just over 2 km which is also much larger than the dish +diameter, 𝐷, of ≈ 14 m. Therefore, the far-field primary beam models +used for astronomical sources are also suitable for use with most +sources of non-local RFI, especially, satellite-based sources. +The geometric delay term, given by Equation (5), that is used in +the RIME, assumes sources are in the far-field of the antenna array, +not just a single receiver. For the MeerKAT telescope, observing in +the L-band, with the longest baseline of ≈ 8 km, 𝑑𝐹 is ≈ 6.4 × 105 +km which is nearly 2 times the distance from the Earth to the Moon. +Therefore, practically all sources of man-made RFI are in the near- +field of the telescope array. For such sources, Equation (5) must +therefore be modified to Equation (7): +𝐾𝑝𝑠(𝑡) = exp +� +− 2𝜋𝑖 +� |�𝑟𝑠(𝑡) − �𝑟 𝑝(𝑡)| +𝜆 +− 𝑤 𝑝(𝑡) +�� +(7) +where we have assumed spherical wave fronts as can be expected +from a point source in the idealised case. In Equation (7), 𝜆 is the +observation wavelength, �𝑟𝑠 is the position of the RFI source, �𝑟 𝑝 is +the antenna position, and 𝑤 𝑝 is the phase tracking delay correction +as used in Equation (5). When combined as a 𝐾𝑝𝐾∗𝑞 term, Equations +(5) & (7) describe the same thing, the path length difference between +antennas 𝑝 and 𝑞, in wavelengths, including the phase tracking cor- +rection. +For a far-field source, relative to the array size, the angle, relative +to the reference direction, at which the signal enters the primary +beam is the same across all antennas. However, this is not the case +for near-field sources. In this case, the angular separation between the +pointing direction and the RFI source is different for each antenna. +This leads to a different angular evaluation of the 𝐸 term, per antenna, +for a given RFI source. The functional form of the 𝐸 term remains +the same though in many cases. +Another consideration for near-field sources is the the spectral flux +density received at the antenna. The spectral flux density, 𝐼RFI +𝑝𝑠 , of an +RFI source 𝑠 at antenna 𝑝 is given in Equation (8). +𝐼RFI +𝑝𝑠 (𝑡) = +𝑃RFI +𝜈 +(𝑡) +4𝜋|�𝑟𝑠(𝑡) − �𝑟 𝑝|2 +(8) +This equation is derived from the free-space path loss formula +assuming spherical wave fronts from an isotropic emitter. In reality, +an RFI source will not be an isotropic emitter however this would +only change the 𝑃RFI +𝜈 +(𝑡) term potentially making it antenna depen- +dent. This is not a problem for our method due to the per antenna +parameterization used in our forward model, refer to Equation (21). +In Equation (8), 𝑃RFI +𝜈 +is the spectral flux of the RFI source in +W.Hz−1 which can be divided by 1026 to make the spectral flux +density in units of Jy, assuming the distance is in metres. Since this +is different for each antenna we adapt the 𝐵𝑠 term in Equation (4) to +a 𝐵𝑝𝑞𝑠 term as defined in Equation (9). +𝐵RFI +𝑝𝑞𝑠 = +√︃ +𝐼RFI +𝑝𝑠 𝐼RFI +𝑞𝑠 +(9) +Finally, to address the moving nature of RFI sources relative to +celestial sources, we need to explicitly perform the integration for +each visibility sample. Typically for radio interferometry simulations, +one can evaluate all terms in the model only once per visibility sample +as everything is assumed constant on the time-scale of the visibility +sample integration time. This assumption breaks down for moving +RFI sources, so we must evaluate all terms related to the positional +nature of the RFI at a finer time resolution and then average them +to the cadence of the final visibility samples. The required time +resolution to achieve an accurate result depends on the speed and +direction of movement of the source. +2.4 Data Generating Model +In this section we describe the data generation model. We split this +section into the telescope, DIE, DDE, astronomical and RFI source +models. Together these components fully define our model used to +simulate a calibration observation from the MeerKAT telescope with +basic, but, realistic telescope response, signal corruptions and RFI +contamination. While we choose to simulate MeerKAT observations +for illustrative purposes everything in our algorithm applies to other +radio interferometry observatories. +Table 2 summarizes the parameter values and distributions used. +Throughout our simulations we work with only a single frequency +channel centred at 1.227 GHz and calculate observed visibilities +using Equation (4). Each time sample, 𝑡 𝑗, is an average over a given +integration time, Δ𝑡 = 2s, where the visibility function is sampled +16 times per second. These integration samples are equally spaced +in the range (𝑡 𝑗 − Δ/2𝑡, 𝑡 𝑗 + Δ𝑡/2) where 𝑡 𝑗 is the observation time +centroid of the time sample and Δ𝑡 is the integration time. +2.4.1 Telescope Model +We simulate data for the MeerKAT telescope positioned at a lat- +itude, longitude and elevation of (−30.721◦, 21.411◦, 1054.71m). +We use the East, North, Up (ENU) coordinates obtained from the +South African Radio Astronomy Observatory (SARAO) to simulate +(𝑢, 𝑣, 𝑤) coordinates. We use standard coordinate transformations, as +defined in Thompson et al. 2017, Chapter 4.1, to transform from ENU +coordinates to International Terrestrial Reference Frame (ITRF) co- +ordinates and then to (𝑢, 𝑣, 𝑤) coordinates. +2.4.2 Astronomical Source Model +We simulate both a calibration observation and a target observation. +In the calibration portion we observe the same calibrator source with +known spectral flux density and position. For the calibrator source +we use a 1 Jy point source situated at (𝛼, 𝛿) = (21◦, 10◦). We include +no other astronomical sources in the calibration portion and keep the +source flux density and position fixed in our MCMC sampling. In +the target portion, we observe a 100 point source field, centred on +(𝛼, 𝛿) = (27◦, 15◦), where the positions are uniformly sampled from +a disk with 0.5◦ radius and the spectral flux densities sampled from an +exponential distribution with mean 0.1 Jy. The sources positions are +MNRAS 000, 1–19 (2023) + +tabascal +5 +chosen such that no two sources are closer than ≈ 8 synthesized beam +widths (80"). The simulated visibilities for the calibration portion will +be denoted by 𝑉CAL, and for the target track we will use 𝑉AST. +2.4.3 RFI Source Model +Our satellite-based RFI source is modelled with a spectral flux, 𝑃RFI +𝜈 +, +that is constant in time. This is done for simplicity and is not a +requirement for the functioning of our method as we allow for a +time variable RFI amplitude. We use the near-field expression for the +spectral flux density at a specific antenna 𝑝 as defined in Equation +(8). This leads to the baseline dependent spectral flux density 𝐵𝑝𝑞𝑠 +as defined in Equation (9). +The position of the satellite-based RFI source is modelled using +a circular orbit about the Earth. One could use a more sophisticated +model at the expense of introducing more parameters. Simplified per- +turbation models, such as SGP4/SDP4 (Vallado & Crawford 2008) +would be the preferred model to use, however, since we are currently +testing on simulated data with satellites with very low eccentricity we +decided on a simpler model. For use on real data a more sophisticated +model many very well be needed. For a circular orbit we have four +parameters, namely, the orbit elevation (ℎ), argument of perigee (𝛾), +orbit inclination (𝛽), and the right ascension of the ascending node +( ˜𝛼). The formula for circular motion on an arbitrary plane about the +Earth’s centre of mass, �𝑟Earth CoM as a function of time is given in +Equation (10) (Fitzpatrick 2012). +�𝑟𝑅𝐹𝐼 (𝑡) = 𝑹𝑧( ˜𝛼)𝑹𝑥(𝛽)𝑹𝑧(𝛾) �� +� +(𝑅𝑒 + ℎ) cos(𝜔𝑡) +(𝑅𝑒 + ℎ) sin(𝜔𝑡) +0 +�� +� ++ �𝑟Earth CoM(𝑡) +(10) +In Equation (10) above 𝑹𝑥(𝛽) is a 3D rotation matrix about the axis +𝑥-axis through an angle 𝛽, ℎ is the orbit elevation above the Earth’s +surface in metres, 𝛽 and ˜𝛼 define the orbital plane, 𝛾 is the angular +offset of the orbit and finally 𝜔 = +√︁ +𝐺0𝑀𝑒/(𝑅𝑒 + ℎ)3. Here we have +assumed the satellite’s mass to be negligible compared to the mass of +the Earth. In the equation for 𝜔, 𝑀𝑒 and 𝑅𝑒 are the mass and average +radius of the Earth respectively and 𝐺0 is the gravitational constant. +The angular orbit parameters align with orbital elements used in two- +line element sets (TLE). The orbit elevation, for a circular orbit, is +directly related to the mean motion orbital element, in revolutions +per day, by (86400𝜔/2𝜋. The rotation matrix axes are with respect +to the ITRF frame used for the antennas where +𝑧 points from the +Earth’s centre to the North Pole and +𝑥 points to the intersection of +the Equator and Greenwich meridian. +The time averaging described in the beginning of Section 2.4 is a +fundamental requirement in modelling RFI visibility contributions. +This is because the visibility phases induced by an RFI source are +rapidly varying in time due to their fast movement relative to the +sky reference frame. The resulting effect is called time-smearing and +is especially prominent for moving sources. The magnitude of this +effect increases with the length of the baseline and therefore affects +longer baselines more than shorter baselines, assuming the same +orientation. +2.4.4 Direction Independent Effects (DIE) +We include time-varying complex gains for each antenna. Both the +gain amplitudes and phases are modelled as linear time variates, as +shown in Equations (11). +|𝐺|(𝑡) = |𝐺|(0) + �|𝐺|𝑡 +(11a) +𝜑𝐺(𝑡) = 𝜑(0) +𝐺 + �𝜑𝐺𝑡 +(11b) +𝐺(𝑡) = |𝐺|(𝑡) exp +� +𝑖𝜑𝐺(𝑡) +� +(11c) +The initial values and rates of change are sampled from the distri- +butions described in Table 2 and is done separately for each antenna. +|𝐺|(𝑡) is the gain amplitude and 𝜑𝐺(𝑡) is the gain phase. In Equa- +tions (11), when the parameter has a (0) superscript, it is the initial +value, at 𝑡 = 0, and the overdot is used for the rate of change of the +parameter. The gain phases include the ionospheric effects, a DDE +component, but the spacial scale of variation is assumed to be so large +(> 10◦) that it can be considered a DIE. We therefore only consider +our RFI source to be within this angular distance from the pointing +centre for this approximation to be valid. To extend our method to +such a situation we could explicitly include the ionospheric effects +in the DDE term. This would further require our forward model to +be correspondingly adapted. +2.4.5 Direction Dependent Effects (DDE) +We use the normalised Fourier transform of a circular aperture, the +square of which is the normalized Airy disk, as the primary beam +voltage model. The primary beam is the only DDE that we include. +We keep the primary beam constant in time and the same across all +antennas. The functional form of the primary beam term is given in +Equation (12). +𝐸(𝜃, 𝜈) = 2𝑐0𝐽1 +�𝜋𝑑𝜈 sin 𝜃/𝑐0 +� +𝜋𝑑𝜈 sin 𝜃 +(12) +In Equation (12) 𝐽1 is the Bessel function of the first kind of order +one, 𝜈 is the observation frequency in Hz, sin 𝜃 = +√ +𝑙2 + 𝑚2 where +𝜃 is the angular separation between our pointing direction and the +source and 𝑐0 is the speed of light in a vacuum. +We only consider a real-valued primary beam voltage model and +leave complex voltage patterns and other DDEs for further study. The +inclusion of complex DDEs, such as a complex voltage pattern and +ionospheric effects, create a degeneracy in the forward model that +would need to be broken by the inclusion of appropriate priors in the +probabilistic model. +2.4.6 Noise Model +We add circularly-symmetric complex normally distributed noise to +the visibilities as defined in Thompson et al. (2017). Each baseline is +an independent measurement with independent noise. The standard +deviation, 𝜎𝑛, of the noise is the same for each baseline. There- +fore, our modelled visibility data are independent and identically +distributed (i.i.d.). The noisy data is generated using Equation (13), +where 𝜂𝑝𝑞 is the noise term. +ˆ𝑉𝑝𝑞 = 𝐺 𝑝 +� ∑︁ +𝑠 +𝐸𝑝𝑠𝐾∗ +𝑝𝑠𝐵𝑠𝐾∗ +𝑞𝑠𝐸𝑞𝑠 +� +𝐺∗ +𝑞 + 𝜂𝑝𝑞 +(13) +In Jonas & Team (2016), measurements show that the System +Equivalent Flux Density (SEFD) of a single MeerKAT receptor is +approximately 420 Jy at 1.227 GHz. Using Equation (14), this implies +a per visibility noise level, 𝜎𝑛, of about 0.65 Jy using a 2 s integration +time, Δ𝑡, and 209 kHz bandwidth, Δ𝜈. We therefore model the noise +MNRAS 000, 1–19 (2023) + +6 +Chris Finlay et al. +Parameter Description +Symbol +Units +Value/Distribution +Dish Diameter +𝑑 +m +13.965 +Observation Frequency +𝜈0 +GHz +1.227 +Channel Bandwidth +Δ𝜈 +kHz +209 +Sampling Rate +- +Hz +16 +Integration Time +Δ𝑡 +s +2.0 +Noise Amplitude +𝜎𝑛 +Jy +0.65 +Calibrator Spectral +Flux Density +𝑆CAL +𝜈 +Jy +1.0 +Calibrator Position +(𝛼, 𝛿) +(deg,deg) +(21.0, 10.0) +RFI Spectral Flux +𝑃RFI +𝜈 +𝜇W.Hz−6 +5.8 +Orbit Elevation +ℎ +km +20,200 +Argument of Perigee +𝛾 +deg +5.0 +Orbit Inclination +𝛽 +deg +55.0 +Right Ascension of +the Ascending Node +˜𝛼 +deg +21.0 +Initial Gain Amplitude +|𝐺|(0) +- +N(1.0, 0.052) +Gain Amplitude Drift +� +|𝐺| +10−5.s−1 +N(0.0, 1.02) +Initial Gain Phase +𝜑(0) +𝐺 +deg +U[−𝜋/2, 𝜋/2] +Gain Phase Drift +� +𝜑𝐺 +10−3deg.s−1 +N(0.0, 1.02) +Noise distribution +𝜂𝑝𝑞 +Jy +CN(0.0, 𝜎2𝑛) +Earth Radius +𝑅𝑒 +km +6, 371 +Earth Mass +𝑀𝑒 +kg +5.9722 × 1024 +Gravitational Constant +𝐺0 +N.m2.kg−2 +6.67408 × 10−11 +Speed of Light +𝑐0 +m.s−1 +2.99792458 × 108 +Table 2. Table of parameter values and distributions for the data generating +model. Values have been chosen to, at best, mirror what we have found from +various sources including the MeerKAT Specifications web page. +as CN (0, 0.652) which is equivalent to N +� +0, 0.652/2 +� +in both the +real and imaginary parts of the visibility independently. +𝜎𝑛 = SEFD +√ +Δ𝜈Δ𝑡 +(14) +2.4.7 Summary of Parameter Values +We chose the values in Table 2 to represent worse or equivalent +performance to the real world. We found values for these parameters +from a number of sources. On the MeerKAT Specifications web +page, gain amplitude stability was found to be < 3% over 3 hours +resulting in ≈ 3 × 10−6s−1 we use 10−5s−1. On the RAGAVI (Andati +et al. 2022) package web page we found gain amplitudes across +antennas to be within 5% where we have used this value as our +standard deviation. On the same web page we found the gain phases +between antennas to lie within a 40◦ band about 0◦ and we have used +180◦. The gain phase stability was estimated form the MeerKAT +examples on the RAGAVI(Andati et al. 2022) web page to be less +than 10◦ over 2 hours resulting in 1.4 × 10−3 deg.s−1. We have used +×10−3 deg.s−1 as the standard deviation of the gain phase drift. The +MeerKAT dish diameter is taken from Jonas & Team (2016). The +observation frequency is chosen to be in the middle of the MeerKAT +L-band corrupted by most GNSS signals as is shown in Figure 1. +The channel bandwidth is taken from standard L-band 4k mode for +MeerKAT. The visibility sampling rate is chosen to be as fast as +possible while being computationally viable on a laptop with 16GB +of RAM. The integration time was chosen to be 2 seconds which +is one of the options provided by SARAO. The noise is calculated +from the estimated SEFD as shown in Section 2.4.6. The calibrator +flux density was chosen to be on the weaker end of the L-band +calibrators that SARAO provides on their MeerKAT Service Desk +web page. The calibrator position was chosen for convenience in +finding a suitable satellite orbit passing within 10◦. Finally, the RFI +orbit parameters are chosen to align with what is publicly available +from example TLEs with the argument of perigee and right ascension +of the ascending node tuned so that the satellites pass within 10◦ of +both the calibrator and target fields. +2.5 Bayesian Inference +Bayesian inference is a paradigm of statistical inference which uses +Bayes’ theorem, Equation (15), +P(Θ|D, M) = L(Θ|D, M)Π(Θ, M) +𝑍(D, M) +(15) +to update our knowledge about some parameters/hypothesis given +new information. The goal of Bayesian inference is to acquire the +posterior probability distribution, P(Θ|D, M), of our model pa- +rameters, Θ, given some data, D, and the model, M. The poste- +rior distribution is comprised of three components, the likelihood, +L(Θ|D, M), the prior, Π(Θ, M), and the evidence 𝑍(D, M). The +likelihood is the probability of seeing the data given the param- +eters in our model. The prior distribution encodes any prior in- +formation we have about the parameters of our model. The prior +is defined by the user and can include information about the +parameters from data not included in the likelihood as well as +heuristics or physical limitations of the parameters. The evidence, +𝑍(D, M) = +∫ +L(Θ|D, M)Π(Θ, M)𝑑Θ, is a normalizing factor but +is also used in model selection problems when deciding between two +models, M1 and M2, by looking at their ratio. An example would +be choosing between a model that contains one satellite compared to +two satellites. +In Section 2.7 we carefully construct a prior that guides our model +parameters, Θ, toward desirable solutions and provides suitable ini- +tial conditions to reliably find maximum a posteriori (MAP) points +through optimization. +An important concept when dealing with a multivariate probability +distribution is marginalization. We may consider a subset of our +model parameters to be so called nuisance parameters. If we integrate +the probability distribution over the nuisance parameters we are left +with a marginal distribution over our parameters of interest. The +marginal distribution in this case, is a distribution over our parameters +of interest taking into consideration all possible values of the nuisance +parameters simultaneously. Letting Θ = (Θ𝐼 , Θ𝑁 ), where Θ𝐼 are our +parameters of interest and Θ𝑁 are our nuisance parameters, we obtain +our marginalized posterior over Θ𝐼 by Equation (16). +P(Θ𝐼 |D, M) = +∫ +P(Θ𝐼 , Θ𝑁 |D, M) 𝑑Θ𝑁 +(16) +2.6 Likelihood +The likelihood is determined by a combination of our noise model, +described in Section 2.4.6, our forward model and the observed data. +Since visibilities have additive noise that is independent and normally +distributed in both the real and imaginary parts then our likelihood is +the product of the individual likelihood terms for each data point. We +further assume that the noise is identically distributed in each data +MNRAS 000, 1–19 (2023) + +tabascal +7 +point. The expression for the total likelihood is given in Equation +(17). +L(Θ|𝑉𝑜𝑏𝑠) = (𝜋𝜎2 +𝑛)−𝑁D exp +� +− +𝑁D +∑︁ +𝑗 +��𝑉OBS +𝑗 +− ˜𝑉𝑗 (Θ) +��2/𝜎2 +𝑛 +� +(17) +Here we use Θ to denote the column vector of model parameters, +𝑉OBS for the observed visibilities, ˜𝑉 for the noiseless model visibil- +ities, 𝑗 to index each data point, and 𝜎𝑛 for the standard deviation +of the additive complex noise. Our total number of complex-valued +data points is 𝑁D = 𝑁𝑡 𝑁𝑎(𝑁𝑎 − 1)/2, 𝑁𝑡 is the number of time +steps and 𝑁𝑎 is the number of antennas in the telescope array. For +our problem, we only have one frequency channel and polarization. +Each likelihood term is a circularly-symmetric complex normal +distribution. Its functional form differs slightly from a (real-valued) +normal distribution in that factors of 2 are missing. Since the real +and imaginary components are independent with identical variance +(circularly symmetric) the variance of the complex visibility sample +is twice that of the individual real or imaginary component. For- +mulating the likelihood in terms of real and imaginary components +individually would lead to the factors of 2 returning to the functional +form. We therefore consider one complex visibility as one data point +as opposed to two, as would be the case for separating the real and +imaginary components. Both formulations are equivalent when using +the appropriate noise variance, 𝜎2𝑛. +2.7 Probabilistic Model +In Section 2.6 we have described the likelihood. We will now discuss +all of the parameters of the forward model and the priors we set +on them. Figure 2 shows a Bayesian factor graph (model diagram) +that summarizes the entire probabilistic model. This problem is fully +constrained and does permit the usage of a purely likelihood based +approach or the use of wide, uniform priors. We make use of semi- +informative priors based on real-world assumptions that can be made. +This improves the consistency and stability of convergence in finding +a solution both for optimization and MCMC by regularizing the +problem. +2.7.1 Gains +Our gain parameters are composed of amplitudes and phases per +antenna, per time step. We have a parameter for the gain amplitude on +each antenna at each time step. We have the same for the gain phases +except we exclude phases for the last antenna using it as a reference +antenna. We set these to 0 in the data generation portion as well as in +the forward model. This must be done as our observed visibilities are +composed only of differences in phases. A transformation in all gain +phases of 𝜑′𝑝 = 𝜑𝑝 +𝜑0, where 𝜑𝑝 is the gain phase on antenna 𝑝 and +𝜑0 is a constant, would leave the measurements unchanged. When +𝑁𝑎 is the number of antennas and 𝑁𝑡 is the number of time steps, +we have 𝑁𝑡 (2𝑁𝑎 − 1) real-valued parameters to fully describe the +complex gains. By using a complex gain parameter for each time step +we can model gain variations on shorter time scales than expected +thereby not assuming any specific gain variation beyond stability over +the individual time step integration time. One could assume the gains +to be stable/constant over a 10 second data portion for example and +reduce the number of gain parameters by a factor of 10, assuming a +2 second integration time as we use in this paper. +The priors we have on the gain parameters assume reasonable +estimates have been made on uncontaminated nearby channels such +that the bandpass can be interpolated and provide an estimate in our +contaminated channel. We assume this estimate to be correct with a +10% and 10◦ standard deviation in the gain amplitudes and phases +respectively. We therefore sample from a normal distribution centred +on the true value with a 10% and 10◦ standard deviation for each +antenna and use this sample as the mean of our prior distribution for +all time steps. The prior standard deviation is set to 10% of this value +for the gain amplitudes and 10◦ for the gain phases. The prior on +each parameter is independent and does not include any correlations. +2.7.2 Satellites +To model our satellite-based RFI we have parameters that govern its, +per antenna, signal amplitude and parameters that control its orbit +around the Earth. For the satellite orbit, we have four parameters +as described in Section 2.4.3. These four parameters describe a cir- +cular orbit and are a subset of the six orbital elements needed for +a general orbit in the two-body problem. TLEs(Vallado & Cefola +2012) expand on these parameters to account for atmospheric drag +and the gravitational pull of the moon etc. Space Track provides a +standard catalog of satellites and their TLEs at constantly updated +measurement epochs. In Flohrer et al. (2008), over 11k objects from +the TLE catalogue are analysed to find their positional uncertainties +over a 48 hour period centred on the TLE epoch. These objects have +been categorized according to certain orbit characteristics and the +standard deviations in their orbit determination have been summa- +rized per class. The standard deviations are quoted in radial, in-track +and cross-track (RIC) directions. The RIC coordinates of an orbit, +(�𝑟RIC), are defined with respect to a reference orbit (�𝑟0). The orthog- +onal RIC coordinate unit vectors are defined in Equation (18) and +form the rows of the transformation matrix from an Earth-centred +reference frame, as is used in this paper, to the RIC frame. +ˆ𝑅 = �𝑟0/|�𝑟0| +(18a) +ˆ𝐼 = ˆ𝐶 × ˆ𝑅 +(18b) +ˆ𝐶 = ( ˆ𝑅 × ��𝑟0)/|��𝑟0| +(18c) +�𝑟RIC = +������� +ˆ𝑅 +ˆ𝐼 +ˆ𝐶 +������� +(�𝑟 − �𝑟0) +(19) +Unfortunately TLEs do not include covariance estimates on their +parameters so we make use of those provided in the RIC frame +by Flohrer et al. (2008). To transform the covariance of an orbit +in the RIC frame back to the orbit parameters we make use of the +standard error propagation formula assuming normal errors with a +minor deviation. Let ΣRIC be the covariance of a given orbit in the +RIC frame. ΣRIC is a 3x3 matrix and is diagonal for all work in this +paper. We wish to transform this into ΣΦ, a 4x4 matrix, which is the +prior covariance of our RFI orbit parameters. Firstly, we define our +reference orbit �𝑟0(𝑡) using the true orbit parameters, given by the +TLE in a real-world example. Let �𝑟RIC(𝑡 𝑗) = 𝑇𝑗 (Φ; �𝑟0(𝑡 𝑗)) such that +𝑇 is the function that accepts, Φ, the vector of orbit parameters and +produces 3D positions over time in the RIC frame, given in Equation +(19). +We evaluate this at each of the time steps in the calibration data +portion. Given each time step has a covariance given by ΣRIC, as- +sumed to be the same at all time steps, we take the average of the +transformed precision matrices. The precision matrix is the inverse of +MNRAS 000, 1–19 (2023) + +8 +Chris Finlay et al. +Figure 2. A Bayesian factor graph of the probabilistic model used to estimate uncalibrated and RFI contaminated visibilities. The constants are shown as +diamonds. The free parameters of the model are shown as rectangles with rounded corners. Their distributions are shown in the top left corner with the +parameterization of the distribution given in the smaller rounded rectangles in the top right corner. Repeated parameters that are indexed are placed in a +rectangle, with sharp corners, known as a plate with the index repetition indicated at the bottom centre of the rectangle. The rectangles in the top half of the +diagram form the prior over the parameters and the rectangle in the lower left is the likelihood term. The equation in the lower right of the diagram is the +mathematical model for the observed data. +the covariance matrix. This leads to a ΣΦ that when transformed back +to the RIC frame at best reproduces the original ΣRIC but usually has +worse constraints in the ˆ𝐼 and ˆ𝐶 directions by a factor of 1.12 and 2.16 +respectively. We must do this as using only one time step would not +produce a suitable constraint in a higher dimensional space. Equation +(20) shows the formula to generate our prior covariance on the orbit +parameters using the method just described. +�ΣΦ +� +𝑝𝑞 = �� +� +1 +𝑛 +𝑛 +∑︁ +𝑗 +𝜕𝑇𝑗 +𝜕Φ𝑝 +� +Σ−1 +RIC +� +𝑗 +𝜕𝑇𝑗 +𝜕Φ𝑞 +�� +� +−1 +(20) +In Figure 3 we show the prior distribution used for the RFI orbit +parameters used in the 0-20s data portion. The covariance provided +in Flohrer et al. (2008) for a medium Earth orbit, as is the case for +GNSS satellites, is ΣRIC = +� +diag(73, 131, 54) m +�2 +. We chose the +prior standard deviations to be 10 times larger than this. We did +this to show that our method can handle larger errors in our a prior +knowledge than what is publicly available. +The signal amplitude of the RFI has a parameter per antenna, per +time step. We do this as we are modelling the RFI signal ampli- +tude modulated by the primary beam of each antenna, as defined in +Equation (21). +𝐴RFI +𝑝 +(𝑡) = 𝐸𝑝 +�𝜃(𝑡)�√︃ +𝐼RFI +𝑝 +(𝑡) +(21) +This is a more general approach as compared to assuming some- +thing about the primary beam of the antenna/s and/or the intrinsic +RFI signal. Modelling each of these separately would be degenerate +as we can only constrain the product of the beam and the RFI sig- +nal. If we were to assume the primary beam is the same across all +antennas we could parameterize the primary beam using a Zernike +polynomial based model as in Asad et al. (2021). +For our prior on the modulated RFI amplitudes we choose an +uninformative prior, Equation (22), with a very wide range. +𝐴RFI +𝑝 +(𝑡 𝑗) ∼ N (0, Σ𝐴) +(22) +We chose each parameter prior to be a normal distribution with mean +0 +√︁ +Jy and covariance, Σ𝐴, to be fully independent with diagonal +values of 10 000 Jy. Each parameter prior is fully independent. It is +possible to place a correlated prior on these parameters that leads to +better posterior constraints on the gain amplitude parameters at the +expense of introducing slight biases in the results as this correlation +assumptions is not always true. +A very important aspect of our RFI model is to model the time- +smearing that occurs due to the rapidly varying phases induced by the +fast moving RFI. We, therefore, evaluate the position of the satellite +at multiple time points centred about each time step in our data. +Additionally, we perform a linear interpolation, and extrapolation +at the edges of the data portion, for our RFI amplitudes. This is +needed due to the rapidly varying modulated amplitude caused by, +at minimum, the movement of the RFI source through the primary +beam. Once the amplitudes have been re-sampled at the higher rate, +equal to the position sampling, RFI visibilities are predicted at this +MNRAS 000, 1–19 (2023) + +tabascal +9 + +2 +0 +2 + [arcsec] [1e4] +4000 +2000 +0 +2000 +4000 + [arcsec] +1500 +0 +1500 +h [m] +1500 +0 +1500 + [arcsec] +2 +0 +2 + [arcsec] [1e4] +4000 +2000 +0 +2000 +4000 + [arcsec] +1500 +0 +1500 + [arcsec] +Figure 3. Corner plot showing the prior distribution of the RFI orbit pa- +rameters. The strong parameter correlations should be noted as these are +crucial in choosing initial parameter values for both the optimization and +MCMC routines. The correlations are induced in the error propagation from +the co-moving frame to the orbit model parameters. The correlations change +depending on the time of the observation. These are for the 0-10 second +calibration data portion. The dark and light blue shaded regions show the +68% and 95% prior credible regions. The true value and distribution has been +shifted to 0 to make the contour levels more readable. +finer rate and then averaged back down to the cadence of the observed +visibilities. +2.8 Optimization and Laplace Approximation +We developed our forward and probabilistic models using the +JAXpython library (Frostig et al. 2018; Bradbury et al. 2018) so that +we could make use of just-in-time (JIT) compilation and automatic +differentiation. This allowed us to speed up our computation dramat- +ically and get exact derivatives for our own optimization routine. In +this section we will give a brief explanation of the Laplace approx- +imation and the custom optimization routine that we developed for +this work. +For a quick approximation of the posterior distribution we can +make use of the Laplace approximation (Tierney & Kadane 1986). +The Laplace approximation approximates the posterior distribution +as a multivariate normal distribution centred on the maximum a pos- +teriori (MAP) point, ΘMAP. Since a normal distribution is defined by +its first and second moments we must find these. We therefore make +use of an optimization scheme to find the MAP position (equiva- +lent to the mean, first moment, of a normal distribution). At this +point the covariance of a multivariate normal, its second moment, is +equivalent to the negative inverse of the Hessian of the log posterior. +Symbolically this is described in Equation (23) where 𝑝(Θ|D) is the +posterior distribution density function: +𝑝(Θ|D) ≃ N (Θ; ΘMAP, ΣMAP) , +Σ−1 +MAP = −𝐻 (ln 𝑝 (Θ|D)) +(23) +In Equation (23) we make use of the Hessian, 𝐻, which is the +matrix of second order partial derivatives of a scalar valued function, +𝑓 , as defined in (24) where 𝑗 and 𝑘 are the row and column indices +of the matrix and Θ𝑗 is the 𝑗th parameter in the vector of parameters +Θ. +𝐻 𝑗𝑘 ( 𝑓 ) = +𝜕2 𝑓 +𝜕Θ 𝑗𝜕Θ𝑘 +(24) +Therefore, given the MAP position by running an optimization +scheme using the negative log posterior (NLP) as the minimization +surface and then evaluating the Hessian of this surface at the MAP we +can obtain the Laplace approximation to our posterior distribution. +The Laplace approximation is exact when the posterior distribution +is exactly Normal. For all but the simplest problems this is not the +case, however, in the limit of infinite data with finite variance the +posterior distribution tends toward a normal distribution thanks to +the central limit theorem. Next we describe the optimization routine +used to find the MAP point. In Section 3.2, we show that this is in +fact an excellent approximation of our posterior by comparing the +Laplace approximation to the full posterior obtained using a Markov +Chain Monte Carlo (MCMC) approach. +There are many optimization routines publicly available, however, +many of these did not perform particularly well on our problem out +of the box. Given that we are able to evaluate the Hessian exactly we +developed our own quasi-Newton method. Our method evaluates the +Hessian periodically to save on computation. The update step for a +general quasi-Newton scheme is +Θ𝑘+1 = Θ𝑘 − 𝜖𝐵−1∇ 𝑓 (Θ𝑘) , +(25) +where 𝑓 is the function to minimize, 𝐵 is the Hessian approximation, +Θ𝑘 is the parameter vector at step 𝑘, and 𝜖 is the step size. Since +inverting the Hessian becomes very expensive in a high dimensional +parameter space, and will not always produce a suitable 𝐵−1 in +problematic sections of the parameters space, we block diagonalize +the Hessian before inverting it. This allows us to invert smaller sub +matrices, the blocks, and then recompose them to make a full 𝐵−1. +We do this for the gain amplitudes, phases, RFI amplitudes, RFI +orbit parameters blocks separately. This reduces the computational +expense but more importantly leads to a more robust optimizer while +still efficiently navigating the parameter space. When we reach a +part of the parameter space that is better behaved, we use the full +Hessian and then invert it using an eigendecomposition and applying +the softabs (Betancourt 2013) function to the eigenvalues before +taking their reciprocals. This ensures that we have a 𝐵−1 that is +positive semi-definite. This is usually when 𝜒2 +dof < 1.1. The final +stages of optimization using the full Hessian inverse allow us to +more efficiently converge on the MAP position. +We calculate 𝐵−1 every 250 update steps and find that 500 steps +using the block diagonal version is enough, after which less than 250 +steps are typically needed using the full inverse. We use a decaying +step size 𝜖 that is reduced by an order of magnitude when using the +full inverse in the final optimization steps. Using this scheme we were +able to achieve excellent convergence with 𝜒2 +dof ≈ 1.01 in less than 1 +minute per 10s data portion using a laptop with 16GB of RAM. The +covariance estimation using the Hessian takes around 10 seconds per +data portion of ≈ 1000 parameters. +Optimization routines are typically very sensitive to the initial +parameter values and ours is no exception. We sample 10k points +from a region centred about the true parameter values, for the gains +and RFI orbit parameters, with standard deviations one quarter of +those used for their respective prior distributions. This should not +be a problem when applied to real data as we expect to know these +parameter values to this accuracy or better, we just use especially +MNRAS 000, 1–19 (2023) + +10 +Chris Finlay et al. +wide priors in our analysis. The initial RFI amplitude parameters +are calculated at each time step using the data and the initial gain +parameters, and are the same across antennas. We evaluate the NLP +for each of the 10k parameter sets and start from the best position and +run the optimizer till convergence. Convergence is assumed when, +𝜒2 +dof < 1.05, the improvement in the NLP is less than a set threshold, +and the Hessian at that location is positive definite. If the optimizer +does not converge according this criteria the next best initial position +is used to run a new optimization round. +2.9 MCMC Implementation +In this section we describe the Markov Chain Monte Carlo (MCMC) +implementation we have used in our analysis. We explain the ad- +vantages and highlight some of the specifics of how we obtained +our results that are given in Section 3.2 where we also analyze its +accuracy and convergence for our problem. In this paper we have +made use of Hamiltonian/Hybrid Monte Carlo (HMC) for sampling +the posterior. We designed our own HMC implementation using the +JAXpython library (Frostig et al. 2018; Bradbury et al. 2018) so that +we could customize it as we saw fit as well as make use of just-in-time +(JIT) compilation and automatic differentiation. +The benefit of HMC over a Metropolis-Hastings algorithm using +random walk is that successive proposals in HMC are distant from +one another, significantly reducing their correlation and leading to +higher effective sample sizes for a given MCMC chain length. HMC +is a Monte Carlo method where sample proposals are generated by +treating the parameters as position coordinates of a particle and the +negative log posterior (NLP) as a potential energy function, 𝑈(�𝑥). +New proposals are generated by sampling associated momenta from +a predefined distribution where its negative log represents the kinetic +energy of the particle. A proposal is then formed by evolving the +particle’s position using Hamiltonian dynamics. Equations (26) show +Hamilton’s equations +𝑑�𝑥 +𝑑𝑡 = 𝜕H +𝜕 �𝑝 +(26a) +𝑑 �𝑝 +𝑑𝑡 = − 𝜕H +𝜕�𝑥 +(26b) +where �𝑥 is the position in parameter space, �𝑝 is the momentum +and H is called the Hamiltonian which is defined as the sum of the +potential energy and kinetic energy, Equation (27). +H = 𝑈(�𝑥) + 1 +2 �𝑝𝑀−1 �𝑝 +(27) +The momentum is defined as �𝑝 = 𝑀 · 𝑑�𝑥/𝑑𝑡 where 𝑀 is the +mass matrix. The samples generated from HMC have position and +momenta. We can then marginalize over the momentum variables +leaving our position variables which are our parameters samples. +For all but the simplest of posterior distributions Hamilton’s equa- +tions must be numerically integrated to evolve the position and mo- +mentum variables. A symplectic integrator with time reversibility is +needed for this (Neal et al. 2011). Since our Hamiltonian is separable +we have used the leapfrog integration scheme. By using a numerical +integration scheme an error is introduced that is dependent on the +integration step size. Due to this a Metropolis-Hastings acceptance +test must be introduced. The acceptance probability, 𝛼, of a proposal +is given by Equation (28): +𝛼 = min +� +1, +exp(H 𝑓 ) +exp(H𝑖) +� +, +(28) +where H𝑖 and H 𝑓 are the initial and final values of the Hamilto- +nian in a single proposal evolution. In high dimensions the optimal +acceptance rate for HMC is ≈ 65% (Beskos et al. 2013). +For our particular HMC implementation we have used the standard +kinetic energy function with a mass matrix including off-diagonal +terms. This leads to sampling momenta from a multivariate normal +distribution. We define the mass matrix to be independent of posi- +tion (Euclidean HMC) leading to a simpler implementation that still +allows us to take parameter correlations into consideration. In such +a formulation, parameter correlations that change over the parameter +space cannot be taken into account. As such, the behaviour of the +posterior can significantly affect the efficiency of the sampler. Gener- +ally, when the information content of the data is high, the parameter +space close to the maximum a posteriori (MAP) point is approxi- +mately Gaussian and sampling with HMC is very efficient. Finding +this portion of the parameter space, close to the MAP, can often be +the toughest part of the problem. +More sophisticated HMC routines like Riemannian Manifold +Hamiltonian Monte Carlo (RMHMC, Girolami & Calderhead 2011), +where the mass matrix is a function of position, exist and allow effi- +cient sampling of highly non-Gaussian posteriors. Fortunately, such a +routine is not needed for our problem as our posteriors are very well +approximated by multivariate normal distributions near the maxi- +mum a posteriori (MAP) point, as is shown in Section 3.2. +2.9.1 Initial Conditions +By initial conditions we mean the initial position and mass matrix. +The determination of these is crucial for our problem. When using +unsuitable initial positions the HMC struggles to find the typical +set2 reliably. Additionally, even with a suitable initial position, the +auto-correlation times of many parameters would be unacceptable for +real-world usage/implementation. The solution to this second part is +tuning the mass matrix to include information about the parameters +scales and correlations. Ideally (in the Euclidean HMC formulation), +the mass matrix is chosen to be as close as possible to the posterior +inverse covariance matrix. +The most difficult set of parameters to tune the initial conditions +for are the satellite orbit parameters. Very small changes in these +parameters lead to large differences in the likelihood. Additionally, +these parameters are very strongly correlated leading to very ineffi- +cient sampling when the chosen mass matrix does not incorporate +the correct correlations. +Our initial sampling location is chosen by sampling the gain am- +plitudes, phases and RFI orbit parameters from the prior. The initial +modulated RFI amplitude parameters are estimated by using the +data, the other initial parameters, and the calibrator visibilities and +performing a rough calculation to get RFI amplitudes per time step. +We use this estimate, at each time step, and use the same across all +antennas. The resulting estimate is always positive which is not a +problem as our likelihood cannot tell the difference between positive +and negative RFI amplitudes. +Now that we have a suitable initial position we are left with choos- +ing a suitable mass matrix. We use the block diagonalized inverse +Hessian as described in Section 2.8 for the mass matrix. Using this +mass matrix we can evolve the HMC sampler until it has found the +typical set. To help this process we appropriately vary the integration +step size of the HMC sampler. Once the typical set has been found +2 The typical set of a distribution is the volume of parameter space in which +nearly all of the probability is located. +MNRAS 000, 1–19 (2023) + +tabascal +11 +we can re-evaluate the Hessian at our MAP sample point achieved +thus far. We continue to vary the integration step size until an optimal +value is reached (leading to an acceptance rate of ≈ 65%). Once this +criteria is achieved everything is in place to start efficiently sampling +from the posterior. All samples prior to this point are regarded as +burn-in3. Throughout the sampling the number of integration steps +are kept fixed. Once the burn-in period is complete the integration +step size and the mass matrix are also fixed. This is done to preserve +detailed balance from this point on ensuring that our samples con- +verge to the posterior distribution. We do this for multiple chains in +parallel with different random seeds to attain robust results. +3 RESULTS +The goal of this section is to explore, in detail, the performance of +our methods, showing that the results are as expected. To this end, +the section is split into three subsections. In the first subsection, we +analyze the bias and standard deviations in our parameters of inter- +est. We also compare results derived from MCMC and the Laplace +approximation. In the second subsection, we show the reliability and +accuracy of our MCMC posteriors by calculating the uncertainty in +the posterior standard deviations as well as the degree of conver- +gence in our chains. In the final subsection, we use the estimated +gains and RFI orbit parameters to estimate and subtract the RFI con- +tribution in a succeeding target observation. Using this method, we +show the potential improvements in data retention, after flagging, +and the subsequent reduction in image noise. We also demonstrate +how this results in deeper source extraction with more accurate flux +estimates per source. +Throughout this section we will often reference the bias in a pa- +rameter, both relative and absolute 4. For clarity, the bias refers to the +difference between the estimated value, ˆ𝑥, and the true value, 𝑥true, +i.e. ˆ𝑥 − 𝑥true and the relative bias is then ( ˆ𝑥 − 𝑥true)/𝑥true. Letting +ˆ𝜎𝑥 be the estimated posterior standard deviation/uncertainty in pa- +rameter 𝑥, we formulate the normalised bias as ( ˆ𝑥 − 𝑥true)/ˆ𝜎𝑥. For a +normally distributed, unbiased, estimator with reliable uncertainties +we expect the normalised bias to conform to a standard normal distri- +bution. When referring to the standard deviation on a gain amplitude +it will always be quoted in % as an uncertainty relative to the true +parameter value. +3.1 Posterior Results for Calibration +In this section we analyze the posterior distributions and discuss +how to use their estimates. We show that the estimation biases are +consistent with zero (i.e. are within the uncertainty estimates), and +that the standard deviations reduce with increasing signal-to-noise +ratio (SNR), where the signal is the total observed signal (i.e. the +contaminated visibility). +Figure 4 shows a summary of the data and gain estimates over +a 5 minute calibration observation for which the simulation details +are given in Section 2.4. The estimates in this figure are obtained by +using the Laplace approximation on 10 second portions of the data +in parallel. Each portion uses the same RFI parameter priors. The +3 Burn-in samples are initial samples in an MCMC chain that are discarded as +they would skew our posterior sample estimate. In our case the initial samples +do not conform to detailed balance due to the variation of the integration step +size and mass matrix. +4 What we refer to as bias in this text is often referred to as error in other +texts. +0 +1 +2 +3 +4 +5 +0 +10 +20 +30 +40 +Visibility Amplitude [Jy] +RFI +Astronomical +0 +1 +2 +3 +4 +5 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +Gain Amplitude +Ant. 28 +Ant. 34 +0 +1 +2 +3 +4 +5 +Time [min] +40 +50 +60 +70 +Gain Phase [deg] +Ant. 28 +Ant. 34 +Figure 4. Estimated calibration solutions over time: note how the uncertainties +on the gain amplitude and phase are minimised when the RFI is strong. +The top panel shows the visibility amplitudes over time in the calibration +observation for the RFI and astronomical contributions. The red and blue +dots are the average amplitude across all baselines and the light blue shaded +region shows the amplitude range across all baselines. The strong amplitude +variation of the RFI visibilities is due to the satellite passing through the +primary beam sidelobes. The middle and bottom panels show the Laplace +estimated gain amplitudes and phases respectively. The light blue/orange +shaded error regions in these panels give the 2𝜎 credible intervals from the +posterior distributions. The specific example antennas shown are chosen to +create a more legible plot. The vertical interleaved grey and white shaded +strips show the 10s data portions used in each optimization run. +gain amplitude and phase priors have the same standard deviations +(relative and absolute respectively) across all data portions. The gain +prior means are offset by the same, relative and absolute, quantities +per antenna in all portions. Each data portion used an individually +calculated RFI visibility sampling rate, 𝑓RFI, to optimize run times +and memory usage according to the required accuracy. Equation (29) +shows the calculation used to determine the minimum sampling rate +assuming the average gain amplitudes are 1 and the integration time +is 2 seconds per time step. +𝑓RFI = +� +� +� +mean𝑝𝑞 +� +max𝑡 +���𝑉RFI +𝑝𝑞 (𝑡) +�� +�� +3𝜎𝑛 +Hz +(29) +𝑉RFI +𝑝𝑞 (𝑡) is the RFI visibility on baseline 𝑝𝑞 at time 𝑡 and 𝜎𝑛 is the +standard deviation of the visibility noise. +Due to both the minimum RFI sampling rate and inversion of an +𝑁𝑝 × 𝑁𝑝 matrix, it is more efficient to perform estimation on por- +MNRAS 000, 1–19 (2023) + +12 +Chris Finlay et al. +1 +3 +10 +30 +Mean Observed Visibility Amplitude [Jy] +0.1 +0.3 +1 +3 +Relative Standard Deviation [%] +0.1 +0.3 +1 +3 +Standard Deviation [deg] +Gain Amplitude +Gain Phase +Figure 5. How total signal, including RFI, improves calibration constraints: +using a common gain parameter for both the RFI and astronomical contribu- +tion (since they are within 10◦ on the sky) allows us to leverage the total signal +to improve calibration constraints. The posterior standard deviations in gain +parameters are plotted against the mean (over baseline) observed visibility +amplitude. The dots show the mean standard deviation across antennas and the +shaded region of the corresponding colour show the minimum and maximum +range (in uncertainty) across antennas. The parameter standard deviations are +per time step with a 2 second integration time where the calibrator flux is 1 +Jy and noise level is 0.65 Jy. The prior standard deviations were 10% and 10◦ +for the amplitudes and phases respectively. +tions of data from a memory and computation time stand point. This +parallelization scheme poses many benefits including increasing ro- +bustness to failures in optimization. Failed optimizations can be rerun +with initial positions informed by the successful runs. Additionally, +gain solutions can be estimated on the fly instead of waiting for all +the data to be available. +A notable observation from Figure 4 is that the biases and standard +deviations for the gain estimates decrease for increasing RFI visibility +amplitude. This strongly suggests that the model is able to leverage +the added signal from the RFI to increase the SNR. Figure 5 shows +this relationship clearly. We obtain tighter constraints on the gain +parameters with increasing observed visibility amplitude. This shows +that we are using the total signal to calibrate the antennas. The prior +standard deviations, 5% and 5◦, in the gains are both larger than +the posterior uncertainties, at all signal levels. This shows that our +constraints are not strongly affected by the priors in the weak-RFI +regime. In Figure 5, an increasing spread in posterior uncertainties +for the gain phases can be observed as the mean visibility amplitudes +increase. This is due to the spread in SNR for different antennas. +The antennas in the core of the array contribute mostly to shorter +baselines that have a larger RFI signal compared to longer baselines +due to time-smearing of the signal. +Figure 6 shows that the biases and associated standard devia- +tions, on the gain parameters, are statistically consistent. However, +the gain phases are systematically underestimated. This is shown by +the shifted mean in the normalised bias distribution over all gain +phase estimates. In Section 3.2 the quality of the Laplace approxima- +tion is analyzed and we find the accuracy to be sufficient for practical +purposes. +Typically, the gain estimates from the calibration observation +would be averaged per antenna under the assumption they are con- +stant. Section A2 describes how to combine our posterior estimates +from different data portions as these are independent. The combined +estimate takes into consideration the correlations between our pa- +rameter estimates to maintain reliable uncertainties. Once appropri- +ate data portions are combined, the different time steps within a data +4 +3 +2 +1 +0 +1 +2 +3 +4 +Normalised Bias : (x +xtrue)/ +x +10 +3 +10 +2 +10 +1 +100 +Probability Density +Gain Amplitude +Gain Phase +Standard Normal +Figure 6. Histograms showing tabascal gain estimates are unbiased and +have reliable uncertainties. Distribution in the normalised bias of the estimated +gain amplitudes and phases. These are for the 5 minute calibration observation +displayed in Figure 4. For an unbiased estimation of the parameters we expect +the normalised bias to follow a standard normal distribution, which it does +to good accuracy. The means and standard deviations of the normalised bias +distribution for the gain amplitudes are 0.01 ± 0.97 and the gain phases are +−0.06 ± 0.99. +portion can be combined by following the recipe outlined in Section +A3. The second step requires the extraction of the subcovariance +matrix of the parameter estimates associated with a single antenna. +After this procedure is complete one is left with per antenna gain +estimates using the full calibration observation. We have not shown +such combined estimates as our gains are varying over time at a rate +that would violate the assumption of constant gains. +For our case, the preferred procedure, in our opinion, is to fit +a Gaussian process to the estimates and those of the next calibra- +tion observation simultaneously. The covariances of each data por- +tion would be used as the noise parameter in the Gaussian process. +The resulting gain estimates, with covariances, for the sandwiched +target observation can be used as an informative prior in the 2nd +Generation Calibration5 (2GC) process. Figure 7 shows an exam- +ple Gaussian process for reference where the marginalized posterior +subcovariance and mean has been used to fit gain amplitudes in a +single (5 min.) calibration portion. We find that fitting a Gaussian +process (combining estimates) reduces uncertainties to around 0.5% +and 0.05◦ for the gain amplitudes and phases respectively and the fit- +ted model expects uncertainties to increase by an order of magnitude +20 minutes later. +Figure 8 shows a set of estimates for the RFI orbit parameters from +three different data portions of the 5 minute calibration observation, +as indicated in the upper right of the figure. We see that the individual +estimates are of varying quality that depend on the SNR of the +data used. We also see in a subset of the marginals that include the +inclination parameter that the correlations across portions leading to +greater constraining power when combined. Figure 9 shows the final +posterior estimate after combining the individual posterior estimates +according to the equations in Section A2. Table 3 gives a summary +of the marginal standard deviations for the priors, 10s posteriors, and +5 min. posterior estimates for the orbit parameters. +5 2nd Generation Calibration is another term for selfcal. +MNRAS 000, 1–19 (2023) + +tabascal +13 +0 +1 +2 +3 +4 +5 +Observation Time [min] +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +Gain Amplitude + Mean function +True Gain +Gain Estimates +Figure 7. An example of a Gaussian process that has been fitted to the gain +amplitude estimates of antenna 22 using the full posterior covariance. The +blue dots with error bars are the posterior estimates as shown in Figure 4. +The orange curve and shaded region is the fitted Gaussian process with its +68% credible interval. The black line is the true gain phase used in the data +generation. The root mean squared error (RMSE) for the Gaussian process +is 0.46%, in the calibration portion, aligning well with its mean uncertainty +of 0.45%. 20 minutes after the calibration observation the uncertainty and +RMSE grow to around 2%. + +t = 70-80 s +t = 150-160 s +t = 250-260 s +300 +150 +0 +150 + [arcsec] +40 +20 +0 +20 + [arcsec] +1500 +0 +1500 +h [m] +10 +0 +10 +20 +30 + [arcsec] +300 +150 +0 +150 + [arcsec] +40 +20 +0 +20 + [arcsec] +10 +0 +10 +20 +30 + [arcsec] +Figure 8. Posterior marginal distributions of the RFI orbit parameters from +three different 10s data portions showing the variation in constraints and +parameter correlations. Note the constraints on the angular parameters have +improved by 2 orders of magnitude compared to the prior in Figure 3 and +improve a further 2 orders of magnitudes when combined to form Figure +9. The times for the data portions are shown in the top right corner. The +distributions and true value have been shifted to make the true parameter +values 0. This is done to make uncertainties more legible on the axes. +3.2 Laplace Approximation vs MCMC Analysis +In this section we compare the Laplace approximation with our +MCMC results. We use the MCMC results as the true posterior for +comparison. Later in this section, we give evidence for this claim by +analyzing the MCMC chains and show their reliability as a posterior +benchmark. +Parameter Name +Prior +10s Posterior +5 min. Posterior +Orbit Elevation (m) +730.000 +700.000 +31.464 +Argument of Perigee (arcsec) +10106.391 +81.932 +0.414 +Orbit Inclination (arcsec) +1349.979 +11.076 +0.062 +Right Ascension of +the Ascending Node (arcsec) +774.384 +6.589 +0.068 +Table 3. The mean marginal standard deviations for the priors and posteriors +in each 10s data portion, as well as the final posterior marginal errors. The final +posterior is the combination of all data portions in the 5 minute calibration +observation. + +0.8 +0.0 +0.8 + [arcsec] +0.1 +0.0 +0.1 +0.2 + [arcsec] +150 +100 +50 +0 +50 +h [m] +0.30 +0.15 +0.00 + [arcsec] +0.8 +0.0 +0.8 + [arcsec] +0.1 +0.0 +0.1 +0.2 + [arcsec] +0.30 +0.15 +0.00 + [arcsec] +Figure 9. The combined posterior distribution of the RFI orbit parameters +from 5 minutes of calibration data. Note the improvement in constraints on the +angular parameters compared to the prior in Figure 3 (4 orders of magnitude) +and the posteriors from 10s of data in figure 8 (2 orders of magnitude). The +potential estimation bias can be considered a statistical fluctuation as this +disappears in when changing any aspect of the simulations. The distributions +and true value have been shifted to make the true parameter values 0. This is +done to make uncertainties more legible on the axes. +Figures 10 & 11 show the bias and posterior standard deviations +on the gain parameter estimates from the Laplace approximation +and MCMC. In both figures, the upper panels show the biases and +the lower panels show the standard deviations. The bias tells us +how accurate our best estimate is and the standard deviation is the +uncertainty in the estimate. A larger bias is not necessarily a problem +as long as its associated uncertainty is proportional such that the +normalised bias follows the correct statistics. Figure 6 shows this to +be true. These are the results for the 3 initial 20s portions (0-60s) of +the calibration observation, described in the Section 2. +As seen in the upper panel of Figure 10, the Laplace approxi- +mation/optimization routine shows a negligible underestimate of the +gain amplitudes in comparison to MCMC. This is not present in +Figure 6 so we can safely assume this to be a statistical fluctuation +rather than a systematic effect. The posterior standard deviations, in +the lower panel of Figure 10, overlap so well that one only sees a +muddy brown color everywhere as opposed to sections with distinct +blue or orange. +Figure 11 shows that the Laplace approximation is an excellent +MNRAS 000, 1–19 (2023) + +14 +Chris Finlay et al. +8 +6 +4 +2 +0 +2 +4 +6 +8 +10 +Relative Bias in Gain Amplitudes [%] +0 +100 +200 +300 +400 +500 +600 +Number of Parameters +1920 Gain Amplitudes +MCMC +Laplace +0 +1 +2 +3 +4 +Relative Standard Deviation +of Gain Amplitudes [%] +0 +100 +200 +300 +400 +500 +600 +Number of Parameters +MCMC +Laplace +Figure 10. Comparison of biases and posterior uncertainties in our gain +amplitude estimates from the Laplace approximation and MCMC. The brown +region is where the distributions overlap and only in a couple bins near the +centre of the top panel can we see a discrepancy. This indicates excellent +agreement between the Laplace approximated posterior and the true (MCMC) +posterior. The top panel shows the relative biases and the bottom panel shows +the posterior standard deviations of these estimates. Each estimate is for a +specific antenna and time step. The mean of each distribution is indicated by +the dashed vertical line of the corresponding colour. +estimate of the posterior distribution over the gain phases. There is +near perfect agreement in both biases and the posterior errors. +Figure 12 shows the marginalised posterior over the RFI orbit pa- +rameters for the 0-20s portion of the calibration observation. The +Laplace approximation shows excellent agreement with the true +(MCMC) posterior. Only when looking very closely can one see +that two distributions have been plotted. +Next, we analyze the convergence of our MCMC chains to gauge +the reliability of the MCMC derived posterior as a benchmark. Un- +fortunately, no analytical solution is available for our problem, as +for most real world problems, and MCMC is the gold standard for +estimating the posterior distribution. An MCMC routine that follows +detailed-balance and is ergodic (Neal et al. 2011), as our routine does, +is guaranteed to converge to the true posterior distribution in the limit +of infinitely many samples. Since this is computationally intractable +we must rely on ‘sufficiently many samples’. There are standard tools +available to gauge if we have ‘sufficiently many samples’. The pri- +mary tool used to gauge the efficiency of an MCMC sampler is the +lag-autocorrelation, denoted by 𝜌𝑡, of individual parameter sample +chains. From this the effective sample size (ESS) of a chain, or set +of chains, can be calculated. This is then used in the estimation of +the MCMC standard error on individual parameters. The MCMC +standard error gives an estimate of the uncertainty in the posterior +8 +6 +4 +2 +0 +2 +4 +6 +8 +10 +Bias in Gain Phases [deg] +0 +200 +400 +600 +Number of Parameters +1890 Gain Phases +MCMC +Laplace +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Standard Deviation of Gain Phases [deg] +0 +200 +400 +600 +800 +Number of Parameters +MCMC +Laplace +Figure 11. Comparison of biases and posterior uncertainties in our gain phase +estimates from the Laplace approximation and MCMC. The distributions are +indistinguishable, resulting in a brown colour as opposed to the separated blue +and orange regions. This shows exceptional agreement between our Laplace +approximated posterior and the true (MCMC) posterior. The top panel shows +the biases in our gain phase estimates and the bottom panel shows the posterior +standard deviations of these estimates. Each estimate is for a specific antenna +and time step. The mean of each distribution is indicated by the dashed vertical +line of the corresponding colour. +standard deviation estimate, 𝜎𝑝, of parameter 𝑝. +𝜎 ˆ𝜎𝑥 = +ˆ𝜎𝑥 +√𝑁eff +, +𝑁eff = +𝑁 +1 + 2 �∞ +1 𝜌𝑡 +(30) +It is the uncertainty on the uncertainty. Equation (30) gives its defini- +tion and Figure 4 shows the distributions of these, as relative errors, +for our different categories of parameters. The median relative error +was ≈0.5% with less than 1% of the estimates being worse than 2%. +It is also standard to check the convergence of our chains to make +sure this estimate would not get worse with more samples. The po- +tential scale reduction factor (Gelman & Rubin 1992), ˆ𝑅, also know +as the Gelman-Rubin (GR) statistic, is a commonly used measure of +convergence for a set of MCMC chains. The closer ˆ𝑅 is to 1 the better +converged it is. In Vats & Knudson (2021) 99% of a random sample +of papers from 2017 use an ˆ𝑅 cut-off of 1.01 or greater, the remaining +1% used 1.003. In Table 4 we give the minimum, median, maximum +and the first last percentiles of the calculated Gelman-Rubin and split +Gelman-Rubin statistics. These are calculated for all estimated pa- +rameters (≈6000 parameters) in the 0-60s data portions. Given that +the 99th Percentile values for both standard and split are below 1.01 +we can confidently say that our MCMC chains are well converged. +3.3 Application to a Target Observation +In this section we analyze our method with application to a 7.5 minute +target observation that takes place shortly after the calibration obser- +MNRAS 000, 1–19 (2023) + +tabascal +15 + +Laplace Posterior +MCMC Posterior +80 +40 +0 +40 +80 + [arcsec] +8 +4 +0 +4 +8 + [arcsec] +600 +0 +600 +1200 +h [m] +5.0 +2.5 +0.0 +2.5 +5.0 + [arcsec] +80 +40 +0 +40 +80 + [arcsec] +8 +4 +0 +4 +8 + [arcsec] +5.0 +2.5 +0.0 +2.5 +5.0 + [arcsec] +Figure 12. A corner plot of the marginalised posterior distribution of the +RFI orbit parameters for a 20 second data portion. The Laplace approximated +posterior is barely visible underneath the true (MCMC) posterior showing +excellent agreement between the two. The 68% and 95% credible regions are +shown for the Laplace approximated posterior (blue) and MCMC posterior +(green) distributions. The distribution and true value has been shifted such +that the true value is defined to be 0. +Minimum +1st Percentile +Median +99th Percentile +Maximum +Relative MCMC +Standard Error +0.183% +0.190% +0.553% +1.787% +19.144% +Gelman-Rubin +Statistic +1.0000 +1.0000 +1.0001 +1.0045 +1.0442 +Split Gelman-Rubin +Statistic +1.0000 +1.00000 +1.0001 +1.0040 +1.0356 +Table 4. Distribution statistics for standard accuracy and convergence tests on +MCMC posterior chains. The 99th Percentile figures show excellent accuracy +and convergence of our chains for ≈6000 parameters with only a couple of +outliers. +vation. The target phase centre is (𝛼, 𝛿) = (27◦, 15◦). This position +keeps the RFI location to within 10◦ on the sky. The target observa- +tion has 100 uniformly distributed point sources where intensities are +drawn from an exponential distribution with mean of 100 mJy. The +gains used carry on from the model used for the calibration observa- +tion with the appropriate time steps and the primary beam model is +identical to the calibration observation. Figure 13 shows the visibility +amplitudes (averaged over baseline) for the astronomical component +and the combined contaminated visibilities. No noise or gains are +included to clearly show the difference in the time variability. +In the basic standard approach of 1st Generation Calibration +(1GC), a target observation is initially flagged for RFI followed by ap- +plying calibration solutions, from the calibration observation. After +this stage various methods may be used to try and flag any remain- +ing RFI that was missed in the first round of flagging. Hopefully, +after these steps the astronomer is left with calibrated visibilities, +that are free from RFI, that can go on to be imaged or as input for +2nd Generation Calibration (2GC). One of the troubles with this ap- +proach is that good calibration solutions are rarely available for RFI +0 +1 +2 +3 +4 +5 +6 +7 +Observation Time [min] +100 +101 +102 +Visibility Amplitude [Jy] +All Baselines +AST+RFI +AST +Figure 13. The visibility amplitudes in the target observation. We can see the +near constant nature of the astronomical contribution (in red) in comparison +to the RFI visibilities that vary by orders of magnitude over a 1 minute period. +These are the average visibility magnitudes across all baselines. +contaminated channels. This is due to the lack of uncontaminated +data, in the calibration observation, available to calculate the gain +solutions. Channels with persistent RFI may be flagged entirely for +all times. Such is the case for the MeerKAT calibration pipeline on +the baselines in the array core (|(𝑢, 𝑣)| < 1 km) as seen in Figure +1. This problem is solved by our method directly as we are able to +accurately estimate gains in the calibration observation in the pres- +ence of RFI. Of course having good gain solutions in contaminated +channels is not enough as the the visibilities in the target observation +are still contaminated. Here we describe and analyze a basic method +to remove the visibility contribution from the RFI in a target ob- +servation that follows the calibration observation. This method has +many similarities with those used in Cotton (2009) except we predict +the RFI visibility fringes from the orbit model. We assume the RFI +contribution is only from the same source that was present in the +calibration observation. This demonstrates the advantage of having +estimates of the parameters that describe the RFI’s motion, the orbit +parameters in our case. +To predict the RFI visibility component we take advantage of the +expected phase wrapping caused by the movement of the RFI rela- +tive to the image frame. Since we have a good estimate of the RFI +orbit parameters we can reliably predict the position of the satellite +and therefore the visibility phases it would induce. We do this for +the entire target observation at the ‘true’ telescope sampling rate and +calculate the expected visibilities. The high resolution visibilities are +then averaged down to the 2s integration time. We infer a normalized +sub integration time, time variation per integration window. This is +done by linearly interpolating the mean observed visibility magni- +tude and then normalizing the mean of each integration window to +1. Using these simulated visibilities we can determine specific base- +lines on which the RFI visibility phases wrap close to an integer +number of times. We want this as when assuming a constant ampli- +tude the visibility should average to zero. This approximation tends +to zero when considering a higher number of wraps. This should +leave us with only the astronomical visibility contribution. We found +that choosing the 10 baselines with phase wraps closest to 10 (in a +seven minute target observation) worked quite well. We then take the +magnitude after subtracting the time averaged visibilities on these 10 +baselines. This gives us an estimated RFI visibility amplitude over +time on the 10 baselines. We average across the 10 baselines leaving +MNRAS 000, 1–19 (2023) + +16 +Chris Finlay et al. +1 +10 +100 +1000 +Mean RFI Visibility Amplitude [Jy] +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Flag Rate [%] +18x data +3x data +Noise +Mean + AST + Vis. +Baselines <1000m 3 Clip +Standard Best +tabascal 10 sec +tabascal 5 min +tabascal Best +Figure 14. The percentage of data loss on baselines shorter than 1km. A +dramatic improvement is seen in the 5-100 Jy region where less than 10% +data loss is observed when using tabascal (red and orange )compared to a +60-90% data loss when only performing calibration (blue). In the MeerKAT +pipeline these baselines are flagged without question using a static mask. The +red curve shows the performance of our method using the true gains and RFI +orbit parameters. The shaded regions give the 68% credible interval obtained +from using estimates from different calibration data portions. +us with a time varying RFI visibility amplitude. To estimate the RFI +visibility component on all baselines and time steps, we multiply the +estimated RFI visibilities, that assumed 1 Jy, from before by the time +varying RFI visibility amplitude. This method depends on the idea +that we expect the RFI visibility phases to wrap over a given period on +which the astronomical visibilities to remain relatively constant. This +is not a perfect estimate for the RFI visibility component, however, +it produces surprisingly good results. We will refer to this combined +method of calibration and RFI subtraction as tabascal or simply +our method for the entirety of this section. +3.3.1 Flagging Improvement +In this section we perform a flagging comparison on the target ob- +servation data. For the standard case, i.e. 1GC, we have calibrated +the data using the true gain solutions, averaged over time, from the +calibration observation. Flagging is then performed using 𝜎 thresh- +olding, where 𝜎 = 0.65 Jy is equal to the visibility noise, after sub- +tracting the true astronomical visibilities. For our method, tabascal, +we use our averaged estimated gains from the calibration observa- +tion. After this we apply our RFI subtraction technique described in +Section 3.3 above. tabascal best or ideal refers to the use of the true +gain and orbit parameters and when we reference a time it indicates +the amount of data that was used to form the estimate. +In Figure 14 we show a comparison of the percentage of data +that is flagged using a 3𝜎 flagging threshold on baselines shorter +than 1km in the target observation data. We have varied the scale +of the RFI visibilities in the target observation to show how our +method compares when applied to data that has varying levels of +RFI contamination. The best performance, when compared to the +standard approach, is in the 5-100 Jy range leading to an approximate +70 percentage point decrease in data being flagged. This translates to +approximately 3-18 times more data, in the short baselines, available +for imaging. It should be noted however that this is in an optimal +flagging situation where the true astronomical visibilities are known. +In practice, for the case of MeerKAT, the data on these baselines +would be completely flagged meaning our method is opening up the +0.4 +0.2 +0.0 +0.2 +0.4 +DEC Offset [deg] +Uncontaminated + +I: 0.98 mJy/beam +tabascal + Flagging (Ideal) + +I: 1.21 mJy/beam +0.4 +0.2 +0.0 +0.2 +0.4 +RA Offset [deg] +0.4 +0.2 +0.0 +0.2 +0.4 +DEC Offset [deg] +Flagging only + +I: 2.38 mJy/beam +0.4 +0.2 +0.0 +0.2 +0.4 +RA Offset [deg] +tabascal + Flagging (5 min.) + +I: 1.25 mJy/beam +4 +2 +0 +2 +4 +Flux mJy/beam +Mean RFI Amplitude: 69.0 Jy +Figure 15. Imaging comparisons of a target observation of 100 point sources. +tabascalmanages to reduce the RFI contamination significantly resulting in +around half the image noise compared to flagging alone and only 25% more +noise than an image form uncontaminated data. Additionally, only 5 minutes +of calibration data is needed to achieve near optimal results for the method. +The top left image uses true astronomical visibilities with noise added. The +top right image is our method, tabascal, using the true RFI orbit and gains +after 11% of the data is flagged. The bottom left image is the standard method, +1GC only, using the true gains with 89% of the data flagged. The bottom right +image is our method, tabascal, using the estimated RFI orbit and gains +from the 5 minute calibration observation. In this case 15% of the data was +flagged. For all flagging a 3𝜎 threshold was used. +possibility of science in an untapped domain. With the effective use +of the short baselines, astronomical large scale structure could now be +accessed in the contaminated frequency bands. HI intensity mapping, +with radio interferometers, could now probe the small scale structures +in the 1-40 arcminute range at redshift of z = 0.092 - 0.2356. +In Figure 14 the flag rate is compared between the standard method, +1GC only, and our method, tabascal. Our method performs nearly +identically in the best case scenario and using 5 minutes of calibration +data. The confidence intervals, indicated by the shaded regions, are +generated by sampling parameter sets from our marginal posterior +distributions and applying tabascal for each sample and calculating +the flag rate. +The comparison in Figure 14 only takes into account the short +baselines forming the core of the MeerKAT array, however, these +compose over 50% of all the baselines. Looking at all baselines the +results looking remarkably similar. The improvement that tabascal +brings is slightly more pronounced on shorter baselines as the RFI +amplitudes are reduced on longer baselines due to the greater time +smearing/phase wrapping for the RFI contribution. +3.3.2 Imaging Comparison +Imaging was performed using CASA’s tclean with a 4 arcsecond +pixel size, with a Brigg’s weighting scheme with robustness param- +6 These values are calculated using a frequency range of 1150-1300 MHz +and baseline range of 30-1000 m. +MNRAS 000, 1–19 (2023) + +tabascal +17 +10 +30 +100 +300 +True Source Flux [mJy] +30 +20 +10 +0 +10 +20 +Error in Flux Estimation [mJy] +Uncontaminated +Flagging only +tabascal + Flagging (5 min.) +Figure 16. Flux estimation errors for sources found in the images from Figure 15 using pyBDSF. We see that tabascal gives near optimal (uncontaminated) +results while the standard approach is both biased towards lower fluxes and has larger errors. The red shaded area below about 23 mJy is where sources were +completely undetected in the flagging only image. tabascalmanages to find sources down to 16 mJy and the uncontaminated image gets to 13 mJy. The blue +markers are from the image using true astronomical visibilities with noise added. The orange markers are for the best case situation in the standard RFI flagging +only approach and the green markers are for our method, tabascal, using posterior means from 5 minutes of calibration data. +eter of -0.5. We produced 1024×1024 size images with all other +parameters set to default. We image four separate situations for com- +parison. We have an uncontaminated case, a 1GC best case using +perfect calibration and only flagging, and two tabascal cases where +we have also applied flagging after using tabascal. +The uncontaminated case uses purely astronomical visibilities with +noise added. For this case we do not include any telescope response +effects and no RFI contamination. We image only 𝑉AST +𝑝𝑞 ++ 𝜂𝑝𝑞. For +the 1GC standard case we take the observed visibilities, 𝑉OBS +𝑝𝑞 , and +calibrate them using the true gains from the target observation. There- +fore we are imaging 𝑉OBS +𝑝𝑞 /(𝐺 𝑝𝐺∗𝑞) after 3𝜎 threshold flagging is +applied. In the tabascal cases, we initially perform 1GC, then we +subtract an estimated RFI visibility component and finally flag the +data, just as with the standard case. In our best case we use the true +gains and RFI orbit parameters and in our 5 min. case we use our +estimates from the 5 minute calibration observation. +Performing imaging and source extraction with lower levels of RFI +amplitudes leads reduced image noise for both the standard case and +tabascalas would be expected and increases the number of found +sources. Additionally, the bias present in source extraction for the +standard (flagging only) case reduces as the RFI amplitudes decrease. +As RFI amplitudes increase, the bias increases and tabascal also +becomes a victim of this although to a lesser degree. We find that +tabascalconsistently performs better than flagging alone across all +RFI amplitude ranges, both in bias and in image noise. +In Figure 16 we show a comparison on the source finding and +flux recovery. For all cases we used the images in Figure 15. We +used pyBDSF with default settings to perform source finding and +measurement. The same imaging and source extraction settings were +used for each image. pyBDSF gives us source positions and fluxes +all with errors among a number of other source measurements. +Sources were matched to the true source model using astropy’s +match_to_catalog_sky. The error bars are generated by pyBDSF. +None of the ten sources below 13 mJy were found in any of the im- +ages. Our method was comparable to the uncontaminated case only +finding two less source. The standard approach only found sources +down to about 23 mJy. This can be attributed to the higher image +noise as a result of the higher flagging rate. We see that the standard +approach tends to recover less flux compared to tabascal and the +uncontaminated case. The smoothed histograms on the right hand +panel hand panel in Figure 16 shows the distribution of flux estima- +tion errors for each case. The distributions have been weighted by +the source flux uncertainties from pyBDSF. We see that tabascal +performs very similarly in terms of mean error and spread compared +to the uncontaminated case, whereas, the standard (flagging only) +method has both a broader distribution and systematically underes- +timates source fluxes. +4 CONCLUSIONS AND FURTHER WORK +Calibration of radio interferometer arrays is a fundamental step in +radio astronomy and is typically profoundly contaminated by Radio +Frequency Interference (RFI). The usual approach to RFI is to simply +cut out (flag) all obviously contaminated data, leading to significant +data loss. Ideally astronomers would like to be able to effectively re- +move it without losing astronomical signal, but, for moving sources +MNRAS 000, 1–19 (2023) + +18 +Chris Finlay et al. +of RFI, this has not proven possible so far. In this paper we show +that this is possible, at least for a class of RFI moving on predictable +trajectories, such as satellites. The key ideas that allow this progress +are (1) moving sources relative to the phase center coupled with a +curved wave front (near-field) model distinguish RFI from astronom- +ical sources, and (2) we have a good model for the trajectory of the +satellite by using TLE orbit parameters. +Our algorithm, tabascal, starts by building a forward genera- +tive model of the signal parameterized by the antenna gains, satellite +orbital elements, and modulated RFI amplitudes. We then compare +two approaches to estimating a posterior distribution over these pa- +rameters. The fastest method finds the best-fitting parameters (MAP) +by an optimization algorithm followed by using the Laplace (Gaus- +sian) approximation to estimate the parameter uncertainties including +their covariance. The more rigorous approach uses MCMC to find +the full posterior distribution without approximation. We find that +the Laplace approximation works very well on our simulated data +having very good agreement with the full MCMC approach. Addi- +tionally, we find the MCMC approach is computationally feasible on +realistic data sizes in case it is required for dealing with real-world +data complexities. +One of the most interesting results of our analysis is that tabascal +is able to calibrate using the combined astronomical + RFI signal, +thus turning the contamination into an advantage to yield more pre- +cise calibration with reliable uncertainties. In application to an adja- +cent target observation, tabascal uses the estimated RFI trajectory +and calibration parameters to estimate and subtract the RFI signal. +The residual data is then flagged using sigma clipping. +For a simulated MeerKAT target observation and looking at all +baselines shorter than 1km we find that for a mean RFI amplitude +of 17 Jy, using tabascal leads to less than 1% loss of data com- +pared to ∼ 75% data loss from an ideal 3𝜎 flagging algorithm. +At 69 Jy the loss is 89% for the standard method and ∼ 11% for +tabascal, a nearly 9× increase in data available for science. Once +imaged, tabascal processed data allows recovery of faint sources +that are completely missed in images from purely flagged data, i.e. +the standard method. Empirically we found that using tabascal +halves the detection threshold relative to the standard method, bring- +ing it near the ideal detection threshold for data uncontaminated by +any RFI. Furthermore, the recovered source flux distribution from +tabascalprocessed data was in line with the uncontaminated data +while source fluxes recovered through flagging alone were biased +towards fainter fluxes. +In this work we have used tabascal in only a single frequency +channel. However, it is trivially applied to multiple frequencies by +running it in parallel across frequency channels. Currently tabascal +does not formulate the recovery of astronomical signal in the target +observation as an inverse problem as it does for the calibration obser- +vation. This is why flagging is still required after the application of +tabascal to target observation data. The extension of this work to- +wards this is already in progress and will be presented in a follow-up +paper. So far we have worked with the total intensity signal, however, +it is straightforward to extend this to the full polarization domain as +most RFI signals are strongly polarized due to their antenna geome- +try. +Finally, to extend this work to real observations, it is expected +that a simplified perturbation model (SGP4/SDP4) may be needed +to model satellite trajectories with sufficient accuracy. These are +publicly available and can be re-implemented in JAX. +ACKNOWLEDGEMENTS +We thank members of the SARAO Data Science team, Radio As- +tronomy Research Group at SARAO and Niruj Mohan for useful +discussions. CF and MK acknowledge funding by the Swiss Na- +tional Science Foundation. We also acknowledge the support of the +South African Radio Astronomy Observatory. This research has been +conducted using resources provided by the Science and Technology +Facilities Council (STFC) through the Newton Fund and SARAO. +DATA AVAILABILITY +The data and code for recreating Figures 3 to 16 and Ta- +bles 3 & 4 is made available through the following url: +https://doi.org/10.5281/zenodo.7516960. It consists of three Python +notebooks that are reasonably documented and reproduce the exact +plots used in this paper. The code for simulating the data, running the +optimization and MCMC routines, as well as, the Laplace approxi- +mation will be made available in the future on the GitHub repository +for tabascal at the url: https://github.com/chrisfinlay/tabascal. +REFERENCES +Aghanim N., et al., 2020, A&A, 641, A5 +Andati L., Smirnov O., Makhathini S., 2022, in Astronomical Society of the +Pacific Conference Series. p. 529 +Arras P., Bester H. L., Perley R. A., Leike R., Smirnov O., Westermann R., +Enßlin T. 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M., et al., 2011, Handbook of markov chain monte carlo, 2, 2 +Perley R., Cornwell T., 2003, EVLA Memo Series, 61 +MNRAS 000, 1–19 (2023) + +tabascal +19 +Sihlangu I., Oozeer N., Bassett B. A., 2021, Journal of Astronomical Tele- +scopes, Instruments, and Systems, 8, 011003 +Smirnov O. M., 2011, A&A, 527, A107 +Sun H., Deng H., Wang F., Mei Y., Xu T., Smirnov O., Deng L., Wei S., 2022, +MNRAS, 512, 2025 +Thompson A. R., Moran J. M., Swenson G. W., 2017, Interferometry and +synthesis in radio astronomy. Springer Nature +Tierney L., Kadane J. B., 1986, Journal of the american statistical association, +81, 82 +Vafaei Sadr A., Bassett B. A., Oozeer N., Fantaye Y., Finlay C., 2020, MN- +RAS, 499, 379 +Vallado D. A., Cefola P. J., 2012, in 63rd International Astronautical Congress, +Naples, Italy. pp 1–14 +Vallado D., Crawford P., 2008, in AIAA/AAS Astrodynamics Specialist Con- +ference and Exhibit. p. 6770 +Vats D., Knudson C., 2021, Statistical Science, 36, 518 +Wells D., 1985, in , Data analysis in astronomy. Springer, pp 195–209 +APPENDIX A: COMBINING MEASUREMENTS +A1 Correlation Structure in the Prior +By using a correlated prior for the modulated RFI amplitudes one is +able to improve the constraints on the gain amplitude estimates. this +is because there is a pathway for information to be shared between +antennas much like when we assume the astronomical source bright- +ness is the same for all antennas. For our prior on the modulated RFI +amplitudes across antennas, at each time step, we can assume the +amplitudes are drawn from a multivariate normal distribution with +mean 0. We then set a prior covariance, that includes correlation +across antennas, for each time step. We do this as we expect the RFI +amplitudes, across antennas, within a single time step to be very +similar to one another with the assumptions that the primary beam +patterns of the antennas are similar. We formulate the covariance ma- +trix using a covariance function with a squared exponential kernel +in the same way as a Gaussian process. We use the same covariance +at each time step. Equation (A1) gives the prior distribution of the +modulated RFI amplitudes, where 𝑨RFI(𝑡 𝑗) is the vector of ampli- +tudes at time step 𝑡 𝑗 and 0 is the zero vector. The covariance, Σ𝐴, +of the prior distribution is defined in Equation (A2) where �𝑥𝑝 is the +position of antennas 𝑝. For example, with the variance, 𝜎2 +𝐴, and the +length scale, 𝑙𝐴, of the covariance function set to 10 000 Jy and 10 +km respectively, we allow a large variation in amplitude but impose +strong correlations across the entire antenna array. +𝑨RFI(𝑡 𝑗) ∼ N (0, Σ𝐴) +(A1) +(Σ𝐴) 𝑝𝑞 +� +𝜎2 +𝐴, 𝑙𝐴 +� += 𝜎2 +𝐴 exp +� +− |�𝑥𝑝 − �𝑥𝑞|2 +2𝑙2 +𝐴 +� +(A2) +A2 Independent Measurements +Let us have 𝑁 independent measurements of an 𝑚-dimensional vari- +able 𝑋. Each measurement, 𝑥𝑛, is normally distributed with mean +𝜇𝑥𝑛 and covariance 𝐶𝑥𝑛. The combination of all N measurements is +simply the normalised product of their probability densities. Since +each measurement is normally distributed the combined probability +density will also be a normal distribution. Note that 𝑛 is used to index +individual measurements. +𝑥𝑛 ∼ N (𝜇𝑥𝑛, 𝐶𝑥𝑛) +(A3) +𝑥 ∼ N (𝜇𝑥, 𝐶𝑥) = +� +𝑛 +N �𝜇𝑥𝑛, 𝐶𝑥𝑛 +� +(A4) +𝐶𝑥 = +�∑︁ +𝑛 +𝐶−1 +𝑥𝑛 +�−1 +(A5) +𝜇𝑥 = 𝐶𝑥 +�∑︁ +𝑛 +𝐶−1 +𝑥𝑛 𝜇𝑥𝑛 +� +(A6) +A3 Correlated Measurements +Let us have 𝑁 correlated measurements of a 1-dimensional variable. +Each measurement is denoted by 𝑥 𝑗 and the total error covariance +of all measurements is 𝐶𝑥, of dimension 𝑁 × 𝑁. The combined +measurement ¯𝑥 is calculated using Equation (A7) with its associated +error variance 𝜎2 +¯𝑥 as calculated using Equation (A8). Note the 𝑗, 𝑘 +and 𝑙 subscripts are used for indices of associated measurement vector +and error matrices. The following equations are taken from Avery +(1996). +¯𝑥 +¯𝑥 = +∑︁ +𝑗 +𝑤 𝑗𝑥 𝑗 +(A7) +𝜎2 +¯𝑥 = +∑︁ +𝑗𝑘 +𝑤 𝑗𝑤𝑘 (𝐶𝑥) 𝑗𝑘 +(A8) +where 𝑤 𝑗 is defined by Equation (A9) below +𝑤 𝑗 = +� +𝑘 +� +𝐶−1 +𝑥 +� +𝑗𝑘 +� +𝑘𝑙 +� +𝐶−1 +𝑥 +� +𝑘𝑙 +(A9) +A4 Independent 1-D Measurements +To check our solutions for both Sections A2 & A3 we can consider +a set of independent 1-dimensional measurements. In this case our +measurement are denoted by 𝑥 𝑗 and our error covariances become +(𝐶𝑥) 𝑗𝑘 = 𝛿 𝑗𝑘𝜎2 +𝑗 and (𝐶𝑥𝑗 ) = 𝜎2 +𝑗 where 𝜎2 +𝑗 are the individual error +variances for each measurement 𝑥 𝑗. The combined measurement ¯𝑥 +with error variance 𝜎2 +¯𝑥 is calculated using Equations (A10) +¯𝑥 = 𝜎2 +¯𝑥 +∑︁ +𝑗 +𝜎−2 +𝑗 𝑥 𝑗 +(A10) +𝜎2 +¯𝑥 = +1 +� +𝑗 𝜎−2 +𝑗 +(A11) +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–19 (2023) + diff --git a/1tE2T4oBgHgl3EQf5Qgr/content/tmp_files/load_file.txt b/1tE2T4oBgHgl3EQf5Qgr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3f1b5810b0350aaf7dc8ac06adb5a742589443a --- /dev/null +++ b/1tE2T4oBgHgl3EQf5Qgr/content/tmp_files/load_file.txt @@ -0,0 +1,1271 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf,len=1270 +page_content='MNRAS 000, 1–19 (2023) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Trajectory Based RFI Subtraction and Calibration for Radio Interferometry Chris Finlay,1,2,3,5★ Bruce A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Bassett,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 Martin Kunz1 and Nadeem Oozeer3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 1Département de Physique Théorique and Center for Astroparticle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Université de Genève,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 24 quai Ernest Ansermet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 1211 Genève 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Switzerland 2Department of Pure and Applied Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' University of Cape Town,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' South Africa 3African Institute for Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 6 Melrose Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Muizenberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 7945,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' South Africa 4South African Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Cape Town,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' South Africa 5South African Radio Astronomy Observatory (SARAO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2 Fir Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Cape Town,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 7925,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' South Africa 6Centre for Radio Astronomy Techniques and Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Department of Physics and Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Rhodes University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Box 94, Makhanda 6140, South Africa Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Radio interferometry calibration and Radio Frequency Interference (RFI) removal are usually done separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Here we show that jointly modelling the antenna gains and RFI has significant benefits when the RFI follows precise trajectories, such as for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' One surprising benefit is improved calibration solutions, by leveraging the RFI signal itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We present tabascal (TrAjectory BAsed RFI Subtraction and CALibration), a new algorithm that jointly models the RFI signal & trajectory as well as the calibration parameters in post-correlation visibilities allowing for curved wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' tabascal can use either optimisation or fully Bayesian statistical methods to find calibration solutions in contaminated data that would otherwise be thrown away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We test tabascal on simulated MeerKAT calibration observations contaminated by satellite-based RFI with amplitudes varying between -20 dB and 15 dB relative to the 1 Jy calibrator source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We obtain gain estimates that are both unbiased and up to an order of magnitude better constrained compared to the case of no RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' tabascal can be further applied to an adjacent target observation: using 5 minutes of calibration data results in a target image with about half the noise achieved when using purely flagged data, and only 23% higher than a completely uncontaminated observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The source detection threshold and recovered flux distribution of tabascal-processed data was on par with uncontaminated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In contrast, the standard method of RFI flagging alone resulted in a higher detection threshold (2×) and led to consistent underestimation of source fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For a mean RFI amplitude of 17 Jy, using tabascal leads to less than 1% loss of data compared to ∼ 75% data loss from an ideal 3𝜎 flagging algorithm, a very significant increase in data available for science analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Although we have examined the case of satellite RFI, tabascal should work for any RFI moving on parameterizable trajectories, relative to the phase centre, such as planes or objects fixed to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Key words: Bayesian Methods – Radio Frequency Interference – Radio Interferometry – Calibration 1 INTRODUCTION A major problem plaguing radio astronomy observatories across the world is the problem of Radio Frequency Interference (RFI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the context of radio astronomy, RFI is generally any unwanted radio signal that can result from both man-made and natural sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The increasing sensitivity of radio telescopes coupled with more RFI sources has led to an exponentially growing number of detected sources of RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Several signal processing methods exist and are used to handle RFI, however, in practice there is no universal fool-proof technique for RFI mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For reviews on RFI mitigation see Kesteven (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Briggs & Kocz (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Fridman & Baan (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Ekers & Bell (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' RFI flagging, the process of identifying data points contaminated by RFI, is the most commonly used post-processing technique for RFI mitigation in use across all observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Though there has ★ E-mail: christopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='finlay@unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='ch been significant progress in applying advances in machine learning to RFI flagging (Vafaei Sadr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Mesarcik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2022), these techniques come at the expense of data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this paper we instead explore statistical methods to filter out the RFI since it is reasonable to expect it to produce advantages similar to that which occurred in analysis of the Cosmic Microwave Background, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In particular, Bayesian methods show significant promise for radio astronomy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Lochner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Arras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' If successful this means losing less useful data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To do so we exploit the key defining property of RFI: namely that it is not stationary in the reference frame of the celestial sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' RFI is always moving relative to the sky, either due to the earth’s rotation or artificial orbits (planes and satellites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Methods for the subtraction of RFI signal have been investigated in the past with limited success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Perley & Cornwell (2003) proposed a method for subtracting RFI visibility contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Cornwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2004) the proposed method was tested on carefully chosen real data and managed to reduce the RFI contribution by up to a factor © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='04188v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='IM] 10 Jan 2023 2 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The fraction of time that is flagged by the MeerKAT RFI flagger in the L-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The blue and orange curves show the fraction of time flagged for baselines shorter than 1 km and longer than 1 km respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The most heavily affected sub-bands have a static mask on the shorter baselines, that is why they are flagged 100 % of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We can see that the majority of the RFI present in this band is from satellite-based RFI of which the Global Navigation Satellite System (GNSS) are the main culprit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Source: SARAO of 1000 in specific channels of a ground-based RFI source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This method therefore showed real promise and appears to have resulted in a task named UVRFI (Kogan & Owen 2010) that is available in the Astronomical Image Processing System (AIPS) software (Wells 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The UVRFI task has two sub tasks: (1) CIRC, an extension of RfiX (Athreya 2009), that is closely related to the original method, is used for ground-based sources and fits a linearly varying amplitude given the implied fringe-frequency, and (2) CEXP, that applies CLEAN (Högbom 1974) in the Fourier domain of the visibility time series, that can be used for any RFI source as long as its fringe frequency is suitably different from that of the celestial signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our proposed method, tabascal, has some minor similarities with the CIRC method from UVRFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' However, it is more general as it includes the ability to work on both stationary and moving sources of RFI, given that the RFI source moves on a fixed trajectory we can parameterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is the case for all stationary sources as well as most moving sources such as planes and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The key differences are (1) the full decomposition of 𝑁(𝑁 − 1)/2 baseline signals into 𝑁 antenna-based signals, (2) the modelling of time-smearing effects on the signal, and (3) the joint estimation of antenna gains and RFI parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this paper we apply tabascal to simulations of MeerKAT (Jonas & Team 2016) observations contaminated by satellite-based RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have chosen this situation as a testbed as MeerKAT is one of the most sensitive radio telescopes in the world, in its operational bands, and is the precursor for the Square Kilometre Array (SKA) Mid tele- scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, its L-band (900 - 1670 MHz) is already severely affected by satellite-based RFI, not to mention the increasing number of satellites yet to be deployed, such as the SpaceX Starlink (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 GHz) and Amazon Kuiper constellations, which will affect radio telescopes in a number of frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The MeerKAT L-band data currently suffers around 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 % data loss due to RFI flagging (Sihlangu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2021) of which the majority is caused by satellite- based sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Furthermore, satellites are a particularly interesting source of RFI from the perspective of RFI subtraction due to their predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The positions of satellites can be predicted based on previous measurements, such as from two-line element sets (TLEs), and for many, the spectral and temporal signal profiles are known a priori (Harper & Dickinson 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The identified satellite constella- tions that currently affect MeerKAT L-band data are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Constellation Frequency Orbit Orbit Bands (MHz) Elevation Inclination GPS L1: 1565 - 1585 L2: 1217 - 1237 L3: 1375 - 1387 L5: 1166 - 1186 20,180 km 55◦ GLONASS L1: 1592 - 1610 L2: 1242 - 1249 L3: 1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='025 19,140 km 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8◦ Galileo E1: 1575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='42 E5a: 1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='45 E5b: 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='14 E5 AltBOC: 1191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='795 E6: 1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='75 23,222 km 56◦ Inmarsat 1526 - 1554 35,785 km 0◦ Iridium 1616 - 1626 780 km 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4◦ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A summary of the characteristics of the satellite-based RFI that affects the MeerKAT L-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' All of the active satellites in these constellations have eccentricities of less than 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Sources: GLONASS ESA, GLONASS Novatel, GPS ESA, Galileo ESA, SARAO RFI Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Iridium ESA This paper is organized as follows: Section 2 describes the data simulations, the probabilistic inverse model and the methods used to recover parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Section 3 we discuss the recovered posterior parameter distributions and how these results are used to improve science performance in a target observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Finally, in Sec- tion 4 we summarize the problem and our findings as well as further directions for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2 METHODS AND SIMULATIONS This section is organised as follows: in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 we discuss the principle of radio interferometry and how we model the response of the telescope to the sky brightness distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 we discuss modifications to the standard model necessary for most types of RFI and the specific implementation of our MeerKAT data simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 start with a brief introduction to Bayesian concepts used in this paper, and then go on to describe our likelihood term and forward model along with the associated priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We finish off with Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='9 where we describe the methods used to recover the posterior distributions of our model parameters by means of a Laplace approximation and Markov chain Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We begin by describing the problem of transfer calibration, also referred to as 1st Generation Calibration (1GC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 1GC is the first step in the calibration of an interferometer that provides a starting point for further calibration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2GC/selfcal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In 1GC observations of a calibrator source, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' a source of known position and flux, is used to estimate the antenna gains which can then be used do an initial calibration on a following target observation where the sky distribution is not known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A suitable calibrator source is a bright, unresolved (smaller then the synthesized beam) source, is isolated in the sky (relative to the primary beam width) and should be close to the target (within 10◦ on the sky) (Mauch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The brighter the calibrator source, the greater the signal-to-noise ratio (SNR) achieved leading to better constrained antenna gain estimates for a given observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The source should be isolated in the sky so that the apparent sky model can be estimated as only consisting of the calibrator source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Finally, the requirement of the calibrator MNRAS 000, 1–19 (2023) Fraction of time flagged forbaselines <1 km and >1km for 4hr track (Xx pol) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 ged flagge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 Aircraft transponders GSM down GLONASS L2 Fraction of time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 GSM up Galileo 2 Inmarsat GPS L5 Galileo 3 GPS L1 GLONASS Iridium GPS L GPS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 bl <1 km MMUM bl >1 km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 900 1000 1100 1200 1300 1400 1500 1600 1700 Frequency (MHz)tabascal 3 lying within 10◦ of the target field comes from the angular scale of ionospheric fluctuations which affect the gain phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A list of suitable calibrators is available from the SARAO External Service Desk1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Radio Interferometry A radio interferometer measures the sky brightness distribution (spectral radiance) by taking samples in visibility space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The do- main of visibility space is denoted by the coordinates (𝑢, 𝑣, 𝑤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To sample a point in visibility space, the signal from two antennas with some spatial separation, measured in wavelengths, is correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The separation vector, given by the components (𝑢, 𝑣, 𝑤)𝑝𝑞, referred to as a baseline, points from antenna 𝑝 to antenna 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is done for all antenna pair combinations in an interferometric array at multi- ple time steps leading to samples of many locations in the visibility space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' One is then able to infer the sky brightness distribution from the visibility samples using the formalism described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For an ideal fringe-stopping interferometer, the visibility distribu- tion is defined by Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝑉(𝑢, 𝑣, 𝑤) = ∬ 𝑙𝑚 𝐼(𝑙, 𝑚) exp � −2𝜋𝑖�𝑢𝑙 +𝑣𝑚+𝑤(𝑛−1)�� 𝑑𝑙𝑑𝑚 𝑛 (1) In Equation (1) 𝑖 is the imaginary unit and 𝐼(𝑙, 𝑚) is the brightness distribution on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The sky coordinates, (𝑙, 𝑚, 𝑛), are unitless direction cosines lying in the range [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since the domain of the sky brightness distribution, 𝐼, lies on a sphere of fixed arbitrary radius, the celestial sphere, the third direction cosine, 𝑛, is fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝑛 = √ 1 − 𝑙2 − 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The origin of the sky coordinates, (0, 0, 1), is called the phase centre and is fixed to a location on the celestial sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is the defining property of a fringe-stopping interferom- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When considering the field of view (FoV) of the telescope to be very small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' the sky signal is dominated by an area of the sky where 𝑙2 + 𝑚2 ≪ 1, Equation (1) reduces to the van Cittert-Zernike (vCZ) theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this case, 𝑛 ≈ 1, producing an exact 2D Fourier relation between the visibilities and the sky brightness distribution (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 Telescope Response The telescope response, Γ𝑝𝑞, to measuring some sky brightness distribution includes the modulation of the primary beam of each of the antennas 𝑝 and 𝑞 and their respective bandpass filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When the primary beam intensity pattern of antenna 𝑝 is |𝐸𝑝(𝑙, 𝑚, 𝜈, 𝑡)|2 and its bandpass is |𝐺 𝑝(𝜈, 𝑡)|, the telescope response is as given in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is very similar to the form given in Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Γ𝑝𝑞 = 1 Δ𝑡 ∬ 𝑡𝑗±Δ𝑡/2 𝜈0±Δ𝜈/2 𝐺 𝑝𝐺∗ 𝑞 ∬ 𝑙𝑚 𝐸𝑝𝐸∗ 𝑞𝐼 exp [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='] 𝑑𝑙𝑑𝑚 𝑛 𝑑𝑡𝑑𝜈 (2) In Equation (2), ∗ indicates the complex conjugate, Δ𝑡 ≫ Δ𝜈−1 is the integration time in the correlator for a single sample at time 𝑡 𝑗 and Δ𝜈 is the bandwidth of a single frequency channel centred on 𝜈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have emitted the arguments of 𝐺, 𝐸 & 𝐼 but in principle all of these depend on 𝜈 and 𝑡 and are generally assumed constant 1 https://skaafrica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='atlassian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='net/wiki/spaces/ESDKB/overview over the intervals Δ𝜈 and Δ𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Only 𝐸 and 𝐼 are functions of 𝑙 and 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The expression inside the exponential denoted by [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='] is the same as in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once the integrals are calculated we are left with Equation (3) Γ𝑝𝑞(𝑢, 𝑣, 𝑤) = |𝐺 𝑝||𝐺𝑞|𝑒𝑖(𝜑𝑝−𝜑𝑞)Δ𝜈 √︃ 𝐴𝑝 𝐴𝑞𝑉𝑝𝑞(𝑢, 𝑣, 𝑤) (3) where 𝑉𝑝𝑞 is the true visibility at the point (𝑢, 𝑣, 𝑤)𝑝𝑞, 𝐴𝑝 is the collecting area of the dish on antenna 𝑝 in the direction of the phase centre, |𝐺 𝑝| is the magnitude of the bandpass/gain at antenna 𝑝 and 𝜑𝑝 − 𝜑𝑞 is the phase difference of the gains between antennas 𝑝 and 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equation (3) we have assumed that the gains are constant over the integration time and the channel bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is a standard assumption as telescopes are designed to have such stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Addi- tionally, due to the rotation of the Earth and frequency dependence of the (𝑢, 𝑣, 𝑤) coordinates, the visibility phases (from the exponential term in Equation (1)) vary over time and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This variation cause the phases to change slightly over the integration window lead- ing to decorrelation of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This results in a reduction in the amplitude of the visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This effect is known as time/frequency smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The strength of this effect increases with baseline length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Therefore, integration windows and frequency channels are chosen to be small enough to minimize this effect for astronomical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Typically, the integration windows are still too large for signals from RFI sources to correlate fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is treated as a feature to reduce the level of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The telescope response Γ𝑝𝑞 is in units of Watts when the sky brightness distribution, 𝐼(𝑙, 𝑚), is in units of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When modelling the visibilities we discretize the sky brightness distribution as a collection of point sources which are summed over, as in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ˜𝑉𝑝𝑞 = 𝐺 𝑝 �∑︁ 𝑠 𝐸𝑝𝑠𝐾∗ 𝑝𝑠𝐵𝑠𝐾∗ 𝑞𝑠𝐸𝑞𝑠 � 𝐺∗ 𝑞 (4) where the measured visibility, ˜𝑉𝑝𝑞 = Γ𝑝𝑞, the 𝐸 terms are nor- malized as 𝐸𝑝 → 𝐸𝑝/√︁𝐴𝑝 and the 𝐺 𝑝 → 𝐺 𝑝/ √ Δ𝜈 making them dimensionless in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This way we have both the point source brightnesses, 𝐵𝑠, and the measured visibilities, ˜𝑉𝑝𝑞, in the same units, Jansky (Jy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Those familiar with the Radio Interferom- etry Measurement Equation (RIME) (Smirnov 2011) will recognize Equation (4), however, here we use it in a scalar form and do not consider polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equation (4) we use the subscript 𝑠 to label a point source at position (𝑙𝑠, 𝑚𝑠, 𝑛𝑠) in the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equation (4) the 𝐾 terms, as defined in Equation (5), 𝐾𝑝𝑠 = exp �−2𝜋𝑖 �𝑢𝑝𝑙𝑠 + 𝑣 𝑝𝑚𝑠 + 𝑤 𝑝(𝑛𝑠 − 1)�� (5) are the geometric delay between an antenna and an arbitrary refer- ence position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Their combination 𝐾𝑝𝑠𝐾∗𝑞𝑠 produce the exponential term in Equation (1) for a specific location 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The 𝐵𝑠 term is the spec- tral flux density of the point source 𝑠 in units of Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Extended sources are represented by a discretized/pixelized version where each pixel is treated as a point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The 𝐸 terms are the direction-dependent ef- fects (DDEs), previously used for the primary beam in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The 𝐺 terms are the direction-independent effects (DIEs), previously used for the gains in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equation (5) the (𝑙, 𝑚, 𝑛)𝑠 are the coordinates of the point source 𝑠 and (𝑢, 𝑣, 𝑤)𝑝 are the coordinates of antenna 𝑝 relative to a global reference position, in units of wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 4 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 Modifications for RFI A number of features make signals from RFI sources distinct from astronomical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' RFI sources are much closer to us than astro- nomical sources and move relative to the celestial sphere on which astronomical sources, outside our galaxy, remain stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We will start by discussing the implications of receiving signal from a source closer than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Spherical waves, originating from a source, resemble plane waves when the source is very far away from the receiver, relative to the receiver dimensions and wavelength of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is known as the far-field regime and is defined in Equation (6): 𝑑𝐹 = 2𝐷2 𝜆 , given 𝑑𝐹 ≫ 𝐷 & 𝑑𝐹 ≫ 𝜆 (6) For a single MeerKAT receptor observing in the L-band (𝜆 ≈ 21 cm), 𝑑𝐹 is just over 2 km which is also much larger than the dish diameter, 𝐷, of ≈ 14 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Therefore, the far-field primary beam models used for astronomical sources are also suitable for use with most sources of non-local RFI, especially, satellite-based sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The geometric delay term, given by Equation (5), that is used in the RIME, assumes sources are in the far-field of the antenna array, not just a single receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For the MeerKAT telescope, observing in the L-band, with the longest baseline of ≈ 8 km, 𝑑𝐹 is ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 × 105 km which is nearly 2 times the distance from the Earth to the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Therefore, practically all sources of man-made RFI are in the near- field of the telescope array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For such sources, Equation (5) must therefore be modified to Equation (7): 𝐾𝑝𝑠(𝑡) = exp � − 2𝜋𝑖 � |�𝑟𝑠(𝑡) − �𝑟 𝑝(𝑡)| 𝜆 − 𝑤 𝑝(𝑡) �� (7) where we have assumed spherical wave fronts as can be expected from a point source in the idealised case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equation (7), 𝜆 is the observation wavelength, �𝑟𝑠 is the position of the RFI source, �𝑟 𝑝 is the antenna position, and 𝑤 𝑝 is the phase tracking delay correction as used in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When combined as a 𝐾𝑝𝐾∗𝑞 term, Equations (5) & (7) describe the same thing, the path length difference between antennas 𝑝 and 𝑞, in wavelengths, including the phase tracking cor- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For a far-field source, relative to the array size, the angle, relative to the reference direction, at which the signal enters the primary beam is the same across all antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' However, this is not the case for near-field sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this case, the angular separation between the pointing direction and the RFI source is different for each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This leads to a different angular evaluation of the 𝐸 term, per antenna, for a given RFI source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The functional form of the 𝐸 term remains the same though in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Another consideration for near-field sources is the the spectral flux density received at the antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The spectral flux density, 𝐼RFI 𝑝𝑠 , of an RFI source 𝑠 at antenna 𝑝 is given in Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝐼RFI 𝑝𝑠 (𝑡) = 𝑃RFI 𝜈 (𝑡) 4𝜋|�𝑟𝑠(𝑡) − �𝑟 𝑝|2 (8) This equation is derived from the free-space path loss formula assuming spherical wave fronts from an isotropic emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In reality, an RFI source will not be an isotropic emitter however this would only change the 𝑃RFI 𝜈 (𝑡) term potentially making it antenna depen- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is not a problem for our method due to the per antenna parameterization used in our forward model, refer to Equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equation (8), 𝑃RFI 𝜈 is the spectral flux of the RFI source in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='Hz−1 which can be divided by 1026 to make the spectral flux density in units of Jy, assuming the distance is in metres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since this is different for each antenna we adapt the 𝐵𝑠 term in Equation (4) to a 𝐵𝑝𝑞𝑠 term as defined in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝐵RFI 𝑝𝑞𝑠 = √︃ 𝐼RFI 𝑝𝑠 𝐼RFI 𝑞𝑠 (9) Finally, to address the moving nature of RFI sources relative to celestial sources, we need to explicitly perform the integration for each visibility sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Typically for radio interferometry simulations, one can evaluate all terms in the model only once per visibility sample as everything is assumed constant on the time-scale of the visibility sample integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This assumption breaks down for moving RFI sources, so we must evaluate all terms related to the positional nature of the RFI at a finer time resolution and then average them to the cadence of the final visibility samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The required time resolution to achieve an accurate result depends on the speed and direction of movement of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 Data Generating Model In this section we describe the data generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We split this section into the telescope, DIE, DDE, astronomical and RFI source models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Together these components fully define our model used to simulate a calibration observation from the MeerKAT telescope with basic, but, realistic telescope response, signal corruptions and RFI contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' While we choose to simulate MeerKAT observations for illustrative purposes everything in our algorithm applies to other radio interferometry observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Table 2 summarizes the parameter values and distributions used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Throughout our simulations we work with only a single frequency channel centred at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='227 GHz and calculate observed visibilities using Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each time sample, 𝑡 𝑗, is an average over a given integration time, Δ𝑡 = 2s, where the visibility function is sampled 16 times per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These integration samples are equally spaced in the range (𝑡 𝑗 − Δ/2𝑡, 𝑡 𝑗 + Δ𝑡/2) where 𝑡 𝑗 is the observation time centroid of the time sample and Δ𝑡 is the integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Telescope Model We simulate data for the MeerKAT telescope positioned at a lat- itude, longitude and elevation of (−30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='721◦, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='411◦, 1054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='71m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use the East, North, Up (ENU) coordinates obtained from the South African Radio Astronomy Observatory (SARAO) to simulate (𝑢, 𝑣, 𝑤) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use standard coordinate transformations, as defined in Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2017, Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1, to transform from ENU coordinates to International Terrestrial Reference Frame (ITRF) co- ordinates and then to (𝑢, 𝑣, 𝑤) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 Astronomical Source Model We simulate both a calibration observation and a target observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the calibration portion we observe the same calibrator source with known spectral flux density and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For the calibrator source we use a 1 Jy point source situated at (𝛼, 𝛿) = (21◦, 10◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We include no other astronomical sources in the calibration portion and keep the source flux density and position fixed in our MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the target portion, we observe a 100 point source field, centred on (𝛼, 𝛿) = (27◦, 15◦), where the positions are uniformly sampled from a disk with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5◦ radius and the spectral flux densities sampled from an exponential distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The sources positions are MNRAS 000, 1–19 (2023) tabascal 5 chosen such that no two sources are closer than ≈ 8 synthesized beam widths (80").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The simulated visibilities for the calibration portion will be denoted by 𝑉CAL, and for the target track we will use 𝑉AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 RFI Source Model Our satellite-based RFI source is modelled with a spectral flux, 𝑃RFI 𝜈 , that is constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is done for simplicity and is not a requirement for the functioning of our method as we allow for a time variable RFI amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use the near-field expression for the spectral flux density at a specific antenna 𝑝 as defined in Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This leads to the baseline dependent spectral flux density 𝐵𝑝𝑞𝑠 as defined in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The position of the satellite-based RFI source is modelled using a circular orbit about the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' One could use a more sophisticated model at the expense of introducing more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Simplified per- turbation models, such as SGP4/SDP4 (Vallado & Crawford 2008) would be the preferred model to use, however, since we are currently testing on simulated data with satellites with very low eccentricity we decided on a simpler model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For use on real data a more sophisticated model many very well be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For a circular orbit we have four parameters, namely, the orbit elevation (ℎ), argument of perigee (𝛾), orbit inclination (𝛽), and the right ascension of the ascending node ( ˜𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The formula for circular motion on an arbitrary plane about the Earth’s centre of mass, �𝑟Earth CoM as a function of time is given in Equation (10) (Fitzpatrick 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' �𝑟𝑅𝐹𝐼 (𝑡) = 𝑹𝑧( ˜𝛼)𝑹𝑥(𝛽)𝑹𝑧(𝛾) �� � (𝑅𝑒 + ℎ) cos(𝜔𝑡) (𝑅𝑒 + ℎ) sin(𝜔𝑡) 0 �� � + �𝑟Earth CoM(𝑡) (10) In Equation (10) above 𝑹𝑥(𝛽) is a 3D rotation matrix about the axis 𝑥-axis through an angle 𝛽, ℎ is the orbit elevation above the Earth’s surface in metres, 𝛽 and ˜𝛼 define the orbital plane, 𝛾 is the angular offset of the orbit and finally 𝜔 = √︁ 𝐺0𝑀𝑒/(𝑅𝑒 + ℎ)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Here we have assumed the satellite’s mass to be negligible compared to the mass of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the equation for 𝜔, 𝑀𝑒 and 𝑅𝑒 are the mass and average radius of the Earth respectively and 𝐺0 is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The angular orbit parameters align with orbital elements used in two- line element sets (TLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The orbit elevation, for a circular orbit, is directly related to the mean motion orbital element, in revolutions per day, by (86400𝜔/2𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The rotation matrix axes are with respect to the ITRF frame used for the antennas where +𝑧 points from the Earth’s centre to the North Pole and +𝑥 points to the intersection of the Equator and Greenwich meridian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The time averaging described in the beginning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 is a fundamental requirement in modelling RFI visibility contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is because the visibility phases induced by an RFI source are rapidly varying in time due to their fast movement relative to the sky reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The resulting effect is called time-smearing and is especially prominent for moving sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The magnitude of this effect increases with the length of the baseline and therefore affects longer baselines more than shorter baselines, assuming the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 Direction Independent Effects (DIE) We include time-varying complex gains for each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Both the gain amplitudes and phases are modelled as linear time variates, as shown in Equations (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' |𝐺|(𝑡) = |𝐺|(0) + �|𝐺|𝑡 (11a) 𝜑𝐺(𝑡) = 𝜑(0) 𝐺 + �𝜑𝐺𝑡 (11b) 𝐺(𝑡) = |𝐺|(𝑡) exp � 𝑖𝜑𝐺(𝑡) � (11c) The initial values and rates of change are sampled from the distri- butions described in Table 2 and is done separately for each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' |𝐺|(𝑡) is the gain amplitude and 𝜑𝐺(𝑡) is the gain phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Equa- tions (11), when the parameter has a (0) superscript, it is the initial value, at 𝑡 = 0, and the overdot is used for the rate of change of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The gain phases include the ionospheric effects, a DDE component, but the spacial scale of variation is assumed to be so large (> 10◦) that it can be considered a DIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We therefore only consider our RFI source to be within this angular distance from the pointing centre for this approximation to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To extend our method to such a situation we could explicitly include the ionospheric effects in the DDE term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This would further require our forward model to be correspondingly adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 Direction Dependent Effects (DDE) We use the normalised Fourier transform of a circular aperture, the square of which is the normalized Airy disk, as the primary beam voltage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The primary beam is the only DDE that we include.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We keep the primary beam constant in time and the same across all antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The functional form of the primary beam term is given in Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝐸(𝜃, 𝜈) = 2𝑐0𝐽1 �𝜋𝑑𝜈 sin 𝜃/𝑐0 � 𝜋𝑑𝜈 sin 𝜃 (12) In Equation (12) 𝐽1 is the Bessel function of the first kind of order one, 𝜈 is the observation frequency in Hz, sin 𝜃 = √ 𝑙2 + 𝑚2 where 𝜃 is the angular separation between our pointing direction and the source and 𝑐0 is the speed of light in a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We only consider a real-valued primary beam voltage model and leave complex voltage patterns and other DDEs for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The inclusion of complex DDEs, such as a complex voltage pattern and ionospheric effects, create a degeneracy in the forward model that would need to be broken by the inclusion of appropriate priors in the probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 Noise Model We add circularly-symmetric complex normally distributed noise to the visibilities as defined in Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each baseline is an independent measurement with independent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The standard deviation, 𝜎𝑛, of the noise is the same for each baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' There- fore, our modelled visibility data are independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The noisy data is generated using Equation (13), where 𝜂𝑝𝑞 is the noise term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ˆ𝑉𝑝𝑞 = 𝐺 𝑝 � ∑︁ 𝑠 𝐸𝑝𝑠𝐾∗ 𝑝𝑠𝐵𝑠𝐾∗ 𝑞𝑠𝐸𝑞𝑠 � 𝐺∗ 𝑞 + 𝜂𝑝𝑞 (13) In Jonas & Team (2016), measurements show that the System Equivalent Flux Density (SEFD) of a single MeerKAT receptor is approximately 420 Jy at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='227 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Using Equation (14), this implies a per visibility noise level, 𝜎𝑛, of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='65 Jy using a 2 s integration time, Δ𝑡, and 209 kHz bandwidth, Δ𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We therefore model the noise MNRAS 000, 1–19 (2023) 6 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Parameter Description Symbol Units Value/Distribution Dish Diameter 𝑑 m 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='965 Observation Frequency 𝜈0 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='227 Channel Bandwidth Δ𝜈 kHz 209 Sampling Rate Hz 16 Integration Time Δ𝑡 s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Noise Amplitude 𝜎𝑛 Jy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='65 Calibrator Spectral Flux Density 𝑆CAL 𝜈 Jy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Calibrator Position (𝛼, 𝛿) (deg,deg) (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0) RFI Spectral Flux 𝑃RFI 𝜈 𝜇W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='Hz−6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 Orbit Elevation ℎ km 20,200 Argument of Perigee 𝛾 deg 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Orbit Inclination 𝛽 deg 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Right Ascension of the Ascending Node ˜𝛼 deg 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Initial Gain Amplitude |𝐺|(0) N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='052) Gain Amplitude Drift � |𝐺| 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='s−1 N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='02) Initial Gain Phase 𝜑(0) 𝐺 deg U[−𝜋/2, 𝜋/2] Gain Phase Drift � 𝜑𝐺 10−3deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='s−1 N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='02) Noise distribution 𝜂𝑝𝑞 Jy CN(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0, 𝜎2𝑛) Earth Radius 𝑅𝑒 km 6, 371 Earth Mass 𝑀𝑒 kg 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='9722 × 1024 Gravitational Constant 𝐺0 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='kg−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='67408 × 10−11 Speed of Light 𝑐0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='s−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='99792458 × 108 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Table of parameter values and distributions for the data generating model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Values have been chosen to, at best, mirror what we have found from various sources including the MeerKAT Specifications web page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' as CN (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='652) which is equivalent to N � 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='652/2 � in both the real and imaginary parts of the visibility independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝜎𝑛 = SEFD √ Δ𝜈Δ𝑡 (14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 Summary of Parameter Values We chose the values in Table 2 to represent worse or equivalent performance to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We found values for these parameters from a number of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' On the MeerKAT Specifications web page, gain amplitude stability was found to be < 3% over 3 hours resulting in ≈ 3 × 10−6s−1 we use 10−5s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' On the RAGAVI (Andati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2022) package web page we found gain amplitudes across antennas to be within 5% where we have used this value as our standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' On the same web page we found the gain phases between antennas to lie within a 40◦ band about 0◦ and we have used 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The gain phase stability was estimated form the MeerKAT examples on the RAGAVI(Andati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2022) web page to be less than 10◦ over 2 hours resulting in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 × 10−3 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have used ×10−3 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='s−1 as the standard deviation of the gain phase drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The MeerKAT dish diameter is taken from Jonas & Team (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The observation frequency is chosen to be in the middle of the MeerKAT L-band corrupted by most GNSS signals as is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The channel bandwidth is taken from standard L-band 4k mode for MeerKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The visibility sampling rate is chosen to be as fast as possible while being computationally viable on a laptop with 16GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The integration time was chosen to be 2 seconds which is one of the options provided by SARAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The noise is calculated from the estimated SEFD as shown in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The calibrator flux density was chosen to be on the weaker end of the L-band calibrators that SARAO provides on their MeerKAT Service Desk web page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The calibrator position was chosen for convenience in finding a suitable satellite orbit passing within 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Finally, the RFI orbit parameters are chosen to align with what is publicly available from example TLEs with the argument of perigee and right ascension of the ascending node tuned so that the satellites pass within 10◦ of both the calibrator and target fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 Bayesian Inference Bayesian inference is a paradigm of statistical inference which uses Bayes’ theorem, Equation (15), P(Θ|D, M) = L(Θ|D, M)Π(Θ, M) 𝑍(D, M) (15) to update our knowledge about some parameters/hypothesis given new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The goal of Bayesian inference is to acquire the posterior probability distribution, P(Θ|D, M), of our model pa- rameters, Θ, given some data, D, and the model, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The poste- rior distribution is comprised of three components, the likelihood, L(Θ|D, M), the prior, Π(Θ, M), and the evidence 𝑍(D, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The likelihood is the probability of seeing the data given the param- eters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The prior distribution encodes any prior in- formation we have about the parameters of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The prior is defined by the user and can include information about the parameters from data not included in the likelihood as well as heuristics or physical limitations of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The evidence, 𝑍(D, M) = ∫ L(Θ|D, M)Π(Θ, M)𝑑Θ, is a normalizing factor but is also used in model selection problems when deciding between two models, M1 and M2, by looking at their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' An example would be choosing between a model that contains one satellite compared to two satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 we carefully construct a prior that guides our model parameters, Θ, toward desirable solutions and provides suitable ini- tial conditions to reliably find maximum a posteriori (MAP) points through optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' An important concept when dealing with a multivariate probability distribution is marginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We may consider a subset of our model parameters to be so called nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' If we integrate the probability distribution over the nuisance parameters we are left with a marginal distribution over our parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The marginal distribution in this case, is a distribution over our parameters of interest taking into consideration all possible values of the nuisance parameters simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Letting Θ = (Θ𝐼 , Θ𝑁 ), where Θ𝐼 are our parameters of interest and Θ𝑁 are our nuisance parameters, we obtain our marginalized posterior over Θ𝐼 by Equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' P(Θ𝐼 |D, M) = ∫ P(Θ𝐼 , Θ𝑁 |D, M) 𝑑Θ𝑁 (16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 Likelihood The likelihood is determined by a combination of our noise model, described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6, our forward model and the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since visibilities have additive noise that is independent and normally distributed in both the real and imaginary parts then our likelihood is the product of the individual likelihood terms for each data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We further assume that the noise is identically distributed in each data MNRAS 000, 1–19 (2023) tabascal 7 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The expression for the total likelihood is given in Equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' L(Θ|𝑉𝑜𝑏𝑠) = (𝜋𝜎2 𝑛)−𝑁D exp � − 𝑁D ∑︁ 𝑗 ��𝑉OBS 𝑗 − ˜𝑉𝑗 (Θ) ��2/𝜎2 𝑛 � (17) Here we use Θ to denote the column vector of model parameters, 𝑉OBS for the observed visibilities, ˜𝑉 for the noiseless model visibil- ities, 𝑗 to index each data point, and 𝜎𝑛 for the standard deviation of the additive complex noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our total number of complex-valued data points is 𝑁D = 𝑁𝑡 𝑁𝑎(𝑁𝑎 − 1)/2, 𝑁𝑡 is the number of time steps and 𝑁𝑎 is the number of antennas in the telescope array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For our problem, we only have one frequency channel and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each likelihood term is a circularly-symmetric complex normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Its functional form differs slightly from a (real-valued) normal distribution in that factors of 2 are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since the real and imaginary components are independent with identical variance (circularly symmetric) the variance of the complex visibility sample is twice that of the individual real or imaginary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For- mulating the likelihood in terms of real and imaginary components individually would lead to the factors of 2 returning to the functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We therefore consider one complex visibility as one data point as opposed to two, as would be the case for separating the real and imaginary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Both formulations are equivalent when using the appropriate noise variance, 𝜎2𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 Probabilistic Model In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='6 we have described the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We will now discuss all of the parameters of the forward model and the priors we set on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 2 shows a Bayesian factor graph (model diagram) that summarizes the entire probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This problem is fully constrained and does permit the usage of a purely likelihood based approach or the use of wide, uniform priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We make use of semi- informative priors based on real-world assumptions that can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This improves the consistency and stability of convergence in finding a solution both for optimization and MCMC by regularizing the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Gains Our gain parameters are composed of amplitudes and phases per antenna, per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have a parameter for the gain amplitude on each antenna at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have the same for the gain phases except we exclude phases for the last antenna using it as a reference antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We set these to 0 in the data generation portion as well as in the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This must be done as our observed visibilities are composed only of differences in phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A transformation in all gain phases of 𝜑′𝑝 = 𝜑𝑝 +𝜑0, where 𝜑𝑝 is the gain phase on antenna 𝑝 and 𝜑0 is a constant, would leave the measurements unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When 𝑁𝑎 is the number of antennas and 𝑁𝑡 is the number of time steps, we have 𝑁𝑡 (2𝑁𝑎 − 1) real-valued parameters to fully describe the complex gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' By using a complex gain parameter for each time step we can model gain variations on shorter time scales than expected thereby not assuming any specific gain variation beyond stability over the individual time step integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' One could assume the gains to be stable/constant over a 10 second data portion for example and reduce the number of gain parameters by a factor of 10, assuming a 2 second integration time as we use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The priors we have on the gain parameters assume reasonable estimates have been made on uncontaminated nearby channels such that the bandpass can be interpolated and provide an estimate in our contaminated channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We assume this estimate to be correct with a 10% and 10◦ standard deviation in the gain amplitudes and phases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We therefore sample from a normal distribution centred on the true value with a 10% and 10◦ standard deviation for each antenna and use this sample as the mean of our prior distribution for all time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The prior standard deviation is set to 10% of this value for the gain amplitudes and 10◦ for the gain phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The prior on each parameter is independent and does not include any correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 Satellites To model our satellite-based RFI we have parameters that govern its, per antenna, signal amplitude and parameters that control its orbit around the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For the satellite orbit, we have four parameters as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These four parameters describe a cir- cular orbit and are a subset of the six orbital elements needed for a general orbit in the two-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' TLEs(Vallado & Cefola 2012) expand on these parameters to account for atmospheric drag and the gravitational pull of the moon etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Space Track provides a standard catalog of satellites and their TLEs at constantly updated measurement epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Flohrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2008), over 11k objects from the TLE catalogue are analysed to find their positional uncertainties over a 48 hour period centred on the TLE epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These objects have been categorized according to certain orbit characteristics and the standard deviations in their orbit determination have been summa- rized per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The standard deviations are quoted in radial, in-track and cross-track (RIC) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The RIC coordinates of an orbit, (�𝑟RIC), are defined with respect to a reference orbit (�𝑟0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The orthog- onal RIC coordinate unit vectors are defined in Equation (18) and form the rows of the transformation matrix from an Earth-centred reference frame, as is used in this paper, to the RIC frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ˆ𝑅 = �𝑟0/|�𝑟0| (18a) ˆ𝐼 = ˆ𝐶 × ˆ𝑅 (18b) ˆ𝐶 = ( ˆ𝑅 × ��𝑟0)/|��𝑟0| (18c) �𝑟RIC = ������� ˆ𝑅 ˆ𝐼 ˆ𝐶 ������� (�𝑟 − �𝑟0) (19) Unfortunately TLEs do not include covariance estimates on their parameters so we make use of those provided in the RIC frame by Flohrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To transform the covariance of an orbit in the RIC frame back to the orbit parameters we make use of the standard error propagation formula assuming normal errors with a minor deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Let ΣRIC be the covariance of a given orbit in the RIC frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ΣRIC is a 3x3 matrix and is diagonal for all work in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We wish to transform this into ΣΦ, a 4x4 matrix, which is the prior covariance of our RFI orbit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Firstly, we define our reference orbit �𝑟0(𝑡) using the true orbit parameters, given by the TLE in a real-world example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Let �𝑟RIC(𝑡 𝑗) = 𝑇𝑗 (Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' �𝑟0(𝑡 𝑗)) such that 𝑇 is the function that accepts, Φ, the vector of orbit parameters and produces 3D positions over time in the RIC frame, given in Equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We evaluate this at each of the time steps in the calibration data portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Given each time step has a covariance given by ΣRIC, as- sumed to be the same at all time steps, we take the average of the transformed precision matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The precision matrix is the inverse of MNRAS 000, 1–19 (2023) 8 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A Bayesian factor graph of the probabilistic model used to estimate uncalibrated and RFI contaminated visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The constants are shown as diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The free parameters of the model are shown as rectangles with rounded corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Their distributions are shown in the top left corner with the parameterization of the distribution given in the smaller rounded rectangles in the top right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Repeated parameters that are indexed are placed in a rectangle, with sharp corners, known as a plate with the index repetition indicated at the bottom centre of the rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The rectangles in the top half of the diagram form the prior over the parameters and the rectangle in the lower left is the likelihood term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The equation in the lower right of the diagram is the mathematical model for the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This leads to a ΣΦ that when transformed back to the RIC frame at best reproduces the original ΣRIC but usually has worse constraints in the ˆ𝐼 and ˆ𝐶 directions by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='16 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We must do this as using only one time step would not produce a suitable constraint in a higher dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Equation (20) shows the formula to generate our prior covariance on the orbit parameters using the method just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' �ΣΦ � 𝑝𝑞 = �� � 1 𝑛 𝑛 ∑︁ 𝑗 𝜕𝑇𝑗 𝜕Φ𝑝 � Σ−1 RIC � 𝑗 𝜕𝑇𝑗 𝜕Φ𝑞 �� � −1 (20) In Figure 3 we show the prior distribution used for the RFI orbit parameters used in the 0-20s data portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The covariance provided in Flohrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2008) for a medium Earth orbit, as is the case for GNSS satellites, is ΣRIC = � diag(73, 131, 54) m �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We chose the prior standard deviations to be 10 times larger than this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We did this to show that our method can handle larger errors in our a prior knowledge than what is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The signal amplitude of the RFI has a parameter per antenna, per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We do this as we are modelling the RFI signal ampli- tude modulated by the primary beam of each antenna, as defined in Equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝐴RFI 𝑝 (𝑡) = 𝐸𝑝 �𝜃(𝑡)�√︃ 𝐼RFI 𝑝 (𝑡) (21) This is a more general approach as compared to assuming some- thing about the primary beam of the antenna/s and/or the intrinsic RFI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Modelling each of these separately would be degenerate as we can only constrain the product of the beam and the RFI sig- nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' If we were to assume the primary beam is the same across all antennas we could parameterize the primary beam using a Zernike polynomial based model as in Asad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For our prior on the modulated RFI amplitudes we choose an uninformative prior, Equation (22), with a very wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝐴RFI 𝑝 (𝑡 𝑗) ∼ N (0, Σ𝐴) (22) We chose each parameter prior to be a normal distribution with mean 0 √︁ Jy and covariance, Σ𝐴, to be fully independent with diagonal values of 10 000 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each parameter prior is fully independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' It is possible to place a correlated prior on these parameters that leads to better posterior constraints on the gain amplitude parameters at the expense of introducing slight biases in the results as this correlation assumptions is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A very important aspect of our RFI model is to model the time- smearing that occurs due to the rapidly varying phases induced by the fast moving RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We, therefore, evaluate the position of the satellite at multiple time points centred about each time step in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, we perform a linear interpolation, and extrapolation at the edges of the data portion, for our RFI amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is needed due to the rapidly varying modulated amplitude caused by, at minimum, the movement of the RFI source through the primary beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once the amplitudes have been re-sampled at the higher rate, equal to the position sampling, RFI visibilities are predicted at this MNRAS 000, 1–19 (2023) tabascal 9 2 0 2 [arcsec] [1e4] 4000 2000 0 2000 4000 [arcsec] 1500 0 1500 h [m] 1500 0 1500 [arcsec] 2 0 2 [arcsec] [1e4] 4000 2000 0 2000 4000 [arcsec] 1500 0 1500 [arcsec] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Corner plot showing the prior distribution of the RFI orbit pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The strong parameter correlations should be noted as these are crucial in choosing initial parameter values for both the optimization and MCMC routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The correlations are induced in the error propagation from the co-moving frame to the orbit model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The correlations change depending on the time of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These are for the 0-10 second calibration data portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The dark and light blue shaded regions show the 68% and 95% prior credible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The true value and distribution has been shifted to 0 to make the contour levels more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' finer rate and then averaged back down to the cadence of the observed visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 Optimization and Laplace Approximation We developed our forward and probabilistic models using the JAXpython library (Frostig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Bradbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2018) so that we could make use of just-in-time (JIT) compilation and automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This allowed us to speed up our computation dramat- ically and get exact derivatives for our own optimization routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this section we will give a brief explanation of the Laplace approx- imation and the custom optimization routine that we developed for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For a quick approximation of the posterior distribution we can make use of the Laplace approximation (Tierney & Kadane 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The Laplace approximation approximates the posterior distribution as a multivariate normal distribution centred on the maximum a pos- teriori (MAP) point, ΘMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since a normal distribution is defined by its first and second moments we must find these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We therefore make use of an optimization scheme to find the MAP position (equiva- lent to the mean, first moment, of a normal distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' At this point the covariance of a multivariate normal, its second moment, is equivalent to the negative inverse of the Hessian of the log posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Symbolically this is described in Equation (23) where 𝑝(Θ|D) is the posterior distribution density function: 𝑝(Θ|D) ≃ N (Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ΘMAP, ΣMAP) , Σ−1 MAP = −𝐻 (ln 𝑝 (Θ|D)) (23) In Equation (23) we make use of the Hessian, 𝐻, which is the matrix of second order partial derivatives of a scalar valued function, 𝑓 , as defined in (24) where 𝑗 and 𝑘 are the row and column indices of the matrix and Θ𝑗 is the 𝑗th parameter in the vector of parameters Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝐻 𝑗𝑘 ( 𝑓 ) = 𝜕2 𝑓 𝜕Θ 𝑗𝜕Θ𝑘 (24) Therefore, given the MAP position by running an optimization scheme using the negative log posterior (NLP) as the minimization surface and then evaluating the Hessian of this surface at the MAP we can obtain the Laplace approximation to our posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The Laplace approximation is exact when the posterior distribution is exactly Normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For all but the simplest problems this is not the case, however, in the limit of infinite data with finite variance the posterior distribution tends toward a normal distribution thanks to the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Next we describe the optimization routine used to find the MAP point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2, we show that this is in fact an excellent approximation of our posterior by comparing the Laplace approximation to the full posterior obtained using a Markov Chain Monte Carlo (MCMC) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' There are many optimization routines publicly available, however, many of these did not perform particularly well on our problem out of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Given that we are able to evaluate the Hessian exactly we developed our own quasi-Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our method evaluates the Hessian periodically to save on computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The update step for a general quasi-Newton scheme is Θ𝑘+1 = Θ𝑘 − 𝜖𝐵−1∇ 𝑓 (Θ𝑘) , (25) where 𝑓 is the function to minimize, 𝐵 is the Hessian approximation, Θ𝑘 is the parameter vector at step 𝑘, and 𝜖 is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since inverting the Hessian becomes very expensive in a high dimensional parameter space, and will not always produce a suitable 𝐵−1 in problematic sections of the parameters space, we block diagonalize the Hessian before inverting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This allows us to invert smaller sub matrices, the blocks, and then recompose them to make a full 𝐵−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We do this for the gain amplitudes, phases, RFI amplitudes, RFI orbit parameters blocks separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This reduces the computational expense but more importantly leads to a more robust optimizer while still efficiently navigating the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When we reach a part of the parameter space that is better behaved, we use the full Hessian and then invert it using an eigendecomposition and applying the softabs (Betancourt 2013) function to the eigenvalues before taking their reciprocals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This ensures that we have a 𝐵−1 that is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is usually when 𝜒2 dof < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The final stages of optimization using the full Hessian inverse allow us to more efficiently converge on the MAP position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We calculate 𝐵−1 every 250 update steps and find that 500 steps using the block diagonal version is enough, after which less than 250 steps are typically needed using the full inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use a decaying step size 𝜖 that is reduced by an order of magnitude when using the full inverse in the final optimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Using this scheme we were able to achieve excellent convergence with 𝜒2 dof ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='01 in less than 1 minute per 10s data portion using a laptop with 16GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The covariance estimation using the Hessian takes around 10 seconds per data portion of ≈ 1000 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Optimization routines are typically very sensitive to the initial parameter values and ours is no exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We sample 10k points from a region centred about the true parameter values, for the gains and RFI orbit parameters, with standard deviations one quarter of those used for their respective prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This should not be a problem when applied to real data as we expect to know these parameter values to this accuracy or better, we just use especially MNRAS 000, 1–19 (2023) 10 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' wide priors in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The initial RFI amplitude parameters are calculated at each time step using the data and the initial gain parameters, and are the same across antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We evaluate the NLP for each of the 10k parameter sets and start from the best position and run the optimizer till convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Convergence is assumed when, 𝜒2 dof < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='05, the improvement in the NLP is less than a set threshold, and the Hessian at that location is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' If the optimizer does not converge according this criteria the next best initial position is used to run a new optimization round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='9 MCMC Implementation In this section we describe the Markov Chain Monte Carlo (MCMC) implementation we have used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We explain the ad- vantages and highlight some of the specifics of how we obtained our results that are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 where we also analyze its accuracy and convergence for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this paper we have made use of Hamiltonian/Hybrid Monte Carlo (HMC) for sampling the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We designed our own HMC implementation using the JAXpython library (Frostig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Bradbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2018) so that we could customize it as we saw fit as well as make use of just-in-time (JIT) compilation and automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The benefit of HMC over a Metropolis-Hastings algorithm using random walk is that successive proposals in HMC are distant from one another, significantly reducing their correlation and leading to higher effective sample sizes for a given MCMC chain length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' HMC is a Monte Carlo method where sample proposals are generated by treating the parameters as position coordinates of a particle and the negative log posterior (NLP) as a potential energy function, 𝑈(�𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' New proposals are generated by sampling associated momenta from a predefined distribution where its negative log represents the kinetic energy of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A proposal is then formed by evolving the particle’s position using Hamiltonian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Equations (26) show Hamilton’s equations 𝑑�𝑥 𝑑𝑡 = 𝜕H 𝜕 �𝑝 (26a) 𝑑 �𝑝 𝑑𝑡 = − 𝜕H 𝜕�𝑥 (26b) where �𝑥 is the position in parameter space, �𝑝 is the momentum and H is called the Hamiltonian which is defined as the sum of the potential energy and kinetic energy, Equation (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' H = 𝑈(�𝑥) + 1 2 �𝑝𝑀−1 �𝑝 (27) The momentum is defined as �𝑝 = 𝑀 · 𝑑�𝑥/𝑑𝑡 where 𝑀 is the mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The samples generated from HMC have position and momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We can then marginalize over the momentum variables leaving our position variables which are our parameters samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For all but the simplest of posterior distributions Hamilton’s equa- tions must be numerically integrated to evolve the position and mo- mentum variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A symplectic integrator with time reversibility is needed for this (Neal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since our Hamiltonian is separable we have used the leapfrog integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' By using a numerical integration scheme an error is introduced that is dependent on the integration step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Due to this a Metropolis-Hastings acceptance test must be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The acceptance probability, 𝛼, of a proposal is given by Equation (28): 𝛼 = min � 1, exp(H 𝑓 ) exp(H𝑖) � , (28) where H𝑖 and H 𝑓 are the initial and final values of the Hamilto- nian in a single proposal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In high dimensions the optimal acceptance rate for HMC is ≈ 65% (Beskos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For our particular HMC implementation we have used the standard kinetic energy function with a mass matrix including off-diagonal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This leads to sampling momenta from a multivariate normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We define the mass matrix to be independent of posi- tion (Euclidean HMC) leading to a simpler implementation that still allows us to take parameter correlations into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In such a formulation, parameter correlations that change over the parameter space cannot be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' As such, the behaviour of the posterior can significantly affect the efficiency of the sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Gener- ally, when the information content of the data is high, the parameter space close to the maximum a posteriori (MAP) point is approxi- mately Gaussian and sampling with HMC is very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Finding this portion of the parameter space, close to the MAP, can often be the toughest part of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' More sophisticated HMC routines like Riemannian Manifold Hamiltonian Monte Carlo (RMHMC, Girolami & Calderhead 2011), where the mass matrix is a function of position, exist and allow effi- cient sampling of highly non-Gaussian posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Fortunately, such a routine is not needed for our problem as our posteriors are very well approximated by multivariate normal distributions near the maxi- mum a posteriori (MAP) point, as is shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Initial Conditions By initial conditions we mean the initial position and mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The determination of these is crucial for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When using unsuitable initial positions the HMC struggles to find the typical set2 reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, even with a suitable initial position, the auto-correlation times of many parameters would be unacceptable for real-world usage/implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The solution to this second part is tuning the mass matrix to include information about the parameters scales and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Ideally (in the Euclidean HMC formulation), the mass matrix is chosen to be as close as possible to the posterior inverse covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The most difficult set of parameters to tune the initial conditions for are the satellite orbit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Very small changes in these parameters lead to large differences in the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, these parameters are very strongly correlated leading to very ineffi- cient sampling when the chosen mass matrix does not incorporate the correct correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our initial sampling location is chosen by sampling the gain am- plitudes, phases and RFI orbit parameters from the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The initial modulated RFI amplitude parameters are estimated by using the data, the other initial parameters, and the calibrator visibilities and performing a rough calculation to get RFI amplitudes per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use this estimate, at each time step, and use the same across all antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The resulting estimate is always positive which is not a problem as our likelihood cannot tell the difference between positive and negative RFI amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Now that we have a suitable initial position we are left with choos- ing a suitable mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use the block diagonalized inverse Hessian as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 for the mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Using this mass matrix we can evolve the HMC sampler until it has found the typical set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To help this process we appropriately vary the integration step size of the HMC sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once the typical set has been found 2 The typical set of a distribution is the volume of parameter space in which nearly all of the probability is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) tabascal 11 we can re-evaluate the Hessian at our MAP sample point achieved thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We continue to vary the integration step size until an optimal value is reached (leading to an acceptance rate of ≈ 65%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once this criteria is achieved everything is in place to start efficiently sampling from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' All samples prior to this point are regarded as burn-in3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Throughout the sampling the number of integration steps are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once the burn-in period is complete the integration step size and the mass matrix are also fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is done to preserve detailed balance from this point on ensuring that our samples con- verge to the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We do this for multiple chains in parallel with different random seeds to attain robust results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 3 RESULTS The goal of this section is to explore, in detail, the performance of our methods, showing that the results are as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To this end, the section is split into three subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the first subsection, we analyze the bias and standard deviations in our parameters of inter- est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We also compare results derived from MCMC and the Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the second subsection, we show the reliability and accuracy of our MCMC posteriors by calculating the uncertainty in the posterior standard deviations as well as the degree of conver- gence in our chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the final subsection, we use the estimated gains and RFI orbit parameters to estimate and subtract the RFI con- tribution in a succeeding target observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Using this method, we show the potential improvements in data retention, after flagging, and the subsequent reduction in image noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We also demonstrate how this results in deeper source extraction with more accurate flux estimates per source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Throughout this section we will often reference the bias in a pa- rameter, both relative and absolute 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For clarity, the bias refers to the difference between the estimated value, ˆ𝑥, and the true value, 𝑥true, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ˆ𝑥 − 𝑥true and the relative bias is then ( ˆ𝑥 − 𝑥true)/𝑥true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Letting ˆ𝜎𝑥 be the estimated posterior standard deviation/uncertainty in pa- rameter 𝑥, we formulate the normalised bias as ( ˆ𝑥 − 𝑥true)/ˆ𝜎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For a normally distributed, unbiased, estimator with reliable uncertainties we expect the normalised bias to conform to a standard normal distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' When referring to the standard deviation on a gain amplitude it will always be quoted in % as an uncertainty relative to the true parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Posterior Results for Calibration In this section we analyze the posterior distributions and discuss how to use their estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We show that the estimation biases are consistent with zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' are within the uncertainty estimates), and that the standard deviations reduce with increasing signal-to-noise ratio (SNR), where the signal is the total observed signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' the contaminated visibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 4 shows a summary of the data and gain estimates over a 5 minute calibration observation for which the simulation details are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The estimates in this figure are obtained by using the Laplace approximation on 10 second portions of the data in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each portion uses the same RFI parameter priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The 3 Burn-in samples are initial samples in an MCMC chain that are discarded as they would skew our posterior sample estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In our case the initial samples do not conform to detailed balance due to the variation of the integration step size and mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 4 What we refer to as bias in this text is often referred to as error in other texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 0 1 2 3 4 5 0 10 20 30 40 Visibility Amplitude [Jy] RFI Astronomical 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 Gain Amplitude Ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 28 Ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 34 0 1 2 3 4 5 Time [min] 40 50 60 70 Gain Phase [deg] Ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 28 Ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 34 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Estimated calibration solutions over time: note how the uncertainties on the gain amplitude and phase are minimised when the RFI is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The top panel shows the visibility amplitudes over time in the calibration observation for the RFI and astronomical contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The red and blue dots are the average amplitude across all baselines and the light blue shaded region shows the amplitude range across all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The strong amplitude variation of the RFI visibilities is due to the satellite passing through the primary beam sidelobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The middle and bottom panels show the Laplace estimated gain amplitudes and phases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The light blue/orange shaded error regions in these panels give the 2𝜎 credible intervals from the posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The specific example antennas shown are chosen to create a more legible plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The vertical interleaved grey and white shaded strips show the 10s data portions used in each optimization run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' gain amplitude and phase priors have the same standard deviations (relative and absolute respectively) across all data portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The gain prior means are offset by the same, relative and absolute, quantities per antenna in all portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each data portion used an individually calculated RFI visibility sampling rate, 𝑓RFI, to optimize run times and memory usage according to the required accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Equation (29) shows the calculation used to determine the minimum sampling rate assuming the average gain amplitudes are 1 and the integration time is 2 seconds per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝑓RFI = � � � mean𝑝𝑞 � max𝑡 ���𝑉RFI 𝑝𝑞 (𝑡) �� �� 3𝜎𝑛 Hz (29) 𝑉RFI 𝑝𝑞 (𝑡) is the RFI visibility on baseline 𝑝𝑞 at time 𝑡 and 𝜎𝑛 is the standard deviation of the visibility noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Due to both the minimum RFI sampling rate and inversion of an 𝑁𝑝 × 𝑁𝑝 matrix, it is more efficient to perform estimation on por- MNRAS 000, 1–19 (2023) 12 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 1 3 10 30 Mean Observed Visibility Amplitude [Jy] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 1 3 Relative Standard Deviation [%] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 1 3 Standard Deviation [deg] Gain Amplitude Gain Phase Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' How total signal, including RFI, improves calibration constraints: using a common gain parameter for both the RFI and astronomical contribu- tion (since they are within 10◦ on the sky) allows us to leverage the total signal to improve calibration constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The posterior standard deviations in gain parameters are plotted against the mean (over baseline) observed visibility amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The dots show the mean standard deviation across antennas and the shaded region of the corresponding colour show the minimum and maximum range (in uncertainty) across antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The parameter standard deviations are per time step with a 2 second integration time where the calibrator flux is 1 Jy and noise level is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='65 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The prior standard deviations were 10% and 10◦ for the amplitudes and phases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' tions of data from a memory and computation time stand point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This parallelization scheme poses many benefits including increasing ro- bustness to failures in optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Failed optimizations can be rerun with initial positions informed by the successful runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, gain solutions can be estimated on the fly instead of waiting for all the data to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A notable observation from Figure 4 is that the biases and standard deviations for the gain estimates decrease for increasing RFI visibility amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This strongly suggests that the model is able to leverage the added signal from the RFI to increase the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 5 shows this relationship clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We obtain tighter constraints on the gain parameters with increasing observed visibility amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This shows that we are using the total signal to calibrate the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The prior standard deviations, 5% and 5◦, in the gains are both larger than the posterior uncertainties, at all signal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This shows that our constraints are not strongly affected by the priors in the weak-RFI regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Figure 5, an increasing spread in posterior uncertainties for the gain phases can be observed as the mean visibility amplitudes increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is due to the spread in SNR for different antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The antennas in the core of the array contribute mostly to shorter baselines that have a larger RFI signal compared to longer baselines due to time-smearing of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 6 shows that the biases and associated standard devia- tions, on the gain parameters, are statistically consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' However, the gain phases are systematically underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is shown by the shifted mean in the normalised bias distribution over all gain phase estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 the quality of the Laplace approxima- tion is analyzed and we find the accuracy to be sufficient for practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Typically, the gain estimates from the calibration observation would be averaged per antenna under the assumption they are con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Section A2 describes how to combine our posterior estimates from different data portions as these are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The combined estimate takes into consideration the correlations between our pa- rameter estimates to maintain reliable uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once appropri- ate data portions are combined, the different time steps within a data 4 3 2 1 0 1 2 3 4 Normalised Bias : (x xtrue)/ x 10 3 10 2 10 1 100 Probability Density Gain Amplitude Gain Phase Standard Normal Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Histograms showing tabascal gain estimates are unbiased and have reliable uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Distribution in the normalised bias of the estimated gain amplitudes and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These are for the 5 minute calibration observation displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For an unbiased estimation of the parameters we expect the normalised bias to follow a standard normal distribution, which it does to good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The means and standard deviations of the normalised bias distribution for the gain amplitudes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='97 and the gain phases are −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' portion can be combined by following the recipe outlined in Section A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The second step requires the extraction of the subcovariance matrix of the parameter estimates associated with a single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' After this procedure is complete one is left with per antenna gain estimates using the full calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have not shown such combined estimates as our gains are varying over time at a rate that would violate the assumption of constant gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For our case, the preferred procedure, in our opinion, is to fit a Gaussian process to the estimates and those of the next calibra- tion observation simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The covariances of each data por- tion would be used as the noise parameter in the Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The resulting gain estimates, with covariances, for the sandwiched target observation can be used as an informative prior in the 2nd Generation Calibration5 (2GC) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 7 shows an exam- ple Gaussian process for reference where the marginalized posterior subcovariance and mean has been used to fit gain amplitudes in a single (5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=') calibration portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We find that fitting a Gaussian process (combining estimates) reduces uncertainties to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='05◦ for the gain amplitudes and phases respectively and the fit- ted model expects uncertainties to increase by an order of magnitude 20 minutes later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 8 shows a set of estimates for the RFI orbit parameters from three different data portions of the 5 minute calibration observation, as indicated in the upper right of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We see that the individual estimates are of varying quality that depend on the SNR of the data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We also see in a subset of the marginals that include the inclination parameter that the correlations across portions leading to greater constraining power when combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 9 shows the final posterior estimate after combining the individual posterior estimates according to the equations in Section A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Table 3 gives a summary of the marginal standard deviations for the priors, 10s posteriors, and 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' posterior estimates for the orbit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 5 2nd Generation Calibration is another term for selfcal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) tabascal 13 0 1 2 3 4 5 Observation Time [min] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='10 Gain Amplitude Mean function True Gain Gain Estimates Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' An example of a Gaussian process that has been fitted to the gain amplitude estimates of antenna 22 using the full posterior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The blue dots with error bars are the posterior estimates as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The orange curve and shaded region is the fitted Gaussian process with its 68% credible interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The black line is the true gain phase used in the data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The root mean squared error (RMSE) for the Gaussian process is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='46%, in the calibration portion, aligning well with its mean uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='45%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 20 minutes after the calibration observation the uncertainty and RMSE grow to around 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' t = 70-80 s t = 150-160 s t = 250-260 s 300 150 0 150 [arcsec] 40 20 0 20 [arcsec] 1500 0 1500 h [m] 10 0 10 20 30 [arcsec] 300 150 0 150 [arcsec] 40 20 0 20 [arcsec] 10 0 10 20 30 [arcsec] Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Posterior marginal distributions of the RFI orbit parameters from three different 10s data portions showing the variation in constraints and parameter correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Note the constraints on the angular parameters have improved by 2 orders of magnitude compared to the prior in Figure 3 and improve a further 2 orders of magnitudes when combined to form Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The times for the data portions are shown in the top right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The distributions and true value have been shifted to make the true parameter values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is done to make uncertainties more legible on the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 Laplace Approximation vs MCMC Analysis In this section we compare the Laplace approximation with our MCMC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use the MCMC results as the true posterior for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Later in this section, we give evidence for this claim by analyzing the MCMC chains and show their reliability as a posterior benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Parameter Name Prior 10s Posterior 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Posterior Orbit Elevation (m) 730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='000 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='000 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='464 Argument of Perigee (arcsec) 10106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='391 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='414 Orbit Inclination (arcsec) 1349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='979 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='062 Right Ascension of the Ascending Node (arcsec) 774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='384 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='068 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The mean marginal standard deviations for the priors and posteriors in each 10s data portion, as well as the final posterior marginal errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The final posterior is the combination of all data portions in the 5 minute calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 [arcsec] 150 100 50 0 50 h [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='00 [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='8 [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='00 [arcsec] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The combined posterior distribution of the RFI orbit parameters from 5 minutes of calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Note the improvement in constraints on the angular parameters compared to the prior in Figure 3 (4 orders of magnitude) and the posteriors from 10s of data in figure 8 (2 orders of magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The potential estimation bias can be considered a statistical fluctuation as this disappears in when changing any aspect of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The distributions and true value have been shifted to make the true parameter values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is done to make uncertainties more legible on the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figures 10 & 11 show the bias and posterior standard deviations on the gain parameter estimates from the Laplace approximation and MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In both figures, the upper panels show the biases and the lower panels show the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The bias tells us how accurate our best estimate is and the standard deviation is the uncertainty in the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A larger bias is not necessarily a problem as long as its associated uncertainty is proportional such that the normalised bias follows the correct statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 6 shows this to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These are the results for the 3 initial 20s portions (0-60s) of the calibration observation, described in the Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' As seen in the upper panel of Figure 10, the Laplace approxi- mation/optimization routine shows a negligible underestimate of the gain amplitudes in comparison to MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is not present in Figure 6 so we can safely assume this to be a statistical fluctuation rather than a systematic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The posterior standard deviations, in the lower panel of Figure 10, overlap so well that one only sees a muddy brown color everywhere as opposed to sections with distinct blue or orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 11 shows that the Laplace approximation is an excellent MNRAS 000, 1–19 (2023) 14 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 8 6 4 2 0 2 4 6 8 10 Relative Bias in Gain Amplitudes [%] 0 100 200 300 400 500 600 Number of Parameters 1920 Gain Amplitudes MCMC Laplace 0 1 2 3 4 Relative Standard Deviation of Gain Amplitudes [%] 0 100 200 300 400 500 600 Number of Parameters MCMC Laplace Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Comparison of biases and posterior uncertainties in our gain amplitude estimates from the Laplace approximation and MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The brown region is where the distributions overlap and only in a couple bins near the centre of the top panel can we see a discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This indicates excellent agreement between the Laplace approximated posterior and the true (MCMC) posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The top panel shows the relative biases and the bottom panel shows the posterior standard deviations of these estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each estimate is for a specific antenna and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The mean of each distribution is indicated by the dashed vertical line of the corresponding colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' estimate of the posterior distribution over the gain phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' There is near perfect agreement in both biases and the posterior errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 12 shows the marginalised posterior over the RFI orbit pa- rameters for the 0-20s portion of the calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The Laplace approximation shows excellent agreement with the true (MCMC) posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Only when looking very closely can one see that two distributions have been plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Next, we analyze the convergence of our MCMC chains to gauge the reliability of the MCMC derived posterior as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Un- fortunately, no analytical solution is available for our problem, as for most real world problems, and MCMC is the gold standard for estimating the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' An MCMC routine that follows detailed-balance and is ergodic (Neal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 2011), as our routine does, is guaranteed to converge to the true posterior distribution in the limit of infinitely many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since this is computationally intractable we must rely on ‘sufficiently many samples’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' There are standard tools available to gauge if we have ‘sufficiently many samples’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The pri- mary tool used to gauge the efficiency of an MCMC sampler is the lag-autocorrelation, denoted by 𝜌𝑡, of individual parameter sample chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' From this the effective sample size (ESS) of a chain, or set of chains, can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is then used in the estimation of the MCMC standard error on individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The MCMC standard error gives an estimate of the uncertainty in the posterior 8 6 4 2 0 2 4 6 8 10 Bias in Gain Phases [deg] 0 200 400 600 Number of Parameters 1890 Gain Phases MCMC Laplace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 Standard Deviation of Gain Phases [deg] 0 200 400 600 800 Number of Parameters MCMC Laplace Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Comparison of biases and posterior uncertainties in our gain phase estimates from the Laplace approximation and MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The distributions are indistinguishable, resulting in a brown colour as opposed to the separated blue and orange regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This shows exceptional agreement between our Laplace approximated posterior and the true (MCMC) posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The top panel shows the biases in our gain phase estimates and the bottom panel shows the posterior standard deviations of these estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each estimate is for a specific antenna and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The mean of each distribution is indicated by the dashed vertical line of the corresponding colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' standard deviation estimate, 𝜎𝑝, of parameter 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝜎 ˆ𝜎𝑥 = ˆ𝜎𝑥 √𝑁eff , 𝑁eff = 𝑁 1 + 2 �∞ 1 𝜌𝑡 (30) It is the uncertainty on the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Equation (30) gives its defini- tion and Figure 4 shows the distributions of these, as relative errors, for our different categories of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The median relative error was ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5% with less than 1% of the estimates being worse than 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' It is also standard to check the convergence of our chains to make sure this estimate would not get worse with more samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The po- tential scale reduction factor (Gelman & Rubin 1992), ˆ𝑅, also know as the Gelman-Rubin (GR) statistic, is a commonly used measure of convergence for a set of MCMC chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The closer ˆ𝑅 is to 1 the better converged it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Vats & Knudson (2021) 99% of a random sample of papers from 2017 use an ˆ𝑅 cut-off of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='01 or greater, the remaining 1% used 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Table 4 we give the minimum, median, maximum and the first last percentiles of the calculated Gelman-Rubin and split Gelman-Rubin statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These are calculated for all estimated pa- rameters (≈6000 parameters) in the 0-60s data portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Given that the 99th Percentile values for both standard and split are below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='01 we can confidently say that our MCMC chains are well converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 Application to a Target Observation In this section we analyze our method with application to a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 minute target observation that takes place shortly after the calibration obser- MNRAS 000, 1–19 (2023) tabascal 15 Laplace Posterior MCMC Posterior 80 40 0 40 80 [arcsec] 8 4 0 4 8 [arcsec] 600 0 600 1200 h [m] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 [arcsec] 80 40 0 40 80 [arcsec] 8 4 0 4 8 [arcsec] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 [arcsec] Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A corner plot of the marginalised posterior distribution of the RFI orbit parameters for a 20 second data portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The Laplace approximated posterior is barely visible underneath the true (MCMC) posterior showing excellent agreement between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The 68% and 95% credible regions are shown for the Laplace approximated posterior (blue) and MCMC posterior (green) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The distribution and true value has been shifted such that the true value is defined to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Minimum 1st Percentile Median 99th Percentile Maximum Relative MCMC Standard Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='183% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='190% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='553% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='787% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='144% Gelman-Rubin Statistic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0045 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0442 Split Gelman-Rubin Statistic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='00000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0356 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Distribution statistics for standard accuracy and convergence tests on MCMC posterior chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The 99th Percentile figures show excellent accuracy and convergence of our chains for ≈6000 parameters with only a couple of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The target phase centre is (𝛼, 𝛿) = (27◦, 15◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This position keeps the RFI location to within 10◦ on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The target observa- tion has 100 uniformly distributed point sources where intensities are drawn from an exponential distribution with mean of 100 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The gains used carry on from the model used for the calibration observa- tion with the appropriate time steps and the primary beam model is identical to the calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Figure 13 shows the visibility amplitudes (averaged over baseline) for the astronomical component and the combined contaminated visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' No noise or gains are included to clearly show the difference in the time variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the basic standard approach of 1st Generation Calibration (1GC), a target observation is initially flagged for RFI followed by ap- plying calibration solutions, from the calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' After this stage various methods may be used to try and flag any remain- ing RFI that was missed in the first round of flagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Hopefully, after these steps the astronomer is left with calibrated visibilities, that are free from RFI, that can go on to be imaged or as input for 2nd Generation Calibration (2GC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' One of the troubles with this ap- proach is that good calibration solutions are rarely available for RFI 0 1 2 3 4 5 6 7 Observation Time [min] 100 101 102 Visibility Amplitude [Jy] All Baselines AST+RFI AST Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The visibility amplitudes in the target observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We can see the near constant nature of the astronomical contribution (in red) in comparison to the RFI visibilities that vary by orders of magnitude over a 1 minute period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These are the average visibility magnitudes across all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' contaminated channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is due to the lack of uncontaminated data, in the calibration observation, available to calculate the gain solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Channels with persistent RFI may be flagged entirely for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Such is the case for the MeerKAT calibration pipeline on the baselines in the array core (|(𝑢, 𝑣)| < 1 km) as seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This problem is solved by our method directly as we are able to accurately estimate gains in the calibration observation in the pres- ence of RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Of course having good gain solutions in contaminated channels is not enough as the the visibilities in the target observation are still contaminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Here we describe and analyze a basic method to remove the visibility contribution from the RFI in a target ob- servation that follows the calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This method has many similarities with those used in Cotton (2009) except we predict the RFI visibility fringes from the orbit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We assume the RFI contribution is only from the same source that was present in the calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This demonstrates the advantage of having estimates of the parameters that describe the RFI’s motion, the orbit parameters in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To predict the RFI visibility component we take advantage of the expected phase wrapping caused by the movement of the RFI rela- tive to the image frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since we have a good estimate of the RFI orbit parameters we can reliably predict the position of the satellite and therefore the visibility phases it would induce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We do this for the entire target observation at the ‘true’ telescope sampling rate and calculate the expected visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The high resolution visibilities are then averaged down to the 2s integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We infer a normalized sub integration time, time variation per integration window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is done by linearly interpolating the mean observed visibility magni- tude and then normalizing the mean of each integration window to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Using these simulated visibilities we can determine specific base- lines on which the RFI visibility phases wrap close to an integer number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We want this as when assuming a constant ampli- tude the visibility should average to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This approximation tends to zero when considering a higher number of wraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This should leave us with only the astronomical visibility contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We found that choosing the 10 baselines with phase wraps closest to 10 (in a seven minute target observation) worked quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We then take the magnitude after subtracting the time averaged visibilities on these 10 baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This gives us an estimated RFI visibility amplitude over time on the 10 baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We average across the 10 baselines leaving MNRAS 000, 1–19 (2023) 16 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 1 10 100 1000 Mean RFI Visibility Amplitude [Jy] 0 10 20 30 40 50 60 70 80 90 100 Flag Rate [%] 18x data 3x data Noise Mean AST Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Baselines <1000m 3 Clip Standard Best tabascal 10 sec tabascal 5 min tabascal Best Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The percentage of data loss on baselines shorter than 1km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' A dramatic improvement is seen in the 5-100 Jy region where less than 10% data loss is observed when using tabascal (red and orange )compared to a 60-90% data loss when only performing calibration (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the MeerKAT pipeline these baselines are flagged without question using a static mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The red curve shows the performance of our method using the true gains and RFI orbit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The shaded regions give the 68% credible interval obtained from using estimates from different calibration data portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' us with a time varying RFI visibility amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' To estimate the RFI visibility component on all baselines and time steps, we multiply the estimated RFI visibilities, that assumed 1 Jy, from before by the time varying RFI visibility amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This method depends on the idea that we expect the RFI visibility phases to wrap over a given period on which the astronomical visibilities to remain relatively constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is not a perfect estimate for the RFI visibility component, however, it produces surprisingly good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We will refer to this combined method of calibration and RFI subtraction as tabascal or simply our method for the entirety of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='1 Flagging Improvement In this section we perform a flagging comparison on the target ob- servation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For the standard case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 1GC, we have calibrated the data using the true gain solutions, averaged over time, from the calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Flagging is then performed using 𝜎 thresh- olding, where 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='65 Jy is equal to the visibility noise, after sub- tracting the true astronomical visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For our method, tabascal, we use our averaged estimated gains from the calibration observa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' After this we apply our RFI subtraction technique described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' tabascal best or ideal refers to the use of the true gain and orbit parameters and when we reference a time it indicates the amount of data that was used to form the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Figure 14 we show a comparison of the percentage of data that is flagged using a 3𝜎 flagging threshold on baselines shorter than 1km in the target observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have varied the scale of the RFI visibilities in the target observation to show how our method compares when applied to data that has varying levels of RFI contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The best performance, when compared to the standard approach, is in the 5-100 Jy range leading to an approximate 70 percentage point decrease in data being flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This translates to approximately 3-18 times more data, in the short baselines, available for imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' It should be noted however that this is in an optimal flagging situation where the true astronomical visibilities are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In practice, for the case of MeerKAT, the data on these baselines would be completely flagged meaning our method is opening up the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 DEC Offset [deg] Uncontaminated I: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='98 mJy/beam tabascal + Flagging (Ideal) I: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='21 mJy/beam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 RA Offset [deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 DEC Offset [deg] Flagging only I: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='38 mJy/beam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='4 RA Offset [deg] tabascal + Flagging (5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=') I: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='25 mJy/beam 4 2 0 2 4 Flux mJy/beam Mean RFI Amplitude: 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='0 Jy Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Imaging comparisons of a target observation of 100 point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' tabascalmanages to reduce the RFI contamination significantly resulting in around half the image noise compared to flagging alone and only 25% more noise than an image form uncontaminated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, only 5 minutes of calibration data is needed to achieve near optimal results for the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The top left image uses true astronomical visibilities with noise added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The top right image is our method, tabascal, using the true RFI orbit and gains after 11% of the data is flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The bottom left image is the standard method, 1GC only, using the true gains with 89% of the data flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The bottom right image is our method, tabascal, using the estimated RFI orbit and gains from the 5 minute calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this case 15% of the data was flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For all flagging a 3𝜎 threshold was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' possibility of science in an untapped domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' With the effective use of the short baselines, astronomical large scale structure could now be accessed in the contaminated frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' HI intensity mapping, with radio interferometers, could now probe the small scale structures in the 1-40 arcminute range at redshift of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='092 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Figure 14 the flag rate is compared between the standard method, 1GC only, and our method, tabascal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our method performs nearly identically in the best case scenario and using 5 minutes of calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The confidence intervals, indicated by the shaded regions, are generated by sampling parameter sets from our marginal posterior distributions and applying tabascal for each sample and calculating the flag rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The comparison in Figure 14 only takes into account the short baselines forming the core of the MeerKAT array, however, these compose over 50% of all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Looking at all baselines the results looking remarkably similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The improvement that tabascal brings is slightly more pronounced on shorter baselines as the RFI amplitudes are reduced on longer baselines due to the greater time smearing/phase wrapping for the RFI contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='2 Imaging Comparison Imaging was performed using CASA’s tclean with a 4 arcsecond pixel size, with a Brigg’s weighting scheme with robustness param- 6 These values are calculated using a frequency range of 1150-1300 MHz and baseline range of 30-1000 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) tabascal 17 10 30 100 300 True Source Flux [mJy] 30 20 10 0 10 20 Error in Flux Estimation [mJy] Uncontaminated Flagging only tabascal + Flagging (5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=') Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Flux estimation errors for sources found in the images from Figure 15 using pyBDSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We see that tabascal gives near optimal (uncontaminated) results while the standard approach is both biased towards lower fluxes and has larger errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The red shaded area below about 23 mJy is where sources were completely undetected in the flagging only image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' tabascalmanages to find sources down to 16 mJy and the uncontaminated image gets to 13 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The blue markers are from the image using true astronomical visibilities with noise added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The orange markers are for the best case situation in the standard RFI flagging only approach and the green markers are for our method, tabascal, using posterior means from 5 minutes of calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' eter of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We produced 1024×1024 size images with all other parameters set to default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We image four separate situations for com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We have an uncontaminated case, a 1GC best case using perfect calibration and only flagging, and two tabascal cases where we have also applied flagging after using tabascal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The uncontaminated case uses purely astronomical visibilities with noise added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For this case we do not include any telescope response effects and no RFI contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We image only 𝑉AST 𝑝𝑞 + 𝜂𝑝𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For the 1GC standard case we take the observed visibilities, 𝑉OBS 𝑝𝑞 , and calibrate them using the true gains from the target observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' There- fore we are imaging 𝑉OBS 𝑝𝑞 /(𝐺 𝑝𝐺∗𝑞) after 3𝜎 threshold flagging is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In the tabascal cases, we initially perform 1GC, then we subtract an estimated RFI visibility component and finally flag the data, just as with the standard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In our best case we use the true gains and RFI orbit parameters and in our 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' case we use our estimates from the 5 minute calibration observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Performing imaging and source extraction with lower levels of RFI amplitudes leads reduced image noise for both the standard case and tabascalas would be expected and increases the number of found sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Additionally, the bias present in source extraction for the standard (flagging only) case reduces as the RFI amplitudes decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' As RFI amplitudes increase, the bias increases and tabascal also becomes a victim of this although to a lesser degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We find that tabascalconsistently performs better than flagging alone across all RFI amplitude ranges, both in bias and in image noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In Figure 16 we show a comparison on the source finding and flux recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For all cases we used the images in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We used pyBDSF with default settings to perform source finding and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The same imaging and source extraction settings were used for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' pyBDSF gives us source positions and fluxes all with errors among a number of other source measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Sources were matched to the true source model using astropy’s match_to_catalog_sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The error bars are generated by pyBDSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' None of the ten sources below 13 mJy were found in any of the im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our method was comparable to the uncontaminated case only finding two less source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The standard approach only found sources down to about 23 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This can be attributed to the higher image noise as a result of the higher flagging rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We see that the standard approach tends to recover less flux compared to tabascal and the uncontaminated case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The smoothed histograms on the right hand panel hand panel in Figure 16 shows the distribution of flux estima- tion errors for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The distributions have been weighted by the source flux uncertainties from pyBDSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We see that tabascal performs very similarly in terms of mean error and spread compared to the uncontaminated case, whereas, the standard (flagging only) method has both a broader distribution and systematically underes- timates source fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 4 CONCLUSIONS AND FURTHER WORK Calibration of radio interferometer arrays is a fundamental step in radio astronomy and is typically profoundly contaminated by Radio Frequency Interference (RFI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The usual approach to RFI is to simply cut out (flag) all obviously contaminated data, leading to significant data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Ideally astronomers would like to be able to effectively re- move it without losing astronomical signal, but, for moving sources MNRAS 000, 1–19 (2023) 18 Chris Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' of RFI, this has not proven possible so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this paper we show that this is possible, at least for a class of RFI moving on predictable trajectories, such as satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The key ideas that allow this progress are (1) moving sources relative to the phase center coupled with a curved wave front (near-field) model distinguish RFI from astronom- ical sources, and (2) we have a good model for the trajectory of the satellite by using TLE orbit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Our algorithm, tabascal, starts by building a forward genera- tive model of the signal parameterized by the antenna gains, satellite orbital elements, and modulated RFI amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We then compare two approaches to estimating a posterior distribution over these pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The fastest method finds the best-fitting parameters (MAP) by an optimization algorithm followed by using the Laplace (Gaus- sian) approximation to estimate the parameter uncertainties including their covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The more rigorous approach uses MCMC to find the full posterior distribution without approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We find that the Laplace approximation works very well on our simulated data having very good agreement with the full MCMC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Addi- tionally, we find the MCMC approach is computationally feasible on realistic data sizes in case it is required for dealing with real-world data complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' One of the most interesting results of our analysis is that tabascal is able to calibrate using the combined astronomical + RFI signal, thus turning the contamination into an advantage to yield more pre- cise calibration with reliable uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In application to an adja- cent target observation, tabascal uses the estimated RFI trajectory and calibration parameters to estimate and subtract the RFI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The residual data is then flagged using sigma clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For a simulated MeerKAT target observation and looking at all baselines shorter than 1km we find that for a mean RFI amplitude of 17 Jy, using tabascal leads to less than 1% loss of data com- pared to ∼ 75% data loss from an ideal 3𝜎 flagging algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' At 69 Jy the loss is 89% for the standard method and ∼ 11% for tabascal, a nearly 9× increase in data available for science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Once imaged, tabascal processed data allows recovery of faint sources that are completely missed in images from purely flagged data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' the standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Empirically we found that using tabascal halves the detection threshold relative to the standard method, bring- ing it near the ideal detection threshold for data uncontaminated by any RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Furthermore, the recovered source flux distribution from tabascalprocessed data was in line with the uncontaminated data while source fluxes recovered through flagging alone were biased towards fainter fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this work we have used tabascal in only a single frequency channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' However, it is trivially applied to multiple frequencies by running it in parallel across frequency channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Currently tabascal does not formulate the recovery of astronomical signal in the target observation as an inverse problem as it does for the calibration obser- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This is why flagging is still required after the application of tabascal to target observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The extension of this work to- wards this is already in progress and will be presented in a follow-up paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' So far we have worked with the total intensity signal, however, it is straightforward to extend this to the full polarization domain as most RFI signals are strongly polarized due to their antenna geome- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Finally, to extend this work to real observations, it is expected that a simplified perturbation model (SGP4/SDP4) may be needed to model satellite trajectories with sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' These are publicly available and can be re-implemented in JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank members of the SARAO Data Science team, Radio As- tronomy Research Group at SARAO and Niruj Mohan for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' CF and MK acknowledge funding by the Swiss Na- tional Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We also acknowledge the support of the South African Radio Astronomy Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' This research has been conducted using resources provided by the Science and Technology Facilities Council (STFC) through the Newton Fund and SARAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' DATA AVAILABILITY The data and code for recreating Figures 3 to 16 and Ta- bles 3 & 4 is made available through the following url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content='7516960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' It consists of three Python notebooks that are reasonably documented and reproduce the exact plots used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The code for simulating the data, running the optimization and MCMC routines, as well as, the Laplace approxi- mation will be made available in the future on the GitHub repository for tabascal at the url: 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correlated prior for the modulated RFI amplitudes one is able to improve the constraints on the gain amplitude estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' this is because there is a pathway for information to be shared between antennas much like when we assume the astronomical source bright- ness is the same for all antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For our prior on the modulated RFI amplitudes across antennas, at each time step, we can assume the amplitudes are drawn from a multivariate normal distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We then set a prior covariance, that includes correlation across antennas, for each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We do this as we expect the RFI amplitudes, across antennas, within a single time step to be very similar to one another with the assumptions that the primary beam patterns of the antennas are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We formulate the covariance ma- trix using a covariance function with a squared exponential kernel in the same way as a Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' We use the same covariance at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Equation (A1) gives the prior distribution of the modulated RFI amplitudes, where 𝑨RFI(𝑡 𝑗) is the vector of ampli- tudes at time step 𝑡 𝑗 and 0 is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The covariance, Σ𝐴, of the prior distribution is defined in Equation (A2) where �𝑥𝑝 is the position of antennas 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' For example, with the variance, 𝜎2 𝐴, and the length scale, 𝑙𝐴, of the covariance function set to 10 000 Jy and 10 km respectively, we allow a large variation in amplitude but impose strong correlations across the entire antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝑨RFI(𝑡 𝑗) ∼ N (0, Σ𝐴) (A1) (Σ𝐴) 𝑝𝑞 � 𝜎2 𝐴, 𝑙𝐴 � = 𝜎2 𝐴 exp � − |�𝑥𝑝 − �𝑥𝑞|2 2𝑙2 𝐴 � (A2) A2 Independent Measurements Let us have 𝑁 independent measurements of an 𝑚-dimensional vari- able 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each measurement, 𝑥𝑛, is normally distributed with mean 𝜇𝑥𝑛 and covariance 𝐶𝑥𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The combination of all N measurements is simply the normalised product of their probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Since each measurement is normally distributed the combined probability density will also be a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Note that 𝑛 is used to index individual measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' 𝑥𝑛 ∼ N (𝜇𝑥𝑛, 𝐶𝑥𝑛) (A3) 𝑥 ∼ N (𝜇𝑥, 𝐶𝑥) = � 𝑛 N �𝜇𝑥𝑛, 𝐶𝑥𝑛 � (A4) 𝐶𝑥 = �∑︁ 𝑛 𝐶−1 𝑥𝑛 �−1 (A5) 𝜇𝑥 = 𝐶𝑥 �∑︁ 𝑛 𝐶−1 𝑥𝑛 𝜇𝑥𝑛 � (A6) A3 Correlated Measurements Let us have 𝑁 correlated measurements of a 1-dimensional variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Each measurement is denoted by 𝑥 𝑗 and the total error covariance of all measurements is 𝐶𝑥, of dimension 𝑁 × 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The combined measurement ¯𝑥 is calculated using Equation (A7) with its associated error variance 𝜎2 ¯𝑥 as calculated using Equation (A8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' Note the 𝑗, 𝑘 and 𝑙 subscripts are used for indices of associated measurement vector and error matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The following equations are taken from Avery (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' ¯𝑥 ¯𝑥 = ∑︁ 𝑗 𝑤 𝑗𝑥 𝑗 (A7) 𝜎2 ¯𝑥 = ∑︁ 𝑗𝑘 𝑤 𝑗𝑤𝑘 (𝐶𝑥) 𝑗𝑘 (A8) where 𝑤 𝑗 is defined by Equation (A9) below 𝑤 𝑗 = � 𝑘 � 𝐶−1 𝑥 � 𝑗𝑘 � 𝑘𝑙 � 𝐶−1 𝑥 � 𝑘𝑙 (A9) A4 Independent 1-D Measurements To check our solutions for both Sections A2 & A3 we can consider a set of independent 1-dimensional measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' In this case our measurement are denoted by 𝑥 𝑗 and our error covariances become (𝐶𝑥) 𝑗𝑘 = 𝛿 𝑗𝑘𝜎2 𝑗 and (𝐶𝑥𝑗 ) = 𝜎2 𝑗 where 𝜎2 𝑗 are the individual error variances for each measurement 𝑥 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' The combined measurement ¯𝑥 with error variance 𝜎2 ¯𝑥 is calculated using Equations (A10) ¯𝑥 = 𝜎2 ¯𝑥 ∑︁ 𝑗 𝜎−2 𝑗 𝑥 𝑗 (A10) 𝜎2 ¯𝑥 = 1 � 𝑗 𝜎−2 𝑗 (A11) This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE2T4oBgHgl3EQf5Qgr/content/2301.04188v1.pdf'} diff --git a/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf b/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a14fc8a05fe1d9859ff1b35dfe675f873408a821 --- /dev/null +++ b/2dAzT4oBgHgl3EQfRvud/content/2301.01221v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 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a/7dE3T4oBgHgl3EQfRgmQ/content/tmp_files/2301.04422v1.pdf.txt b/7dE3T4oBgHgl3EQfRgmQ/content/tmp_files/2301.04422v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e517885ce33f91113ac8ba72df919efce197dc5 --- /dev/null +++ b/7dE3T4oBgHgl3EQfRgmQ/content/tmp_files/2301.04422v1.pdf.txt @@ -0,0 +1,847 @@ +Optical Flow for Autonomous Driving: Applications, Challenges +and Improvements +Shihao Shen 1, Louis Kerofsky 2 and Senthil Yogamani 3 +1Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S. +2Qualcomm Technologies, Inc., San Diego, California, U.S. +3Automated Driving, QT Technologies Ireland Limited. +ABSTRACT +Optical flow estimation is a well-studied topic for automated +driving applications. Many outstanding optical flow estimation +methods have been proposed, but they become erroneous when +tested in challenging scenarios that are commonly encountered. +Despite the increasing use of fisheye cameras for near-field sens- +ing in automated driving, there is very limited literature on optical +flow estimation with strong lens distortion. Thus we propose and +evaluate training strategies to improve a learning-based optical +flow algorithm by leveraging the only existing fisheye dataset with +optical flow ground truth. While trained with synthetic data, the +model demonstrates strong capabilities to generalize to real world +fisheye data. The other challenge neglected by existing state-of- +the-art algorithms is low light. We propose a novel, generic semi- +supervised framework that significantly boosts performances of +existing methods in such conditions. To the best of our knowl- +edge, this is the first approach that explicitly handles optical flow +estimation in low light. +I +INTRODUCTION +Advancement in the field of computer vision has enabled the +rapid development of perception systems for autonomous vehicles +(AV) in recent years. Optical flow estimation, known as the study +of how to estimate per-pixel 2D motion between two temporally +consecutive frames, is one of the fundamental problems in com- +puter vision that are widely used in autonomous driving. Specif- +ically, optical flow estimation helps vehicles perceive the tempo- +ral continuity of the surrounding environment and hence it plays +significant roles in time-series-based tasks such as object track- +ing [1, 2], visual odometry [3], semantic segmentation [4], motion +segmentation [5], and SLAM systems [6], to point out a few. Horn +and Schunck [7] introduce the first method to compute optical +flow through energy minimization and many excellent methods +obtain better results based on it. However, the optimizing problem +of a complex objective is usually computationally expensive in +terms of real-time applications such as AV. To achieve faster and +more reliable performance, end-to-end neural networks are pro- +posed [8, 9, 10, 11]. These data-driven learning-based methods +are more efficient and robust against challenges, such as occlu- +sions, large displacement and motion blur, that break the bright- +ness constancy and small motion assumptions traditional methods +are built upon. Nevertheless, there are still a few unique chal- +lenges in AV applications that have been neglected by existing +state-of-the-art methods. In this paper, we investigate two com- +Figure 1: Erroneous optical flow estimation by feeding fisheye images into +off-the-shelf RAFT [11]. From left to right in each row: current frame, +next frame, color coded result, sparse vector overlay plots for better visu- +alization. Note how the estimated flow vectors on the ground are either +missing or inconsistent with the vehicle motion. +monly encountered challenges among them and propose the solu- +tions respectively: lens distortion and low-light scenes. +Near-field sensing is a ubiquitous topic for automated driv- +ing. +Some primary use cases are automated parking systems +and traffic jam assistance systems. Near-field sensing is usually +achieved by building a surround-view system with a number of +wide-angle cameras that come with strong radial distortion. For +example, fisheye cameras offer a significantly wider field-of-view +(FoV) than standard pinhole cameras, and in practice four fish- +eye cameras located at the front, rear, and on each wing mirror +are sufficient to build a surround-view system for a full-size ve- +hicle [12]. Although such fisheye systems are widely deployed, +to the best of our knowledge, there is no previous work explic- +itly handling optical flow estimation on images with strong lens +distortion, such as fisheye imagery. As shown in Figure 1, one +of the current state-of-the-art methods [11] shows erroneous re- +sults when taking in fisheye images from WoodScape [13] due to +its focus on narrow field-of-view cameras with mild radial distor- +tion only. An intuitive way to solve this is to correct the distor- +tion in the input images as a preprocessing step before passing +through the neural network. However, this inevitably leads to re- +duced field-of-view and resampling distortion artifacts at the pe- +riphery [14]. Without rectification, building an automotive dataset +is the major bottleneck in optical flow estimation on fisheye im- +agery. Very few synthetic datasets provide optical flow ground +truth associated with fisheye images [15], whereas no real-world +dataset exists with optical flow ground truth. This is due to the +fact that per-pixel motion between every two consecutive frames +is extremely difficult to be manually labelled. Simulators [16, 17] +can readily generate background motion but dynamic foreground +objects need to be explicitly taken care of. In this paper, we inves- +arXiv:2301.04422v1 [cs.CV] 11 Jan 2023 + +tigate and boost the performance of RAFT on strongly distorted +inputs by making use of the only existing dataset with optical flow +groundtruth, SynWoodScape [15]. +Most AV applications are expected to operate not only dur- +ing the day but also at night. Cameras become unreliable and +camera-based computations are prone to failure under low-light +conditions due to its susceptibility to noise and inconsistent expo- +sure. Alternatively, LiDAR sensors can perform robustly in low- +light autonomous driving [18] because active sensors that measure +the time-of-flight of the emitted lasers are independent of illumi- +nation. However, LiDAR is bulky, costly, and requires much more +computation as well as memory resources to process the output, +which makes it inferior to cameras if the latter can provide equiv- +alently reliable results in low light. Thermal cameras [19] provide +robust low light performance but they are not commonly used in +recent automated driving systems. Current optical flow methods +show poor capabilities of dealing with low-light data because low +light is a complex scenario coming with low signal-to-noise ratio, +motion blur and local illumination changes brought by multiple +light sources. In addition, current optical flow datasets [20, 21, 22] +are dominated by daytime images. +In this paper, we propose +a novel, generic architecture that facilitates learning nighttime- +robust representations in a semi-supervised manner, without the +help of any extra data or sacrificing the daytime performance. To +the best of our knowledge, this is the first learning-based method +that explicitly handles optical flow estimation in low light. The +main contributions of this paper are: +1. Introduction and investigation of two challenges in optical +flow estimation for AV applications: strong lens distortion +and low-light scenes. +2. Implementation and improvement of a baseline optical flow +algorithm on fisheye inputs and experimental evaluation. +3. Implementation of an effective but also generic framework +of novel strategies to learn nighttime-robust representations +for learning-based optical flow algorithms. +The paper is organized as follows. Section II discusses re- +lated work on optical flow estimation in the automotive industry +and existing attempts to solve the two aforementioned challenges. +Section III describes the implementation of our proposed flow es- +timation algorithms for fisheye and low-light inputs respectively, +as well as presents the experimental evaluation and results anal- +ysis. Finally, Section IV discusses the remaining challenges for +flow estimation in AV applications and concludes the paper. +II +RELATED WORK +Optical Flow Estimation: Traditional solutions have been stud- +ied and adapted for decades [7, 23]. In order to be robust against +more challenging open world problems including lack of features, +motions in different scales, and occlusions, recent learning-based +methods outperform traditional ones. Dosovitskiy et al. [8] pro- +pose FlowNetS and FlowNetC, which is a pioneer work in show- +ing the feasibility of directly estimating optical flow given im- +ages. Sun et al. [9] design PWC-Net, a much more efficient solu- +tion based on pyramidal processing, warping and the use of a cost +volume. RAFT [11], proposed by Teed and Deng, demonstrates +notable improvement by building multi-scale 4D correlation vol- +umes for all pairs of pixels and iteratively updating flow estimates +through refinement module based on gated recurrent units (GRU). +All these methods are fully supervised and trained using imagery +from a standard pinhole camera. The training data are also col- +lected during the day with sufficient brightness. None of them +pays attention to the performance of optical flow in more chal- +lenging AV applications such as strong lens distortion and driving +at night, which leads to errors and even catastrophic failures. +Strong Lens Distortion: There is very limited work on percep- +tion tasks for strongly distorted images such as fisheye images. +Popular approaches include rectifying the radial distortion before +passing images into any regular perception pipeline. However, +this will inevitably bring reduced field-of-view and resampling +distortion artifacts especially at the image borders [14]. +Spa- +tially variant distortion that makes closer objects appear larger +also poses scaling problems and complexity to geometric percep- +tion tasks. Additionally, Rashed et al. [24] show that the com- +mon use of bounding boxes for object detection no longer fit well +for rectangular objects in distorted images. More sophisticated +representations for detected objects, such as a curved bounding +box exploiting the known radial distortion, are explored in [25]. +Although there is some literature using distorted images with- +out rectification on other perception tasks, such as depth estima- +tion [26, 27], soiling [28], visual odometry [29] and multi-task +models [30, 31], there is no previous work estimating optical flow +due to the difficulty in labeling ground truth. WoodScape [13], +KITTI 360 [32] and Oxford RobotCar [33] are some well-known +autonomous driving datasets containing strongly distorted im- +ages such as fisheye images, but none of them has optical flow +ground truth. In this paper, we take advantage of the synthetic +fisheye dataset published recently, SynWoodScape [15], which is +the first dataset providing optical flow for both foreground and +background motions by computing it analytically using other data +modalities extracted from the simulator. We train our network +using synthetic data from SynWoodScape and evaluate it on real- +world fisheye data from WoodScape. +Low-Light Scenes: Similar to optical flow estimation on strongly +distorted images, there is some work handling low light in a few +perception tasks [18, 34, 35] but none of them has proposed an +optical flow estimation algorithm that is robust against low-light +scenes. Very related to ours, Zheng et al. [36] propose a method +to synthesize low-light optical flow data by simulating the noise +model on dark raw images, which is then used to finetune an off- +the-shelf network. However, their method is not able to synthe- +size more realistic characteristics of real-world low-light scenes +one would observe in AV applications, such as the motion blur +and local illumination changes brought by multiple light sources. +Their improvement is also very limited due to the off-the-shelf +network is not designed nor trained to learn nighttime-robust rep- +resentations. In addition, a variety of techniques have been de- +veloped for low-light image enhancement [37, 38] and image-to- +image translation [39, 40]. The former can preprocess inputs to +a flow estimation network during inference by brightening up a +given low-light image, whereas the latter can translate a daytime +image into its nighttime counterpart so as to complement the lack +of optical flow datasets in low light [18]. But neither approach fa- +cilitates the network training in that the processed data bring in ex- +tra complexities such as additional artificial noise, overexposure, +or inconsistent image translation across frames. Finally, semi- +supervised learning is a common approach to tackling the lack of + +Figure 2: Optical flow estimation (color coded) on real-world automotive data from WoodScape [13]. Input frames are from the fisheye cameras of front +view, right-side view, and left-side view respectively. +optical flow data in particular scenarios, where a set of predefined +transformations are applied to the original labeled data and the +output of the perturbed data are enforced to agree with the out- +puts of the original data [41]. For example, Jeong et al. [42] use +a semi-supervised setup to impose translation and rotation con- +sistency equivariance for optical flow estimation. Yan et al. [43] +synthesize foggy images from clean and labelled images in or- +der to avoid flow estimation errors caused in dense foggy scenes. +Similar to these semi-supervised methods, we incorporate low- +light consistency that facilitates learning explicit nighttime-robust +representations without additional labeling. +III +PROPOSED ALGORITHMS AND RESULTS +In this section, we describe the two proposed optical flow es- +timation algorithms for strongly distorted inputs and low-light in- +puts respectively. We also present the corresponding experimental +evaluation and results analysis. +III. A +Strong Lens Distortion +The limited availability of datasets with strong lens distortion +is the bottleneck that prevents recent methods from generalizing +to more distorted inputs. With the help of SynWoodScape [15], +the first fisheye dataset providing optical flow ground truth for +both foreground and background motions, we are able to train +an optical flow model, using RAFT [11] as the backbone, that +generalizes well on strongly distorted lenses without sacrificing +its original performance on pinhole cameras. +We run the off-the-shelf RAFT on real-world fisheye auto- +motive dataset, e.g. WoodScape [13] and we find sharp and incon- +sistent optical flow estimation, which is especially illustrated on +the ground plane in Figure 1. To solve this, we provide two base- +lines and their qualitative as well as quantitative evaluation. One +is to finetune the pretrained RAFT using SynWoodScape, follow- +ing the training schedule in Table 1a. The other is to jointly train +RAFT on both SynWoodScape and images from pinhole cam- +era that are regularly used in learning-based optical flow meth- +ods [8, 20, 21, 22, 44]. The jointly training baseline follows the +training schedule in Table 1b. +Table 1: Details of the training schedule. Column header abbreviations: +LR: learning rate, BS: batch size, WD: weight decay, CS: crop size. Train- +ing dataset abbreviations: C: FlyingChairs, W: SynWoodScape, S: Sintel, +T: FlyingThings3D, K: KITTI-2015, H: HD1K. +(a) Finetuning baseline. During the Sintel stage, the dataset distribution is +S(.67), T(.12), K(.13), H(.08). +Stage +Weights +Dataset +LR +BS +WD +CS +Chairs +- +C +4e-4 +6 +1e-4 +[368, 496] +Things +Chairs +T +1.2e-4 +3 +1e-4 +[400, 720] +Sintel +Things +S+T+K+H +1.2e-4 +3 +1e-5 +[368, 768] +Finetune +Sintel +W +1e-4 +3 +1e-5 +[600, 800] +(b) Jointly training baseline. During the Joint stage, the dataset distribu- +tion is W(.65), S(.17), T(.13), K(.03), H(.02). +Stage +Weights +Dataset +LR +BS +WD +CS +Chairs +- +C +4e-4 +6 +1e-4 +[368, 496] +Things +Chairs +T +1.2e-4 +3 +1e-4 +[400, 720] +Joint +Things +W+S+T+K+H +1e-4 +3 +1e-5 +[368, 768] + +Finetuned on SynWoodScape +Current Frame +Pretrained on Sintel +Jointly TrainedFigure 3: Overview of our proposed framework. During training, the framework takes two consecutive frames as input and passes them through a set +of low-light-specific data augmentations as well as applies a random illumination mask. Then the optical flow estimator estimates flow on two pairs of +augmented frames in parallel. The network is supervised by two losses: the conventional optical flow loss and the novel brightness consistency loss. +During inference, the input frames are directly passed into the estimator which outputs optical flow, as is the standard way in the existing state of the art. +We then show the quantitative results in Table 2. We use +the endpoint error (EPE) as the metric, which is the standard er- +ror measure for optical flow estimation. It is the Euclidean dis- +tance between the estimated flow vector and the ground truth, +averaged over all pixels. We evaluate the two baselines (Stages +”Finetune” and ”Joint”) described above along with the pretrained +model (Stage ”Sintel”) provided by the author on four hold-out +test sets from SynWoodScape, Sintel (clean and final passes), and +KITTI. SynWoodScape is the only test set of strongly distorted +inputs, while the other three assume a pinhole camera model with +very little distortion. Although the pretrained model gives out- +standing performance on pinhole cameras, its performance sig- +nificantly drops on fisheye inputs. Our first baseline, the one fine- +tuned on fisheye images, gives the best result on SynWoodScape +but has very poor performance on the others. This matches our +expectation because both the pretrained and the finetuned mod- +els are trained to the best for pinhole camera and fisheye camera +respectively, without taking generalization into account. On the +other hand, our second baseline, the jointly trained model, keeps +the second best while being very close to the best score on all four +datasets. Therefore, jointly training provides a straightforward yet +strong baseline that generalizes well over lenses with distinct dis- +tortions. +Table 2: Endpoint-error results on datasets with diverse lens distortion. +Stage +SynWoodScape +Sintel - Clean +Sintel - Final +KITTI +Sintel +5.12 +1.94 +3.18 +5.10 +Finetune +1.40 +5.44 +10.32 +14.34 +Joint +1.48 +2.44 +4.14 +7.31 +In Figure 2, we further show their qualitative results on +WoodScape that support the improvements we obtain by jointly +training RAFT on a mixture of lens distortions. In the front view +case, note how the jointly trained model is able to consistently +estimate the flow on the ground as is the major failure of recent +methods shown in Figure 1. The results on side-view cameras also +show the jointly trained model captures finer details than its fine- +tuned counterpart. For example in the right-side view, not only the +inconsistency on the ground is solved, but optical flow associated +with the bicycle wheel in the upper right corner is also clearly +estimated. In the left-side view, the finetuned model misses the +flow associated with the vehicle’s front wheel, which is captured +by the pretrained model, but the jointly trained model ”regains” +such detailed estimations. In other words, the finetuned model es- +timates more consistent optical flow, which poses a challenge to +the pretrained model due to markedly different projection geome- +tries between fisheye and pinhole cameras, but in return, it loses +some details observed by the pretrained model because interesting +local features become much less significant given the strong lens +distortion. However, the jointly trained model achieves a great +trade-off among the previous two: it re-captures the details lo- +cally while maintaining good performance globally across differ- +ent camera views. +III. B +Low-Light Scenes +We propose a novel and generic semi-supervised framework +that significantly boosts performances of existing state-of-the-art +methods in low light conditions. Figure 3 shows the architec- +ture of the framework. The benefit of our framework is three- +fold. First, it is independent from the design of the existing meth- +ods, so one can apply it generically to an estimator of his choice +(e.g. [8, 9, 11, 45]) and augment its nighttime performance out of +the box. Second, semi-supervised learning does not require any +extra data as the labeling cost for nighttime optical flow datasets is +immense. Lastly, it maintains the estimator’s competitive perfor- +mance on the original daytime data without making any trade-off. +We first break down the root causes of failures in optical flow +estimation under low light and then describe our proposed strate- +gies in the framework that address these root causes accordingly: +1. the complex noise model of images captured at night, +2. severe motion blur caused by longer exposure time, +3. inconsistent local brightness brought by multiple indepen- +dent light sources in the scene. +Images captured in low light tend to have more complex +noises than those captured with sufficient ambient light. Such +noises are never synthesized in the data augmentation step by ex- +isting methods, which is the first reason why the optical flow esti- +mators fail in low light. Similar to [36], we decompose the noise +model in low light as an aggregate of the photon shot noise and + +Ground Truth Flow +Photometric +Motion Blur +Input Pair w/o Brightness Mask +Original Input +Low-light Noise +It+1 +Occlusion +Estimator +Estimated Flow +Supervised Loss +Spatial +Ls +V +Shared Weights +Data +Augmentation +Input Pair w/ Brightness Mask +I! +Local Brightness +Estimator +Estimated Flow +RE +Consistency Loss +Lb +Random MaskFigure 4: Effects of low-light noise augmentation and motion blur aug- +mentation. +thermal noise. The former is due to the changing amount of pho- +tons hitting the sensor with different exposure levels and pixel lo- +cations. The photon shot noise is approximated by a Poisson dis- +tribution. Thermal noise refers to the noise in readout circuitry in +the sensor and is approximated by a Gaussian distribution. There- +fore, we synthesize the low-light noise onto the input frames as +one extra data transform in the data augmentation step. Specif- +ically, we sample Poisson and Gaussian parameters, (a,b), from +ranges observed in real-world low-light images, formulate it into +a single heteroscedastic Gaussian (Equation 1), and apply it to an +input frame I. With probability 0.5, the low-light noise augmen- +tation is performed on each pair of consecutive frames. +I(x) = N +� +µ = x,σ2 = ax+b +� +(1) +Motion blur is another root cause we need to address when +estimating optical flow in low light. In order to mimic the blurring +effects caused by longer exposure length, we generate authentic +motion blur kernels using Point Spread Functions (PSF) at dif- +ferent kernel sizes and intensities. The intensity determines how +non-linear and shaken the motion blur looks. Similar to low-light +noise, we apply the authentic blurring to a pair of input frames +as one extra data augmentation, with probability 0.6. An illustra- +tion of the two introduced data augmentation strategies is shown +in Figure 4. +Inconsistent local brightness is the last but not least root +cause. This is due to multiple independent lighting sources exist- +ing in a low-light scene (street light, headlight, moonlight, etc.), +which leads to uneven bright areas in an image. For example, +the ground plane in the original input in Figure 3 is illuminated +only in front of vehicles’ headlights but remains dark elsewhere. +Unlike in the daytime where sun is the dominant lighting source, +images captured at night have inconsistent local brightness even +on the same object. Because optical flow is estimated by match- +ing pixels across two images, such inconsistencies cause exist- +ing methods to fail easily. For instance, in Figure 5, the first +row shows the catastrophic failure of RAFT when a pedestrian +walks from the dark into the vehicle headlight and his illumina- +tion changes drastically across frames. In order to resolve this, we +resort to semi-supervised learning. Similar to [42], we also adopt +the cow-mask [46] to create sufficiently random yet locally con- +nected illumination patterns as the inconsistent local brightness +occurs in any size, shape and position in images while exhibiting +Figure 5: Optical flow estimation on low-speed sequences from CU- +Lane [47]. +locally explainable structures, depending on the driving environ- +ment and the time. We apply the same binary mask to the original +pair of input frames and randomly adjust the brightness of pix- +els according to the mask. The true area of the binary mask is +uniformly sampled from 40% to 70% of the image. With a proba- +bility of 0.5, we increase the absolute brightness of the true area, +whereas in the remaining time we increase the brightness of the +false area. Finally, we introduce the local brightness consistency +regularization. We use (I′t,I′ +t+1) and (It,It+1) to denote the input +pair after data augmentation with and without applying a random +brightness mask. Both passes in Figure 3 are independent ex- +cept that the spatial transform is shared in order to keep the same +cropped areas for consistency loss calculation. The local bright- +ness consistency loss is calculated as follows +Lb = +��Estimator(It,It+1)−Estimator +� +I′ +t,I′ +t+1 +���2 +2 . +(2) +This regularization explicitly constrains the network to output +consistent optical flow on (I′t,I′ +t+1) as on (It,It+1), which enforces +illumination invariance between the estimated optical flow for the +original pair the estimated optical flow for the randomly trans- +formed pair. Note how this semi-supervised approach is different +from simply adjusting brightness randomly as another data aug- +mentation scheme, which expand training samples without impo- +sition of a sophisticated consistency loss during training. +We choose RAFT [11] as the estimator and we supervise our +network on the aggregated loss L = Ls +Lb. Ls is the l1 distance +between the predicted flow ˜f i(It,It+1) and ground truth flow ft +over all iterations i, as in [11]: +Ls = +N +∑ +i=1 +γN−i ��� ˜f i(It,It+1)− ft +��� +1 . +(3) +Due to the lack of nighttime data with optical flow ground +truth, we are restricted to qualitatively evaluating our approach, +which we call RAFT-Dark for short. We use CULane [47], a large +automotive dataset containing a lot of challenging real-world low +light sequences. In Figure 5, we show the comparison between +vanilla RAFT and RAFT-Dark on some low-speed sequences. +RAFT-Dark demonstrates superior performance to RAFT. In the + +Original ImagePair +After Low-Light Noise Augmentation +Original ImagePair +AfterMotionBlurAugmentationInput +RAFT +RAFT-Dark (Ours)Figure 6: Optical flow estimation on high-speed sequences from CU- +Lane [47]. +first, third and fourth rows, RAFT-Dark is able to detect motions +associated with pedestrians and vehicles that either experience +some drastic illumination change or appear to be too dark and +noisy. In the other cases, note how RAFT-Dark gives a signif- +icantly better estimation on the ground plane as well as the di- +rections and magnitudes that are consistent with the ego vehi- +cle’s motion. For convenience, a color coding wheel to visualize +per-pixel optical flow vectors is attached to the top right corner: +color denotes direction of the flow vector while intensity denotes +length of the displacement. Since the ego vehicle always heads +forward, the ground truth optical flow vectors in the front cam- +era’s image should intuitively point to the image boundaries and +away from the image center. And due to the motion parallax, one +should expect larger magnitudes of flow vectors toward the im- +age boundaries and small magnitudes around the image center. +In other words, although we have no access to numerical ground +truth flow, we know the color coded ground truth should exhibit +the same pattern as the color wheel: bluish or greenish on the +left side while reddish or yellowish on the right side of the image. +With this in mind, RAFT fails to estimate correct optical flow con- +sistent with the vehicle’s motion, especially in background areas +such as the ground plane. On the other hand, RAFT-Dark not only +performs well on these areas but also learns to separate the dark +sky in some cases and to capture details such as the street light +in the second row. Such improvements are further illustrated in +high-speed sequences from CULane in Figure 6. +Our framework of learning strategies enables RAFT to im- +prove estimation accuracy by more than 50% on average (based +on visual observations), and even solves some catastrophic fail- +ures. Although we show our results based on RAFT as the esti- +mator, our framework is generic and one can replace RAFT with +any existing state-of-the-art method of one’s choice. +III. C +Discussion +The goal of this work is to emphasize the importance of +addressing optical flow challenges which are not well explored +in automated driving. We investigate two of them and propose +our solutions accordingly, but the others require further research. +Lack of data tends to be the major bottleneck for most data-driven +optical flow algorithms. We are able to leverage synthetic data to +improve existing methods’ adaptation of various lens distortions +but the sim-to-real gap still exists when these methods are evalu- +ated on real world fisheye data. Optical flow in low light cannot +be addressed in the same way without any synthetic data avail- +able. We experiment image enhancement prior to the network +inference, but it leads to even worse results because enhancement +happens per frame rather than per pair of frames and temporal +consistency is easily broken. Without any extra data, our approach +takes full advantage of publicly available data and simulates three +root causes through novel data augmentation schemes and semi- +supervised learning. However, low light is merely one of many +scenarios that make optical flow estimation harder. Others in- +clude foggy, rainy or snowy weather [48]. A unified and robust +approach aiming for all these cases is encouraged and we see it +also as an opportunity for further investigation by the community. +IV +Conclusion +Both lens distortion and low light are important problems for +higher levels of automated driving, but they are not explored in +detail in the optical flow community as there is no public dataset +available. Thus we propose our approaches to these two respec- +tively. We implement and improve a state-of-the-art optical flow +algorithm by training it on synthetic fisheye data and demonstrat- +ing its adaptation to real-world distorted images as well as gen- +eralizability over various lens distortions. We implement a novel, +generic framework that facilitates learning nighttime-robust rep- +resentations in a semi-supervised manner, which shows superior +performance to the existing state of the art. 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Yogamani, “Weather and +light level classification for autonomous driving: Dataset, baseline +and active learning,” in 2021 IEEE International Intelligent Trans- +portation Systems Conference (ITSC), pp. 2816–2821, IEEE, 2021. +AUTHORS BIOGRAPHY +Shihao Shen is a second-year graduate student in the Robotics +Institute at Carnegie Mellon University and expects to receive his +M.Sc. in Robotic Systems Development in 2023. He worked as +an Interim Engineering Intern in the Multimedia Research and +Development department at Qualcomm in summer 2022 and this +is his work done during his internship. His main research focus +is machine learning with applications in computer vision as well +as simultaneous localization and mapping (SLAM). +Louis Kerofsky is researcher in video compression, video +processing and display. He received M.S. and Ph.D. degrees in +Mathematics from the University of Illinois, Urbana-Champaign +(UIUC). He has over 20 years of experience in research and al- +gorithm development and standardization of video compression. +He has served as an expert in the ITU and ISO video compression +standards committees. He is an author of over 40 publications +which have over 5000 citations. He is an inventor on over 130 +issued US patents. He is a senior member of IEEE, member of +Society for Information Display. +Senthil Yogamani is an artificial intelligence architect for au- +tonomous driving and holds a principal engineer position at +Qualcomm. He leads the research and design of AI algorithms +for various modules of autonomous driving systems. He has over +17 years of experience in computer vision and machine learn- +ing including 14 years of experience in industrial automotive sys- +tems. He is an author of 110+ publications which have 4000+ +citations and 150+ inventions with 85 filed patent families. He +serves on the editorial board of various leading IEEE automotive +conferences including ITSC and IV and advisory board of various +industry consortia including Khronos, Cognitive Vehicles and IS +Auto. He is a recipient of the best associate editor award at ITSC +2015 and best paper award at ITST 2012. + diff --git a/7dE3T4oBgHgl3EQfRgmQ/content/tmp_files/load_file.txt b/7dE3T4oBgHgl3EQfRgmQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..73d4ad4b4d09f741479b1ebd13737e9cb31a1e03 --- /dev/null +++ b/7dE3T4oBgHgl3EQfRgmQ/content/tmp_files/load_file.txt @@ -0,0 +1,632 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf,len=631 +page_content='Optical Flow for Autonomous Driving: Applications, Challenges and Improvements Shihao Shen 1, Louis Kerofsky 2 and Senthil Yogamani 3 1Carnegie Mellon University, Pittsburgh, Pennsylvania, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' 2Qualcomm Technologies, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=', San Diego, California, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' 3Automated Driving, QT Technologies Ireland Limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' ABSTRACT Optical flow estimation is a well-studied topic for automated driving applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Many outstanding optical flow estimation methods have been proposed, but they become erroneous when tested in challenging scenarios that are commonly encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Despite the increasing use of fisheye cameras for near-field sens- ing in automated driving, there is very limited literature on optical flow estimation with strong lens distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Thus we propose and evaluate training strategies to improve a learning-based optical flow algorithm by leveraging the only existing fisheye dataset with optical flow ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' While trained with synthetic data, the model demonstrates strong capabilities to generalize to real world fisheye data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The other challenge neglected by existing state-of- the-art algorithms is low light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We propose a novel, generic semi- supervised framework that significantly boosts performances of existing methods in such conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' To the best of our knowl- edge, this is the first approach that explicitly handles optical flow estimation in low light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' I INTRODUCTION Advancement in the field of computer vision has enabled the rapid development of perception systems for autonomous vehicles (AV) in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Optical flow estimation, known as the study of how to estimate per-pixel 2D motion between two temporally consecutive frames, is one of the fundamental problems in com- puter vision that are widely used in autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Specif- ically, optical flow estimation helps vehicles perceive the tempo- ral continuity of the surrounding environment and hence it plays significant roles in time-series-based tasks such as object track- ing [1, 2], visual odometry [3], semantic segmentation [4], motion segmentation [5], and SLAM systems [6], to point out a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Horn and Schunck [7] introduce the first method to compute optical flow through energy minimization and many excellent methods obtain better results based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, the optimizing problem of a complex objective is usually computationally expensive in terms of real-time applications such as AV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' To achieve faster and more reliable performance, end-to-end neural networks are pro- posed [8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' These data-driven learning-based methods are more efficient and robust against challenges, such as occlu- sions, large displacement and motion blur, that break the bright- ness constancy and small motion assumptions traditional methods are built upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Nevertheless, there are still a few unique chal- lenges in AV applications that have been neglected by existing state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In this paper, we investigate two com- Figure 1: Erroneous optical flow estimation by feeding fisheye images into off-the-shelf RAFT [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' From left to right in each row: current frame, next frame, color coded result, sparse vector overlay plots for better visu- alization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Note how the estimated flow vectors on the ground are either missing or inconsistent with the vehicle motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' monly encountered challenges among them and propose the solu- tions respectively: lens distortion and low-light scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Near-field sensing is a ubiquitous topic for automated driv- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Some primary use cases are automated parking systems and traffic jam assistance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Near-field sensing is usually achieved by building a surround-view system with a number of wide-angle cameras that come with strong radial distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' For example, fisheye cameras offer a significantly wider field-of-view (FoV) than standard pinhole cameras, and in practice four fish- eye cameras located at the front, rear, and on each wing mirror are sufficient to build a surround-view system for a full-size ve- hicle [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Although such fisheye systems are widely deployed, to the best of our knowledge, there is no previous work explic- itly handling optical flow estimation on images with strong lens distortion, such as fisheye imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' As shown in Figure 1, one of the current state-of-the-art methods [11] shows erroneous re- sults when taking in fisheye images from WoodScape [13] due to its focus on narrow field-of-view cameras with mild radial distor- tion only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' An intuitive way to solve this is to correct the distor- tion in the input images as a preprocessing step before passing through the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, this inevitably leads to re- duced field-of-view and resampling distortion artifacts at the pe- riphery [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Without rectification, building an automotive dataset is the major bottleneck in optical flow estimation on fisheye im- agery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Very few synthetic datasets provide optical flow ground truth associated with fisheye images [15], whereas no real-world dataset exists with optical flow ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' This is due to the fact that per-pixel motion between every two consecutive frames is extremely difficult to be manually labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Simulators [16, 17] can readily generate background motion but dynamic foreground objects need to be explicitly taken care of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In this paper, we inves- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='04422v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='CV] 11 Jan 2023 tigate and boost the performance of RAFT on strongly distorted inputs by making use of the only existing dataset with optical flow groundtruth, SynWoodScape [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Most AV applications are expected to operate not only dur- ing the day but also at night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Cameras become unreliable and camera-based computations are prone to failure under low-light conditions due to its susceptibility to noise and inconsistent expo- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Alternatively, LiDAR sensors can perform robustly in low- light autonomous driving [18] because active sensors that measure the time-of-flight of the emitted lasers are independent of illumi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, LiDAR is bulky, costly, and requires much more computation as well as memory resources to process the output, which makes it inferior to cameras if the latter can provide equiv- alently reliable results in low light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Thermal cameras [19] provide robust low light performance but they are not commonly used in recent automated driving systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Current optical flow methods show poor capabilities of dealing with low-light data because low light is a complex scenario coming with low signal-to-noise ratio, motion blur and local illumination changes brought by multiple light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In addition, current optical flow datasets [20, 21, 22] are dominated by daytime images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In this paper, we propose a novel, generic architecture that facilitates learning nighttime- robust representations in a semi-supervised manner, without the help of any extra data or sacrificing the daytime performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' To the best of our knowledge, this is the first learning-based method that explicitly handles optical flow estimation in low light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The main contributions of this paper are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Introduction and investigation of two challenges in optical flow estimation for AV applications: strong lens distortion and low-light scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Implementation and improvement of a baseline optical flow algorithm on fisheye inputs and experimental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Implementation of an effective but also generic framework of novel strategies to learn nighttime-robust representations for learning-based optical flow algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Section II discusses re- lated work on optical flow estimation in the automotive industry and existing attempts to solve the two aforementioned challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Section III describes the implementation of our proposed flow es- timation algorithms for fisheye and low-light inputs respectively, as well as presents the experimental evaluation and results anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Finally, Section IV discusses the remaining challenges for flow estimation in AV applications and concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' II RELATED WORK Optical Flow Estimation: Traditional solutions have been stud- ied and adapted for decades [7, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In order to be robust against more challenging open world problems including lack of features, motions in different scales, and occlusions, recent learning-based methods outperform traditional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [8] pro- pose FlowNetS and FlowNetC, which is a pioneer work in show- ing the feasibility of directly estimating optical flow given im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [9] design PWC-Net, a much more efficient solu- tion based on pyramidal processing, warping and the use of a cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' RAFT [11], proposed by Teed and Deng, demonstrates notable improvement by building multi-scale 4D correlation vol- umes for all pairs of pixels and iteratively updating flow estimates through refinement module based on gated recurrent units (GRU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' All these methods are fully supervised and trained using imagery from a standard pinhole camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The training data are also col- lected during the day with sufficient brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' None of them pays attention to the performance of optical flow in more chal- lenging AV applications such as strong lens distortion and driving at night, which leads to errors and even catastrophic failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Strong Lens Distortion: There is very limited work on percep- tion tasks for strongly distorted images such as fisheye images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Popular approaches include rectifying the radial distortion before passing images into any regular perception pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, this will inevitably bring reduced field-of-view and resampling distortion artifacts especially at the image borders [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Spa- tially variant distortion that makes closer objects appear larger also poses scaling problems and complexity to geometric percep- tion tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Additionally, Rashed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [24] show that the com- mon use of bounding boxes for object detection no longer fit well for rectangular objects in distorted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' More sophisticated representations for detected objects, such as a curved bounding box exploiting the known radial distortion, are explored in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Although there is some literature using distorted images with- out rectification on other perception tasks, such as depth estima- tion [26, 27], soiling [28], visual odometry [29] and multi-task models [30, 31], there is no previous work estimating optical flow due to the difficulty in labeling ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' WoodScape [13], KITTI 360 [32] and Oxford RobotCar [33] are some well-known autonomous driving datasets containing strongly distorted im- ages such as fisheye images, but none of them has optical flow ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In this paper, we take advantage of the synthetic fisheye dataset published recently, SynWoodScape [15], which is the first dataset providing optical flow for both foreground and background motions by computing it analytically using other data modalities extracted from the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We train our network using synthetic data from SynWoodScape and evaluate it on real- world fisheye data from WoodScape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Low-Light Scenes: Similar to optical flow estimation on strongly distorted images, there is some work handling low light in a few perception tasks [18, 34, 35] but none of them has proposed an optical flow estimation algorithm that is robust against low-light scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Very related to ours, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [36] propose a method to synthesize low-light optical flow data by simulating the noise model on dark raw images, which is then used to finetune an off- the-shelf network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, their method is not able to synthe- size more realistic characteristics of real-world low-light scenes one would observe in AV applications, such as the motion blur and local illumination changes brought by multiple light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Their improvement is also very limited due to the off-the-shelf network is not designed nor trained to learn nighttime-robust rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In addition, a variety of techniques have been de- veloped for low-light image enhancement [37, 38] and image-to- image translation [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The former can preprocess inputs to a flow estimation network during inference by brightening up a given low-light image, whereas the latter can translate a daytime image into its nighttime counterpart so as to complement the lack of optical flow datasets in low light [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' But neither approach fa- cilitates the network training in that the processed data bring in ex- tra complexities such as additional artificial noise, overexposure, or inconsistent image translation across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Finally, semi- supervised learning is a common approach to tackling the lack of Figure 2: Optical flow estimation (color coded) on real-world automotive data from WoodScape [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Input frames are from the fisheye cameras of front view, right-side view, and left-side view respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' optical flow data in particular scenarios, where a set of predefined transformations are applied to the original labeled data and the output of the perturbed data are enforced to agree with the out- puts of the original data [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' For example, Jeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [42] use a semi-supervised setup to impose translation and rotation con- sistency equivariance for optical flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [43] synthesize foggy images from clean and labelled images in or- der to avoid flow estimation errors caused in dense foggy scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Similar to these semi-supervised methods, we incorporate low- light consistency that facilitates learning explicit nighttime-robust representations without additional labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' III PROPOSED ALGORITHMS AND RESULTS In this section, we describe the two proposed optical flow es- timation algorithms for strongly distorted inputs and low-light in- puts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We also present the corresponding experimental evaluation and results analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' A Strong Lens Distortion The limited availability of datasets with strong lens distortion is the bottleneck that prevents recent methods from generalizing to more distorted inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' With the help of SynWoodScape [15], the first fisheye dataset providing optical flow ground truth for both foreground and background motions, we are able to train an optical flow model, using RAFT [11] as the backbone, that generalizes well on strongly distorted lenses without sacrificing its original performance on pinhole cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We run the off-the-shelf RAFT on real-world fisheye auto- motive dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' WoodScape [13] and we find sharp and incon- sistent optical flow estimation, which is especially illustrated on the ground plane in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' To solve this, we provide two base- lines and their qualitative as well as quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' One is to finetune the pretrained RAFT using SynWoodScape, follow- ing the training schedule in Table 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The other is to jointly train RAFT on both SynWoodScape and images from pinhole cam- era that are regularly used in learning-based optical flow meth- ods [8, 20, 21, 22, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The jointly training baseline follows the training schedule in Table 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Table 1: Details of the training schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Column header abbreviations: LR: learning rate, BS: batch size, WD: weight decay, CS: crop size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Train- ing dataset abbreviations: C: FlyingChairs, W: SynWoodScape, S: Sintel, T: FlyingThings3D, K: KITTI-2015, H: HD1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' (a) Finetuning baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' During the Sintel stage, the dataset distribution is S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='67), T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='12), K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='13), H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Stage Weights Dataset LR BS WD CS Chairs C 4e-4 6 1e-4 [368, 496] Things Chairs T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='2e-4 3 1e-4 [400, 720] Sintel Things S+T+K+H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='2e-4 3 1e-5 [368, 768] Finetune Sintel W 1e-4 3 1e-5 [600, 800] (b) Jointly training baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' During the Joint stage, the dataset distribu- tion is W(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='65), S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='17), T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='13), K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='03), H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Stage Weights Dataset LR BS WD CS Chairs C 4e-4 6 1e-4 [368, 496] Things Chairs T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='2e-4 3 1e-4 [400, 720] Joint Things W+S+T+K+H 1e-4 3 1e-5 [368, 768] Finetuned on SynWoodScape Current Frame Pretrained on Sintel Jointly TrainedFigure 3: Overview of our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' During training, the framework takes two consecutive frames as input and passes them through a set of low-light-specific data augmentations as well as applies a random illumination mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Then the optical flow estimator estimates flow on two pairs of augmented frames in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The network is supervised by two losses: the conventional optical flow loss and the novel brightness consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' During inference, the input frames are directly passed into the estimator which outputs optical flow, as is the standard way in the existing state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We then show the quantitative results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We use the endpoint error (EPE) as the metric, which is the standard er- ror measure for optical flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' It is the Euclidean dis- tance between the estimated flow vector and the ground truth, averaged over all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We evaluate the two baselines (Stages ”Finetune” and ”Joint”) described above along with the pretrained model (Stage ”Sintel”) provided by the author on four hold-out test sets from SynWoodScape, Sintel (clean and final passes), and KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' SynWoodScape is the only test set of strongly distorted inputs, while the other three assume a pinhole camera model with very little distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Although the pretrained model gives out- standing performance on pinhole cameras, its performance sig- nificantly drops on fisheye inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Our first baseline, the one fine- tuned on fisheye images, gives the best result on SynWoodScape but has very poor performance on the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' This matches our expectation because both the pretrained and the finetuned mod- els are trained to the best for pinhole camera and fisheye camera respectively, without taking generalization into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' On the other hand, our second baseline, the jointly trained model, keeps the second best while being very close to the best score on all four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Therefore, jointly training provides a straightforward yet strong baseline that generalizes well over lenses with distinct dis- tortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Table 2: Endpoint-error results on datasets with diverse lens distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Stage SynWoodScape Sintel - Clean Sintel - Final KITTI Sintel 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='10 Finetune 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='44 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='32 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='34 Joint 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='31 In Figure 2, we further show their qualitative results on WoodScape that support the improvements we obtain by jointly training RAFT on a mixture of lens distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In the front view case, note how the jointly trained model is able to consistently estimate the flow on the ground as is the major failure of recent methods shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The results on side-view cameras also show the jointly trained model captures finer details than its fine- tuned counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' For example in the right-side view, not only the inconsistency on the ground is solved, but optical flow associated with the bicycle wheel in the upper right corner is also clearly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In the left-side view, the finetuned model misses the flow associated with the vehicle’s front wheel, which is captured by the pretrained model, but the jointly trained model ”regains” such detailed estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In other words, the finetuned model es- timates more consistent optical flow, which poses a challenge to the pretrained model due to markedly different projection geome- tries between fisheye and pinhole cameras, but in return, it loses some details observed by the pretrained model because interesting local features become much less significant given the strong lens distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, the jointly trained model achieves a great trade-off among the previous two: it re-captures the details lo- cally while maintaining good performance globally across differ- ent camera views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' B Low-Light Scenes We propose a novel and generic semi-supervised framework that significantly boosts performances of existing state-of-the-art methods in low light conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Figure 3 shows the architec- ture of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The benefit of our framework is three- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' First, it is independent from the design of the existing meth- ods, so one can apply it generically to an estimator of his choice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' [8, 9, 11, 45]) and augment its nighttime performance out of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Second, semi-supervised learning does not require any extra data as the labeling cost for nighttime optical flow datasets is immense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Lastly, it maintains the estimator’s competitive perfor- mance on the original daytime data without making any trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We first break down the root causes of failures in optical flow estimation under low light and then describe our proposed strate- gies in the framework that address these root causes accordingly: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' the complex noise model of images captured at night, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' severe motion blur caused by longer exposure time, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' inconsistent local brightness brought by multiple indepen- dent light sources in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Images captured in low light tend to have more complex noises than those captured with sufficient ambient light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Such noises are never synthesized in the data augmentation step by ex- isting methods, which is the first reason why the optical flow esti- mators fail in low light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Similar to [36], we decompose the noise model in low light as an aggregate of the photon shot noise and Ground Truth Flow Photometric Motion Blur Input Pair w/o Brightness Mask Original Input Low-light Noise It+1 Occlusion Estimator Estimated Flow Supervised Loss Spatial Ls V Shared Weights Data Augmentation Input Pair w/ Brightness Mask I!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Local Brightness Estimator Estimated Flow RE Consistency Loss Lb Random MaskFigure 4: Effects of low-light noise augmentation and motion blur aug- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The former is due to the changing amount of pho- tons hitting the sensor with different exposure levels and pixel lo- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The photon shot noise is approximated by a Poisson dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Thermal noise refers to the noise in readout circuitry in the sensor and is approximated by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' There- fore, we synthesize the low-light noise onto the input frames as one extra data transform in the data augmentation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Specif- ically, we sample Poisson and Gaussian parameters, (a,b), from ranges observed in real-world low-light images, formulate it into a single heteroscedastic Gaussian (Equation 1), and apply it to an input frame I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' With probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='5, the low-light noise augmen- tation is performed on each pair of consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' I(x) = N � µ = x,σ2 = ax+b � (1) Motion blur is another root cause we need to address when estimating optical flow in low light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In order to mimic the blurring effects caused by longer exposure length, we generate authentic motion blur kernels using Point Spread Functions (PSF) at dif- ferent kernel sizes and intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The intensity determines how non-linear and shaken the motion blur looks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Similar to low-light noise, we apply the authentic blurring to a pair of input frames as one extra data augmentation, with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' An illustra- tion of the two introduced data augmentation strategies is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Inconsistent local brightness is the last but not least root cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' This is due to multiple independent lighting sources exist- ing in a low-light scene (street light, headlight, moonlight, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' ), which leads to uneven bright areas in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' For example, the ground plane in the original input in Figure 3 is illuminated only in front of vehicles’ headlights but remains dark elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Unlike in the daytime where sun is the dominant lighting source, images captured at night have inconsistent local brightness even on the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Because optical flow is estimated by match- ing pixels across two images, such inconsistencies cause exist- ing methods to fail easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' For instance, in Figure 5, the first row shows the catastrophic failure of RAFT when a pedestrian walks from the dark into the vehicle headlight and his illumina- tion changes drastically across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In order to resolve this, we resort to semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Similar to [42], we also adopt the cow-mask [46] to create sufficiently random yet locally con- nected illumination patterns as the inconsistent local brightness occurs in any size, shape and position in images while exhibiting Figure 5: Optical flow estimation on low-speed sequences from CU- Lane [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' locally explainable structures, depending on the driving environ- ment and the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We apply the same binary mask to the original pair of input frames and randomly adjust the brightness of pix- els according to the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The true area of the binary mask is uniformly sampled from 40% to 70% of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' With a proba- bility of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='5, we increase the absolute brightness of the true area, whereas in the remaining time we increase the brightness of the false area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Finally, we introduce the local brightness consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We use (I′t,I′ t+1) and (It,It+1) to denote the input pair after data augmentation with and without applying a random brightness mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Both passes in Figure 3 are independent ex- cept that the spatial transform is shared in order to keep the same cropped areas for consistency loss calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' The local bright- ness consistency loss is calculated as follows Lb = ��Estimator(It,It+1)−Estimator � I′ t,I′ t+1 ���2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' (2) This regularization explicitly constrains the network to output consistent optical flow on (I′t,I′ t+1) as on (It,It+1), which enforces illumination invariance between the estimated optical flow for the original pair the estimated optical flow for the randomly trans- formed pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Note how this semi-supervised approach is different from simply adjusting brightness randomly as another data aug- mentation scheme, which expand training samples without impo- sition of a sophisticated consistency loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We choose RAFT [11] as the estimator and we supervise our network on the aggregated loss L = Ls +Lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Ls is the l1 distance between the predicted flow ˜f i(It,It+1) and ground truth flow ft over all iterations i, as in [11]: Ls = N ∑ i=1 γN−i ��� ˜f i(It,It+1)− ft ��� 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' (3) Due to the lack of nighttime data with optical flow ground truth, we are restricted to qualitatively evaluating our approach, which we call RAFT-Dark for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We use CULane [47], a large automotive dataset containing a lot of challenging real-world low light sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In Figure 5, we show the comparison between vanilla RAFT and RAFT-Dark on some low-speed sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' RAFT-Dark demonstrates superior performance to RAFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In the Original ImagePair After Low-Light Noise Augmentation Original ImagePair AfterMotionBlurAugmentationInput RAFT RAFT-Dark (Ours)Figure 6: Optical flow estimation on high-speed sequences from CU- Lane [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' first, third and fourth rows, RAFT-Dark is able to detect motions associated with pedestrians and vehicles that either experience some drastic illumination change or appear to be too dark and noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In the other cases, note how RAFT-Dark gives a signif- icantly better estimation on the ground plane as well as the di- rections and magnitudes that are consistent with the ego vehi- cle’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' For convenience, a color coding wheel to visualize per-pixel optical flow vectors is attached to the top right corner: color denotes direction of the flow vector while intensity denotes length of the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Since the ego vehicle always heads forward, the ground truth optical flow vectors in the front cam- era’s image should intuitively point to the image boundaries and away from the image center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' And due to the motion parallax, one should expect larger magnitudes of flow vectors toward the im- age boundaries and small magnitudes around the image center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In other words, although we have no access to numerical ground truth flow, we know the color coded ground truth should exhibit the same pattern as the color wheel: bluish or greenish on the left side while reddish or yellowish on the right side of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' With this in mind, RAFT fails to estimate correct optical flow con- sistent with the vehicle’s motion, especially in background areas such as the ground plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' On the other hand, RAFT-Dark not only performs well on these areas but also learns to separate the dark sky in some cases and to capture details such as the street light in the second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Such improvements are further illustrated in high-speed sequences from CULane in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Our framework of learning strategies enables RAFT to im- prove estimation accuracy by more than 50% on average (based on visual observations), and even solves some catastrophic fail- ures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Although we show our results based on RAFT as the esti- mator, our framework is generic and one can replace RAFT with any existing state-of-the-art method of one’s choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' C Discussion The goal of this work is to emphasize the importance of addressing optical flow challenges which are not well explored in automated driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We investigate two of them and propose our solutions accordingly, but the others require further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Lack of data tends to be the major bottleneck for most data-driven optical flow algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We are able to leverage synthetic data to improve existing methods’ adaptation of various lens distortions but the sim-to-real gap still exists when these methods are evalu- ated on real world fisheye data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Optical flow in low light cannot be addressed in the same way without any synthetic data avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We experiment image enhancement prior to the network inference, but it leads to even worse results because enhancement happens per frame rather than per pair of frames and temporal consistency is easily broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Without any extra data, our approach takes full advantage of publicly available data and simulates three root causes through novel data augmentation schemes and semi- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' However, low light is merely one of many scenarios that make optical flow estimation harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Others in- clude foggy, rainy or snowy weather [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' A unified and robust approach aiming for all these cases is encouraged and we see it also as an opportunity for further investigation by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' IV Conclusion Both lens distortion and low light are important problems for higher levels of automated driving, but they are not explored in detail in the optical flow community as there is no public dataset available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Thus we propose our approaches to these two respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We implement and improve a state-of-the-art optical flow algorithm by training it on synthetic fisheye data and demonstrat- ing its adaptation to real-world distorted images as well as gen- eralizability over various lens distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' We implement a novel, generic framework that facilitates learning nighttime-robust rep- resentations in a semi-supervised manner, which shows superior performance to the existing state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' In future work, we plan to integrate our current solutions into higher-level pipelines as well as explore other unique challenges of optical flow estima- tion in the context of automated driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Kale, S.' metadata={'source': 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level classification for autonomous driving: Dataset, baseline and active learning,” in 2021 IEEE International Intelligent Trans- portation Systems Conference (ITSC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' 2816–2821, IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' AUTHORS BIOGRAPHY Shihao Shen is a second-year graduate student in the Robotics Institute at Carnegie Mellon University and expects to receive his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' in Robotic Systems Development in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He worked as an Interim Engineering Intern in the Multimedia Research and Development department at Qualcomm in summer 2022 and this is his work done during his internship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' His main research focus is machine learning with applications in computer vision as well as simultaneous localization and mapping (SLAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Louis Kerofsky is researcher in video compression, video processing and display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He received M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' degrees in Mathematics from the University of Illinois, Urbana-Champaign (UIUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He has over 20 years of experience in research and al- gorithm development and standardization of video compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He has served as an expert in the ITU and ISO video compression standards committees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He is an author of over 40 publications which have over 5000 citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He is an inventor on over 130 issued US patents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He is a senior member of IEEE, member of Society for Information Display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' Senthil Yogamani is an artificial intelligence architect for au- tonomous driving and holds a principal engineer position at Qualcomm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He leads the research and design of AI algorithms for various modules of autonomous driving systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He has over 17 years of experience in computer vision and machine learn- ing including 14 years of experience in industrial automotive sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He is an author of 110+ publications which have 4000+ citations and 150+ inventions with 85 filed patent families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He serves on the editorial board of various leading IEEE automotive conferences including ITSC and IV and advisory board of various industry consortia including Khronos, Cognitive Vehicles and IS Auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} +page_content=' He is a recipient of the best associate editor award at ITSC 2015 and best paper award at ITST 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE3T4oBgHgl3EQfRgmQ/content/2301.04422v1.pdf'} diff --git a/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf b/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cdc0a4f650c94f6e194accabea15747115b78f78 --- /dev/null +++ b/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:378a379878cb425c5af4753ec9f7a79d95f5e9bca9a8db2e94cd87b2920f8246 +size 3452062 diff --git a/9NE2T4oBgHgl3EQfQAZl/vector_store/index.faiss b/9NE2T4oBgHgl3EQfQAZl/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8d87116270054ab3c7907f2851e68eef35a9007a --- /dev/null +++ b/9NE2T4oBgHgl3EQfQAZl/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:982ca6ebdfda4927e9a5c8980e697fd7d5d429f64078dcc9d1eea60b46213c80 +size 6225965 diff --git a/9dFST4oBgHgl3EQfbDi0/content/tmp_files/2301.13798v1.pdf.txt b/9dFST4oBgHgl3EQfbDi0/content/tmp_files/2301.13798v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a261f2872c0addf946f4d286dc8fd645b9ddb9f --- /dev/null +++ b/9dFST4oBgHgl3EQfbDi0/content/tmp_files/2301.13798v1.pdf.txt @@ -0,0 +1,757 @@ +EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH +CERN-EP-2023-005 +23 January 2023 +© 2023 CERN for the benefit of the ALICE Collaboration. +Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license. +Exploring the non-universality of charm hadronisation through the +measurement of the fraction of jet longitudinal momentum carried by Λ+ +c +baryons in pp collisions +ALICE Collaboration +Abstract +Recent measurements of charm-baryon production in hadronic collisions have questioned the univer- +sality of charm-quark fragmentation across different collision systems. In this work the fragmentation +of charm quarks into charm baryons is probed, by presenting the first measurement of the longitudinal +jet momentum fraction carried by Λ+ +c baryons, 𝑧ch +|| , in hadronic collisions. The results are obtained +in proton–proton (pp) collisions at √𝑠 = 13 TeV at the LHC, with Λ+ +c baryons and track-based jets +reconstructed in the transverse momentum intervals of 3 ≤ 𝑝Λ+ +c +T < 15 GeV/𝑐 and 7 ≤ 𝑝jet ch +T +< 15 +GeV/𝑐, respectively. The 𝑧ch +|| distribution is compared to a measurement of D0-tagged charged jets in +pp collisions as well as to PYTHIA 8 simulations. The data hints that the fragmentation of charm +quarks into charm baryons is softer with respect to charm mesons, as predicted by hadronisation +models which include colour correlations beyond leading-colour in the string formation. +arXiv:2301.13798v1 [nucl-ex] 31 Jan 2023 + +ALICEIn-jet Λ+ +c production in pp collisions at √𝑠 = 13 TeV +ALICE Collaboration +Heavy-flavour hadrons are produced in high-energy particle collisions through the fragmentation of heavy +(charm and beauty) quarks, which typically originate in hard scattering processes in the early stages of +the collisions. The most common theoretical approach to describe heavy-flavour production in hadronic +collisions is based on the quantum chromodynamics (QCD) factorisation approach [1], and consists of a +convolution of three independent terms: the parton distribution functions of the incoming hadrons, the +cross sections of the partonic scattering producing the heavy quarks, and the fragmentation functions that +parametrise the evolution of a heavy quark into given species of heavy-flavour hadrons. As the transition +of quarks to hadrons cannot be described in perturbation theory, the fragmentation functions cannot be +calculated and must be extracted from data. +Fragmentation functions of charm quarks to charm baryons and mesons have been constrained in e+e− +and e−p collisions [2–5], using a variety of different observables, such as the hadron momentum as a +fraction of its maximum possible momentum, as dictated by the centre-of-mass energy of the collision. +Another method to probe the fragmentation of quarks to hadrons is to parametrise the hadron momentum +in relation to the momentum of jets, which are collimated bunches of hadrons giving experimental access +to the properties of the scattered quark. Recently, the production of charm mesons in jets, probed via the +fractional longitudinal momentum of the jet carried by the D meson, was measured in pp collisions at the +Large Hadron Collider (LHC) [6–8] and appears consistent with Monte Carlo (MC) simulations tuned +on e+e− data. These measurements support the assumption of fragmentation universality across collision +systems in the charm-meson sector. This assumption underpins theoretical calculations describing the +production of heavy-flavour hadrons in hadronic collisions, which make use of fragmentation functions +tuned on e+e− and e−p data. +Measurements of the production cross sections of baryons in pp collisions have questioned the hypothesis +of fragmentation universality across collision systems [9]. In the charm sector, which provides a clean +probe of hadronisation phenomena due to the large mass of the charm quark, recent measurements +performed by the ALICE Collaboration [10–18] in pp collisions have shown that the ratio of the Λ+ +c (and +other charm baryons) and D0 production cross sections measured at low 𝑝T (≲ 12 GeV/𝑐) is significantly +larger than the value expected from MC simulations in which the charm fragmentation is tuned on e+e− +and e−p measurements, such as PYTHIA 8 [19] with the Monash tune [20] or HERWIG 7 [21]. A recent +measurement of the Λ+ +c/D0 ratio in pp collisions, performed by the ALICE Collaboration in intervals +of charged-particle multiplicity, also points to a substantial increase of the Λ+ +c/D0 ratio with increasing +multiplicity, with respect to e+e− collisions, starting at very low multiplicities [14]. +The study of charm-baryon production in jets can provide more differential insights into hadronisation +mechanisms in pp collisions, compared to 𝑝T-differential cross sections and yield ratios of heavy-flavour +hadrons, allowing for a more accurate study of the dynamical properties of baryon production. In this +letter, the first measurement of the longitudinal momentum fraction of the jet carried by Λ+ +c baryons, 𝑧ch +|| , is +presented. The measurement is performed in pp collisions at √𝑠 = 13 TeV in the interval 0.4 ≤ 𝑧ch +|| ≤ 1.0. +The 𝑧ch +|| distribution, fully corrected to particle level, is presented for prompt (charm-quark initiated) +Λ+ +c-tagged jets with 7 ≤ 𝑝jet ch +T +< 15 GeV/𝑐 and 3 ≤ 𝑝Λ+ +c +T < 15 GeV/𝑐. The results are then compared +to PYTHIA 8 simulations [19, 22], including a version where mechanisms beyond the leading-colour +approximation are considered in string formation processes during hadronisation [20], and to an analogous +measurement of the 𝑧ch +|| distribution of D0 mesons, performed by the ALICE Collaboration [6]. +A full description of the ALICE setup and apparatus can be found in Refs. 23, 24. The main detectors +used in this analysis are the Inner Tracking System (ITS), which is used for vertex reconstruction and +tracking; the Time Projection Chamber (TPC), which is used for tracking and particle identification (PID); +and the Time-Of-Flight (TOF) detector, which is used for PID. These detectors cover a pseudorapidity +interval of |𝜂| < 0.9. The analysis was performed on pp collisions at √𝑠 = 13 TeV, collected using a +minimum-bias (MB) trigger during the years 2016, 2017, and 2018. The trigger condition required +2 + +In-jet Λ+ +c production in pp collisions at √𝑠 = 13 TeV +ALICE Collaboration +coincident signals in the two scintillator arrays of the V0 detector, with background events originating +from beam–gas interactions removed offline using timing information from the V0. To mitigate against +pile-up effects, events with multiple reconstructed primary vertices were rejected. To ensure uniform +acceptance, only events with a primary-vertex position along the beam axis direction of |𝑧vtx| < 10 cm +around the nominal interaction point were accepted. After the selections described above, the data sample +consisted of 1.7×109 events, corresponding to an integrated luminosity of Lint = 29 nb−1 [25]. +The Λ+ +c candidates and their charge conjugates were reconstructed via the hadronic Λ+ +c → pK0 +S → pπ+π− +decay channel with a total branching ratio of (1.10 ± 0.06)% [26], in the Λ+ +c transverse-momentum +interval of 3 ≤ 𝑝Λ+ +c +T < 15 GeV/𝑐. Only tracks with |𝜂| < 0.8 and 𝑝T > 0.4 GeV/𝑐, which fulfilled the track +quality selections described in Ref. 13, were considered for the Λ+ +c reconstruction. The Λ+ +c candidates +themselves were reconstructed in the |𝑦Λ+c | < 0.8 rapidity interval. +The Λ+ +c-candidate selection was +performed using a multivariate technique based on the Boosted Decision Tree (BDT) algorithm provided +by the XGBoost package [27]. The features considered in the optimisation include the PID signal for +the proton track, the invariant mass of the K0 +S-meson candidate, and topological variables that exploit the +kinematic properties of the displaced K0 +S-meson decay vertex. The training was performed in intervals +of Λ+ +c-candidate 𝑝T, considering prompt signal candidates from PYTHIA 8 events with the Monash +tune [19, 20], transported through a realistic description of the detector geometry and material budget +using GEANT 3 [28]. Background candidates were extracted from the sidebands of the invariant-mass +distributions in data. The probability thresholds of the BDT selections were tuned, using MC simulations, +to maximise the statistical significance for the signal. Further details on the Λ+ +c-candidate reconstruction +and machine learning procedure are provided in Ref. 14, where the same reconstruction and BDT model +were employed. +For the events where at least one selected Λ+ +c candidate was identified, a jet-finding procedure was +performed, using the FastJet package [29]. Prior to jet clustering, the Λ+ +c-candidate daughter tracks were +replaced by the reconstructed Λ+ +c-candidate four-momentum vector. Track-based jet finding was carried +out on charged tracks with |𝜂| < 0.9 and 𝑝T > 0.15 GeV/𝑐, using the anti-𝑘T algorithm [30], with a +resolution parameter of 𝑅 = 0.4. Tracks were combined using the 𝐸-scheme recombination [31], with the +jet transverse momentum limited to the interval of 5 ≤ 𝑝jet ch +T +< 35 GeV/𝑐. The full jet cone was required +to be within the ALICE central barrel acceptance, limiting the jet axis to the interval |𝜂jet| < 0.5. Only +jets tagged via the presence of a reconstructed Λ+ +c candidate amongst their constituents were considered +for the analysis. For events where more than one Λ+ +c candidate was found, the jet finding and tagging +pass was performed independently for each candidate, with only the daughters of that particular candidate +replaced by the corresponding Λ+ +c four-vector each time. In mechanisms of hadronisation that include +colour correlations beyond the leading-colour approximation [20], which have been shown to be relevant +in hadronic collisions at LHC energies [9], hadrons can be formed in processes that combine quarks from +the parton shower with those from the underlying event [32]. As such, the underlying event is not well +defined with respect to the measured hadron distributions. Therefore no underlying event correction is +implemented in this work. +The fragmentation of charm quarks to Λ+ +c baryons is probed by measuring the fraction of the jet momentum +carried by the Λ+ +c along the direction of the jet axis, 𝑧ch +|| . This is calculated for each jet using +𝑧ch +|| = 𝒑jet · 𝒑Λ+c +𝒑jet · 𝒑jet +, +(1) +where 𝒑jet and 𝒑Λ+c are the jet and Λ+ +c three-momentum vectors, respectively. +The 𝑧ch +|| distributions of true Λ+ +c-tagged jets were extracted in intervals of Λ+ +c 𝑝T and 𝑝jet ch +T +using a sideband +subtraction procedure. To enact this subtraction, the invariant-mass (𝑚inv) distributions of Λ+ +c candidates, +obtained for each Λ+ +c 𝑝T and 𝑝jet ch +T +interval, were fitted with a function comprising a Gaussian for the signal +3 + +In-jet Λ+ +c production in pp collisions at √𝑠 = 13 TeV +ALICE Collaboration +and an exponential for the background. The fit parameters were then used to define signal (containing the +majority of true signal candidates) and sideband (entirely composed of background candidates) regions, +defined by |𝑚inv − 𝜇fit| < 2𝜎fit and 4𝜎fit < |𝑚inv − 𝜇fit| < 9𝜎fit, respectively, where 𝜇fit and 𝜎fit represent +the mean and sigma of the fitted Gaussian distributions. The 𝑧ch +|| (𝑝Λ+ +c +T ,𝑝jet ch +T +) distributions were extracted +in the signal and sideband regions, with the sideband distribution scaled by the ratio of the background +function integrals in the signal and sideband regions. The sideband distribution was then subtracted from +the signal one, with the resulting distribution scaled to account for the fact that the 2𝜎fit width of the +signal region only encompasses approximately 95% of the total signal, to obtain the sideband subtracted +𝑧ch +|| yield in each 𝑝Λ+ +c +T and 𝑝jet ch +T +interval. +To account for the reconstruction and selection efficiency of the Λ+ +c-tagged jet signal, the sideband +subtracted 𝑧ch +|| distributions in each 𝑝Λ+ +c +T and 𝑝jet ch +T +interval, 𝑁(𝑧ch +|| , 𝑝Λ+ +c +T , 𝑝jet ch +T +), were scaled by the recon- +struction efficiency of prompt Λ+ +c-tagged jets, 𝜖prompt, and summed over the entire 𝑝Λ+ +c +T interval to obtain +the efficiency-corrected 𝑧ch +|| yield of Λ+ +c-tagged jets, 𝑁corr(𝑧ch +|| , 𝑝jet ch +T +), given by +𝑁corr(𝑧ch +|| , 𝑝jet ch +T +) = +∑︁ +𝑝Λ+c +T +𝑁(𝑧ch +|| , 𝑝Λ+ +c +T , 𝑝jet ch +T +) +𝜖prompt(𝑝Λ+c +T ) +. +(2) +The 𝜖prompt(𝑝Λ+ +c +T ) efficiency is strongly dependent on 𝑝Λ+ +c +T , ranging from about 20% at 3 < 𝑝Λ+ +c +T < 4 GeV/𝑐 +to 40% at 12 < 𝑝Λ+ +c +T < 24 GeV/𝑐, and was calculated using PYTHIA 8 simulations with the Monash tune +containing prompt Λ+ +c-tagged jets, transported through the detector using GEANT 3. This efficiency does +not exhibit a 𝑝jet ch +T +dependence. +In order to isolate the 𝑁corr(𝑧ch +|| , 𝑝jet ch +T +) distribution of prompt Λ+ +c-tagged jets, a feed-down subtraction +was employed to remove the non-prompt (beauty-quark initiated) contribution. The non-prompt cross +section was obtained from particle level POWHEG [33] + PYTHIA 6 [34] + EvtGen [35] simulations, as +a function of 𝑝jet ch +T +, 𝑝Λ+ +c +T and 𝑧ch +|| , and was scaled according to the integrated luminosity of the analysed +data sample and the branching ratio of the Λ+ +c → pK0 +S → pπ+π− decay channel. The resulting particle- +level yield was multiplied by the ratio of the non-prompt to prompt Λ+ +c-tagged jet reconstruction and +selection efficiency in intervals of 𝑝Λ+c +T +and integrated over the 𝑝Λ+c +T +range. The simulated non-prompt +results were then folded to reconstructed level, using a four-dimensional response matrix generated using +non-prompt Λ+ +c-tagged jets in PYTHIA 8 with the Monash tune, transported through a simulation of +the ALICE detector using GEANT 3. The response matrix was constructed as a function of 𝑝jet ch +T +and +𝑧ch +|| at generator and reconstruction levels. The folded results were then subtracted from the measured +𝑁corr(𝑧ch +|| , 𝑝jet ch +T +) distribution in data, removing the non-prompt contribution. The estimated fraction of +Λ+ +c-tagged jets coming from b-quark fragmentation is found to be about 5%, with no significant 𝑧ch +|| +dependence. +A two-dimensional Bayesian unfolding procedure [36] was performed to correct for detector effects and +obtain the 𝑧ch +|| distribution for prompt Λ+ +c-tagged jets at particle level. +A four-dimensional response +matrix as a function of 𝑝jet ch +T +and 𝑧ch +|| , at generator and reconstruction levels, was populated with prompt +Λ+ +c-tagged jets, obtained with PYTHIA 8 simulations with the Monash tune, passed through a simulation +of the ALICE detector using GEANT 3. The measured data and response matrix were provided in the +intervals of 5 ≤ 𝑝jet ch +T +< 35 GeV/𝑐 and 0.4 ≤ 𝑧ch +|| ≤ 1.0, with the final unfolded results reported in the +intervals 7 ≤ 𝑝jet ch +T +< 15 GeV/𝑐 and 0.4 ≤ 𝑧ch +|| ≤ 1.0. Corrections accounting for migrating entries in +and out of the response matrix ranges, as modelled by the same MC simulation, were also applied. The +corrected 𝑧ch +|| distribution is normalised to the total number of Λ+ +c-tagged jets in the reported 𝑧ch +|| and 𝑝jet ch +T +interval. +4 + +In-jet Λ+ +c production in pp collisions at √𝑠 = 13 TeV +ALICE Collaboration +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +ch +z +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +ch +z +/d +N +) d +jet +N +(1/ +-tagged jets ++ +c +Λ +data +Monash +CR-BLC Mode 2 +PYTHIA 8: + = 13 TeV +s +, pp, +ALICE + = 0.4 +R +, +T +k +charged jets, anti- + 0.5 +≤ + +jet +η +, +c + < 15 GeV/ +jet ch +T +p + +≤ +7 + 0.8 +≤ + ++ +c +Λ +y +, +c + < 15 GeV/ ++ +c +Λ +T +p + +≤ +3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +ch +z +1 +1.5 +2 +MC/data +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +ch +z +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +ch +z +/d +N +) d +jet +N +(1/ +-tagged jets ++ +c +Λ +-tagged jets +0 +D + = 13 TeV +s +, pp, +ALICE + = 0.4 +R +, +T +k +charged jets, anti- + 0.5 +≤ + +jet +η +, +c + < 15 GeV/ +jet ch +T +p + +≤ +7 + 0.8 +≤ + +h +y +, +c + < 15 GeV/ +h +T +p + +≤ +3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +ch +z +0.5 +1 +1.5 +2 +0 +/D ++ +c +Λ +data +PYTHIA 8 Monash +PYTHIA 8 CR-BLC Mode 2 +Figure 1: (Left) Fully corrected 𝑧ch +|| distribution of Λ+ +c-tagged track-based jets (black open circles) measured in the +7 ≤ 𝑝jet ch +T +< 15 GeV/𝑐 and 3 ≤ 𝑝Λ+ +c +T < 15 GeV/𝑐 intervals in pp collisions at √𝑠 = 13 TeV, compared with predictions +from different PYTHIA 8 tunes [19, 20, 22] (red-dotted and green-dashed lines). The ratios of the MC simulations +to the data are shown in the bottom panel. (Right) Comparison of the measured 𝑧ch +|| distribution of Λ+ +c-tagged jets +and the previously measured 𝑧ch +|| distribution of D0-tagged jets [6], obtained in the same kinematic interval. The +ratio of the 𝑧ch +|| distribution of Λ+ +c-tagged and D0-tagged jets is shown in the bottom panel for both the data and the +different PYTHIA tunes. +The systematic uncertainties affecting the measurement were evaluated, in each 𝑧ch +|| interval, by modifying +the strategy adopted at various steps of the analysis procedure and assessing the impact on the unfolded +𝑧ch +|| distribution. The total systematic uncertainty includes contributions from multiple sources. The +first considered source is the sideband subtraction procedure (ranging from 3.7% to 7.6% depending +on the 𝑧ch +|| inteval), whose contribution was estimated by varying the invariant-mass fit parameters as +well as the invariant-mass intervals of the signal and sideband regions. The contribution from the BDT +selection of Λ+ +c candidates (from 7.3% to 19%) was estimated by varying the BDT probability thresholds +to induce a 25% variation in the Λ+ +c-tagged jet reconstruction and selection efficiency. The uncertainty +from the jet energy resolution (from 4.5% to 19%) was estimated by recalculating the response matrix +used for unfolding with a 4% reduced tracking efficiency. The reduction in the tracking efficiency was +evaluated by varying the track-selection criteria and propagating the ITS–TPC track-matching efficiency +uncertainty. The uncertainty on the feed-down subtraction (< 2%) was estimated by varying the choice +of POWHEG parameters considered to generate the feed-down cross section, including the factorisation +and renormalisation scales, as well as the mass of the beauty quark, which were varied according to +theoretical prescriptions [37]. Finally the contribution from the unfolding procedure (from 1.1% to 2.7%) +was estimated by altering the choice of prior, regularisation parameter, and ranges of the response matrix. +For each of the aforementioned categories, several variations were made and the root-mean-square of +the resulting distributions was considered. The exceptions are related to the contribution associated to +the choice of parameters of the POWHEG calculations, where only the largest deviation from the central +result, in each direction, was considered, as well as the uncertainty on the jet energy resolution where +the variation with respect to the central result was taken as the uncertainty. All uncertainties (other than +from the feed-down subtraction) were then symmetrised. The uncertainties were combined in quadrature +to obtain the total systematic uncertainty on the measurement, which ranges from 13% to 28%. +5 + +In-jet Λ+ +c production in pp collisions at √𝑠 = 13 TeV +ALICE Collaboration +The fully corrected 𝑧ch +|| distribution of prompt Λ+ +c-tagged track-based jets in the intervals of 7 ≤ 𝑝jet ch +T +< +15 GeV/𝑐 and 3 ≤ 𝑝Λ+ +c +T < 15 GeV/𝑐 is presented in the left-hand panel of Fig. 1 and compared to PYTHIA 8 +simulations with two different tunes. In PYTHIA 8 the Lund string model of fragmentation is employed, +where endpoints are confined by linear potentials encoded in strings. For the case of heavy quarks, the +Lund fragmentation function is modified to account for the slower propagation of the massive endpoints +compared to their massless counterparts. The Monash tune (red-dotted line) [19], in which the charm +fragmentation is tuned on e+e− measurements, predicts a harder fragmentation than the measurement. +An evaluation of the 𝜒2/ndf between the measured data points and the model was performed, combining +the statistical and systematic uncertainties on the data in quadrature and assuming the uncertainties are +uncorrelated across the 𝑧ch +|| intervals. This exercise determines that there is a 0.4% probability that the +model describes the data. A better agreement is achieved by the PYTHIA 8 with the CR-BLC Mode 2 +tune, that includes colour reconnection mechanisms beyond the leading-colour approximation [22] (green- +dashed line). In this model, the minimisation of the string potential is implemented considering the SU(3) +multiplet structure of QCD in a more realistic way than in the leading-colour approximation, allowing +for the formation of “baryonic” configurations where for example two colours can combine coherently to +form an anti-colour. The same 𝜒2/ndf approach results in a 78% probability that the model describes the +data. The simulation with PYTHIA 8 with the CR-BLC Mode 2 tune also provides a much more accurate +description of the Λ+ +c/D0 cross section ratio, previously measured in pp collisions at the LHC [10–14, 38]. +In the right-hand panel of Fig. 1, a comparison of the 𝑧ch +|| distribution of Λ+ +c-tagged jets and the 𝑧ch +|| distri- +bution previously measured for D0-tagged jets [6] is presented. The latter is consistent with PYTHIA 8 +simulations using both the Monash and CR-BLC Mode 2 tunes. The ratio of the two distributions is +also presented in the bottom panel. The uncertainty from the jet energy resolution was considered to +be correlated between the Λ+ +c-tagged jet and D0-tagged jet measurements and was evaluated directly on +the ratio of the distributions. The remaining uncertainties were considered uncorrelated when taking +the ratio and were then combined in quadrature with the uncertainty of the jet energy resolution. The +uncertainties were considered uncorrelated across the 𝑧ch +|| intervals. The same 𝜒2/ndf exercise described +above determines that there is a 12% probability that the measured ratio is described by a flat distribution +at unity, hinting at a softer fragmentation of charm quarks into charm baryons than charm mesons. The +ratio is better described by the PYTHIA 8 simulations with the CR-BLC Mode 2 compared to the ones +with the Monash tune, with the former describing the data with 88% probability compared to a 0.03% +probability for the latter. +In summary the first measurement in hadronic collisions of the longitudinal momentum fraction of the +jet carried by Λ+ +c baryons was presented for pp collisions at √𝑠 = 13 TeV. The result is fully corrected to +particle level and obtained in the jet and Λ+ +c transverse-momentum intervals of 7 ≤ 𝑝jet ch +T +< 15 GeV/𝑐 +and 3 ≤ 𝑝Λ+c +T < 15 GeV/𝑐, respectively. The measurement presented in this Letter hints that charm quarks +have a softer fragmentation into Λ+ +c baryons compared to D0 mesons, in the measured kinematic interval. +One possible explanation is that charm-baryon production is favoured in the presence of higher particle +multiplicity originating from both the jet fragmentation and the underlying event, which could be tested +with future measurements of the in-jet multiplicity of Λ+ +c-tagged jets. The fragmentation of charm quarks +into Λ+ +c baryons in hadronic collisions exhibits tension with simulations tuned on e+e− data that employ +a leading-colour formalism of hadronisation, such as in the Monash tune of PYTHIA 8. This occurs +despite their successful description of the fragmentation of charm quarks into D0 mesons. However, the +inclusion of mechanisms sensitive to the surrounding partonic density that feature colour reconnection +beyond the leading-colour approximation results in a better agreement with data. 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B 803 (2020) 135328, arXiv:1906.03322 +[hep-ex]. +9 + diff --git a/9dFST4oBgHgl3EQfbDi0/content/tmp_files/load_file.txt b/9dFST4oBgHgl3EQfbDi0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..09ef9ddcef1000b5df92507c25b763c3a03c5f4a --- /dev/null +++ b/9dFST4oBgHgl3EQfbDi0/content/tmp_files/load_file.txt @@ -0,0 +1,456 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf,len=455 +page_content='EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN-EP-2023-005 23 January 2023 © 2023 CERN for the benefit of the ALICE Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Exploring the non-universality of charm hadronisation through the measurement of the fraction of jet longitudinal momentum carried by Λ+ c baryons in pp collisions ALICE Collaboration Abstract Recent measurements of charm-baryon production in hadronic collisions have questioned the univer- sality of charm-quark fragmentation across different collision systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In this work the fragmentation of charm quarks into charm baryons is probed, by presenting the first measurement of the longitudinal jet momentum fraction carried by Λ+ c baryons, 𝑧ch || , in hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The results are obtained in proton–proton (pp) collisions at √𝑠 = 13 TeV at the LHC, with Λ+ c baryons and track-based jets reconstructed in the transverse momentum intervals of 3 ≤ 𝑝Λ+ c T < 15 GeV/𝑐 and 7 ≤ 𝑝jet ch T < 15 GeV/𝑐, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The 𝑧ch || distribution is compared to a measurement of D0-tagged charged jets in pp collisions as well as to PYTHIA 8 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The data hints that the fragmentation of charm quarks into charm baryons is softer with respect to charm mesons, as predicted by hadronisation models which include colour correlations beyond leading-colour in the string formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='13798v1 [nucl-ex] 31 Jan 2023 ALICEIn-jet Λ+ c production in pp collisions at √𝑠 = 13 TeV ALICE Collaboration Heavy-flavour hadrons are produced in high-energy particle collisions through the fragmentation of heavy (charm and beauty) quarks, which typically originate in hard scattering processes in the early stages of the collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The most common theoretical approach to describe heavy-flavour production in hadronic collisions is based on the quantum chromodynamics (QCD) factorisation approach [1], and consists of a convolution of three independent terms: the parton distribution functions of the incoming hadrons, the cross sections of the partonic scattering producing the heavy quarks, and the fragmentation functions that parametrise the evolution of a heavy quark into given species of heavy-flavour hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' As the transition of quarks to hadrons cannot be described in perturbation theory, the fragmentation functions cannot be calculated and must be extracted from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Fragmentation functions of charm quarks to charm baryons and mesons have been constrained in e+e− and e−p collisions [2–5], using a variety of different observables, such as the hadron momentum as a fraction of its maximum possible momentum, as dictated by the centre-of-mass energy of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Another method to probe the fragmentation of quarks to hadrons is to parametrise the hadron momentum in relation to the momentum of jets, which are collimated bunches of hadrons giving experimental access to the properties of the scattered quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Recently, the production of charm mesons in jets, probed via the fractional longitudinal momentum of the jet carried by the D meson, was measured in pp collisions at the Large Hadron Collider (LHC) [6–8] and appears consistent with Monte Carlo (MC) simulations tuned on e+e− data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' These measurements support the assumption of fragmentation universality across collision systems in the charm-meson sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' This assumption underpins theoretical calculations describing the production of heavy-flavour hadrons in hadronic collisions, which make use of fragmentation functions tuned on e+e− and e−p data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Measurements of the production cross sections of baryons in pp collisions have questioned the hypothesis of fragmentation universality across collision systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In the charm sector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' which provides a clean probe of hadronisation phenomena due to the large mass of the charm quark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' recent measurements performed by the ALICE Collaboration [10–18] in pp collisions have shown that the ratio of the Λ+ c (and other charm baryons) and D0 production cross sections measured at low 𝑝T (≲ 12 GeV/𝑐) is significantly larger than the value expected from MC simulations in which the charm fragmentation is tuned on e+e− and e−p measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' such as PYTHIA 8 [19] with the Monash tune [20] or HERWIG 7 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' A recent measurement of the Λ+ c/D0 ratio in pp collisions, performed by the ALICE Collaboration in intervals of charged-particle multiplicity, also points to a substantial increase of the Λ+ c/D0 ratio with increasing multiplicity, with respect to e+e− collisions, starting at very low multiplicities [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The study of charm-baryon production in jets can provide more differential insights into hadronisation mechanisms in pp collisions, compared to 𝑝T-differential cross sections and yield ratios of heavy-flavour hadrons, allowing for a more accurate study of the dynamical properties of baryon production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In this letter, the first measurement of the longitudinal momentum fraction of the jet carried by Λ+ c baryons, 𝑧ch || , is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The measurement is performed in pp collisions at √𝑠 = 13 TeV in the interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 ≤ 𝑧ch || ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The 𝑧ch || distribution, fully corrected to particle level, is presented for prompt (charm-quark initiated) Λ+ c-tagged jets with 7 ≤ 𝑝jet ch T < 15 GeV/𝑐 and 3 ≤ 𝑝Λ+ c T < 15 GeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The results are then compared to PYTHIA 8 simulations [19, 22], including a version where mechanisms beyond the leading-colour approximation are considered in string formation processes during hadronisation [20], and to an analogous measurement of the 𝑧ch || distribution of D0 mesons, performed by the ALICE Collaboration [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' A full description of the ALICE setup and apparatus can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 23, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The main detectors used in this analysis are the Inner Tracking System (ITS), which is used for vertex reconstruction and tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' the Time Projection Chamber (TPC), which is used for tracking and particle identification (PID);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' and the Time-Of-Flight (TOF) detector, which is used for PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' These detectors cover a pseudorapidity interval of |𝜂| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The analysis was performed on pp collisions at √𝑠 = 13 TeV, collected using a minimum-bias (MB) trigger during the years 2016, 2017, and 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The trigger condition required 2 In-jet Λ+ c production in pp collisions at √𝑠 = 13 TeV ALICE Collaboration coincident signals in the two scintillator arrays of the V0 detector, with background events originating from beam–gas interactions removed offline using timing information from the V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' To mitigate against pile-up effects, events with multiple reconstructed primary vertices were rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' To ensure uniform acceptance, only events with a primary-vertex position along the beam axis direction of |𝑧vtx| < 10 cm around the nominal interaction point were accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' After the selections described above, the data sample consisted of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7×109 events, corresponding to an integrated luminosity of Lint = 29 nb−1 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The Λ+ c candidates and their charge conjugates were reconstructed via the hadronic Λ+ c → pK0 S → pπ+π− decay channel with a total branching ratio of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='06)% [26], in the Λ+ c transverse-momentum interval of 3 ≤ 𝑝Λ+ c T < 15 GeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Only tracks with |𝜂| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 and 𝑝T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 GeV/𝑐, which fulfilled the track quality selections described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 13, were considered for the Λ+ c reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The Λ+ c candidates themselves were reconstructed in the |𝑦Λ+c | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 rapidity interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The Λ+ c-candidate selection was performed using a multivariate technique based on the Boosted Decision Tree (BDT) algorithm provided by the XGBoost package [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The features considered in the optimisation include the PID signal for the proton track, the invariant mass of the K0 S-meson candidate, and topological variables that exploit the kinematic properties of the displaced K0 S-meson decay vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The training was performed in intervals of Λ+ c-candidate 𝑝T, considering prompt signal candidates from PYTHIA 8 events with the Monash tune [19, 20], transported through a realistic description of the detector geometry and material budget using GEANT 3 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Background candidates were extracted from the sidebands of the invariant-mass distributions in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The probability thresholds of the BDT selections were tuned, using MC simulations, to maximise the statistical significance for the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Further details on the Λ+ c-candidate reconstruction and machine learning procedure are provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 14, where the same reconstruction and BDT model were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' For the events where at least one selected Λ+ c candidate was identified, a jet-finding procedure was performed, using the FastJet package [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Prior to jet clustering, the Λ+ c-candidate daughter tracks were replaced by the reconstructed Λ+ c-candidate four-momentum vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Track-based jet finding was carried out on charged tracks with |𝜂| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='9 and 𝑝T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='15 GeV/𝑐, using the anti-𝑘T algorithm [30], with a resolution parameter of 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Tracks were combined using the 𝐸-scheme recombination [31], with the jet transverse momentum limited to the interval of 5 ≤ 𝑝jet ch T < 35 GeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The full jet cone was required to be within the ALICE central barrel acceptance, limiting the jet axis to the interval |𝜂jet| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Only jets tagged via the presence of a reconstructed Λ+ c candidate amongst their constituents were considered for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' For events where more than one Λ+ c candidate was found, the jet finding and tagging pass was performed independently for each candidate, with only the daughters of that particular candidate replaced by the corresponding Λ+ c four-vector each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In mechanisms of hadronisation that include colour correlations beyond the leading-colour approximation [20], which have been shown to be relevant in hadronic collisions at LHC energies [9], hadrons can be formed in processes that combine quarks from the parton shower with those from the underlying event [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' As such, the underlying event is not well defined with respect to the measured hadron distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Therefore no underlying event correction is implemented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The fragmentation of charm quarks to Λ+ c baryons is probed by measuring the fraction of the jet momentum carried by the Λ+ c along the direction of the jet axis, 𝑧ch || .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' This is calculated for each jet using 𝑧ch || = 𝒑jet · 𝒑Λ+c 𝒑jet · 𝒑jet , (1) where 𝒑jet and 𝒑Λ+c are the jet and Λ+ c three-momentum vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The 𝑧ch || distributions of true Λ+ c-tagged jets were extracted in intervals of Λ+ c 𝑝T and 𝑝jet ch T using a sideband subtraction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' To enact this subtraction, the invariant-mass (𝑚inv) distributions of Λ+ c candidates, obtained for each Λ+ c 𝑝T and 𝑝jet ch T interval, were fitted with a function comprising a Gaussian for the signal 3 In-jet Λ+ c production in pp collisions at √𝑠 = 13 TeV ALICE Collaboration and an exponential for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The fit parameters were then used to define signal (containing the majority of true signal candidates) and sideband (entirely composed of background candidates) regions, defined by |𝑚inv − 𝜇fit| < 2𝜎fit and 4𝜎fit < |𝑚inv − 𝜇fit| < 9𝜎fit, respectively, where 𝜇fit and 𝜎fit represent the mean and sigma of the fitted Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The 𝑧ch || (𝑝Λ+ c T ,𝑝jet ch T ) distributions were extracted in the signal and sideband regions, with the sideband distribution scaled by the ratio of the background function integrals in the signal and sideband regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The sideband distribution was then subtracted from the signal one, with the resulting distribution scaled to account for the fact that the 2𝜎fit width of the signal region only encompasses approximately 95% of the total signal, to obtain the sideband subtracted 𝑧ch || yield in each 𝑝Λ+ c T and 𝑝jet ch T interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' To account for the reconstruction and selection efficiency of the Λ+ c-tagged jet signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' the sideband subtracted 𝑧ch || distributions in each 𝑝Λ+ c T and 𝑝jet ch T interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑁(𝑧ch || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑝Λ+ c T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑝jet ch T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' were scaled by the recon- struction efficiency of prompt Λ+ c-tagged jets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝜖prompt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' and summed over the entire 𝑝Λ+ c T interval to obtain the efficiency-corrected 𝑧ch || yield of Λ+ c-tagged jets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑁corr(𝑧ch || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑝jet ch T ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' given by 𝑁corr(𝑧ch || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑝jet ch T ) = ∑︁ 𝑝Λ+c T 𝑁(𝑧ch || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑝Λ+ c T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 𝑝jet ch T ) 𝜖prompt(𝑝Λ+c T ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' (2) The 𝜖prompt(𝑝Λ+ c T ) efficiency is strongly dependent on 𝑝Λ+ c T , ranging from about 20% at 3 < 𝑝Λ+ c T < 4 GeV/𝑐 to 40% at 12 < 𝑝Λ+ c T < 24 GeV/𝑐, and was calculated using PYTHIA 8 simulations with the Monash tune containing prompt Λ+ c-tagged jets, transported through the detector using GEANT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' This efficiency does not exhibit a 𝑝jet ch T dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In order to isolate the 𝑁corr(𝑧ch || , 𝑝jet ch T ) distribution of prompt Λ+ c-tagged jets, a feed-down subtraction was employed to remove the non-prompt (beauty-quark initiated) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The non-prompt cross section was obtained from particle level POWHEG [33] + PYTHIA 6 [34] + EvtGen [35] simulations, as a function of 𝑝jet ch T , 𝑝Λ+ c T and 𝑧ch || , and was scaled according to the integrated luminosity of the analysed data sample and the branching ratio of the Λ+ c → pK0 S → pπ+π− decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The resulting particle- level yield was multiplied by the ratio of the non-prompt to prompt Λ+ c-tagged jet reconstruction and selection efficiency in intervals of 𝑝Λ+c T and integrated over the 𝑝Λ+c T range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The simulated non-prompt results were then folded to reconstructed level, using a four-dimensional response matrix generated using non-prompt Λ+ c-tagged jets in PYTHIA 8 with the Monash tune, transported through a simulation of the ALICE detector using GEANT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The response matrix was constructed as a function of 𝑝jet ch T and 𝑧ch || at generator and reconstruction levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The folded results were then subtracted from the measured 𝑁corr(𝑧ch || , 𝑝jet ch T ) distribution in data, removing the non-prompt contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The estimated fraction of Λ+ c-tagged jets coming from b-quark fragmentation is found to be about 5%, with no significant 𝑧ch || dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' A two-dimensional Bayesian unfolding procedure [36] was performed to correct for detector effects and obtain the 𝑧ch || distribution for prompt Λ+ c-tagged jets at particle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' A four-dimensional response matrix as a function of 𝑝jet ch T and 𝑧ch || , at generator and reconstruction levels, was populated with prompt Λ+ c-tagged jets, obtained with PYTHIA 8 simulations with the Monash tune, passed through a simulation of the ALICE detector using GEANT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The measured data and response matrix were provided in the intervals of 5 ≤ 𝑝jet ch T < 35 GeV/𝑐 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 ≤ 𝑧ch || ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='0, with the final unfolded results reported in the intervals 7 ≤ 𝑝jet ch T < 15 GeV/𝑐 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 ≤ 𝑧ch || ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Corrections accounting for migrating entries in and out of the response matrix ranges, as modelled by the same MC simulation, were also applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The corrected 𝑧ch || distribution is normalised to the total number of Λ+ c-tagged jets in the reported 𝑧ch || and 𝑝jet ch T interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 4 In-jet Λ+ c production in pp collisions at √𝑠 = 13 TeV ALICE Collaboration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='9 1 ch z 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 ch z /d N ) d jet N (1/ tagged jets + c Λ data Monash CR-BLC Mode 2 PYTHIA 8: = 13 TeV s , pp, ALICE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 R , T k charged jets, anti- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 ≤ jet η , c < 15 GeV/ jet ch T p ≤ 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 ≤ + c Λ y , c < 15 GeV/ + c Λ T p ≤ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='9 1 ch z 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 2 MC/data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='9 1 ch z 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 5 ch z /d N ) d jet N (1/ tagged jets + c Λ tagged jets 0 D = 13 TeV s , pp, ALICE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 R , T k charged jets, anti- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 ≤ jet η , c < 15 GeV/ jet ch T p ≤ 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 ≤ h y , c < 15 GeV/ h T p ≤ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='9 1 ch z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5 2 0 /D + c Λ data PYTHIA 8 Monash PYTHIA 8 CR-BLC Mode 2 Figure 1: (Left) Fully corrected 𝑧ch || distribution of Λ+ c-tagged track-based jets (black open circles) measured in the 7 ≤ 𝑝jet ch T < 15 GeV/𝑐 and 3 ≤ 𝑝Λ+ c T < 15 GeV/𝑐 intervals in pp collisions at √𝑠 = 13 TeV, compared with predictions from different PYTHIA 8 tunes [19, 20, 22] (red-dotted and green-dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The ratios of the MC simulations to the data are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' (Right) Comparison of the measured 𝑧ch || distribution of Λ+ c-tagged jets and the previously measured 𝑧ch || distribution of D0-tagged jets [6], obtained in the same kinematic interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The ratio of the 𝑧ch || distribution of Λ+ c-tagged and D0-tagged jets is shown in the bottom panel for both the data and the different PYTHIA tunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The systematic uncertainties affecting the measurement were evaluated, in each 𝑧ch || interval, by modifying the strategy adopted at various steps of the analysis procedure and assessing the impact on the unfolded 𝑧ch || distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The total systematic uncertainty includes contributions from multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The first considered source is the sideband subtraction procedure (ranging from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7% to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='6% depending on the 𝑧ch || inteval), whose contribution was estimated by varying the invariant-mass fit parameters as well as the invariant-mass intervals of the signal and sideband regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The contribution from the BDT selection of Λ+ c candidates (from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='3% to 19%) was estimated by varying the BDT probability thresholds to induce a 25% variation in the Λ+ c-tagged jet reconstruction and selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The uncertainty from the jet energy resolution (from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='5% to 19%) was estimated by recalculating the response matrix used for unfolding with a 4% reduced tracking efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The reduction in the tracking efficiency was evaluated by varying the track-selection criteria and propagating the ITS–TPC track-matching efficiency uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The uncertainty on the feed-down subtraction (< 2%) was estimated by varying the choice of POWHEG parameters considered to generate the feed-down cross section, including the factorisation and renormalisation scales, as well as the mass of the beauty quark, which were varied according to theoretical prescriptions [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' Finally the contribution from the unfolding procedure (from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='1% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='7%) was estimated by altering the choice of prior, regularisation parameter, and ranges of the response matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' For each of the aforementioned categories, several variations were made and the root-mean-square of the resulting distributions was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The exceptions are related to the contribution associated to the choice of parameters of the POWHEG calculations, where only the largest deviation from the central result, in each direction, was considered, as well as the uncertainty on the jet energy resolution where the variation with respect to the central result was taken as the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' All uncertainties (other than from the feed-down subtraction) were then symmetrised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The uncertainties were combined in quadrature to obtain the total systematic uncertainty on the measurement, which ranges from 13% to 28%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 5 In-jet Λ+ c production in pp collisions at √𝑠 = 13 TeV ALICE Collaboration The fully corrected 𝑧ch || distribution of prompt Λ+ c-tagged track-based jets in the intervals of 7 ≤ 𝑝jet ch T < 15 GeV/𝑐 and 3 ≤ 𝑝Λ+ c T < 15 GeV/𝑐 is presented in the left-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 1 and compared to PYTHIA 8 simulations with two different tunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In PYTHIA 8 the Lund string model of fragmentation is employed, where endpoints are confined by linear potentials encoded in strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' For the case of heavy quarks, the Lund fragmentation function is modified to account for the slower propagation of the massive endpoints compared to their massless counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The Monash tune (red-dotted line) [19], in which the charm fragmentation is tuned on e+e− measurements, predicts a harder fragmentation than the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' An evaluation of the 𝜒2/ndf between the measured data points and the model was performed, combining the statistical and systematic uncertainties on the data in quadrature and assuming the uncertainties are uncorrelated across the 𝑧ch || intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' This exercise determines that there is a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='4% probability that the model describes the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' A better agreement is achieved by the PYTHIA 8 with the CR-BLC Mode 2 tune, that includes colour reconnection mechanisms beyond the leading-colour approximation [22] (green- dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In this model, the minimisation of the string potential is implemented considering the SU(3) multiplet structure of QCD in a more realistic way than in the leading-colour approximation, allowing for the formation of “baryonic” configurations where for example two colours can combine coherently to form an anti-colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The same 𝜒2/ndf approach results in a 78% probability that the model describes the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The simulation with PYTHIA 8 with the CR-BLC Mode 2 tune also provides a much more accurate description of the Λ+ c/D0 cross section ratio, previously measured in pp collisions at the LHC [10–14, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In the right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' 1, a comparison of the 𝑧ch || distribution of Λ+ c-tagged jets and the 𝑧ch || distri- bution previously measured for D0-tagged jets [6] is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The latter is consistent with PYTHIA 8 simulations using both the Monash and CR-BLC Mode 2 tunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The ratio of the two distributions is also presented in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The uncertainty from the jet energy resolution was considered to be correlated between the Λ+ c-tagged jet and D0-tagged jet measurements and was evaluated directly on the ratio of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The remaining uncertainties were considered uncorrelated when taking the ratio and were then combined in quadrature with the uncertainty of the jet energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The uncertainties were considered uncorrelated across the 𝑧ch || intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The same 𝜒2/ndf exercise described above determines that there is a 12% probability that the measured ratio is described by a flat distribution at unity, hinting at a softer fragmentation of charm quarks into charm baryons than charm mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The ratio is better described by the PYTHIA 8 simulations with the CR-BLC Mode 2 compared to the ones with the Monash tune, with the former describing the data with 88% probability compared to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content='03% probability for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' In summary the first measurement in hadronic collisions of the longitudinal momentum fraction of the jet carried by Λ+ c baryons was presented for pp collisions at √𝑠 = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The result is fully corrected to particle level and obtained in the jet and Λ+ c transverse-momentum intervals of 7 ≤ 𝑝jet ch T < 15 GeV/𝑐 and 3 ≤ 𝑝Λ+c T < 15 GeV/𝑐, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The measurement presented in this Letter hints that charm quarks have a softer fragmentation into Λ+ c baryons compared to D0 mesons, in the measured kinematic interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' One possible explanation is that charm-baryon production is favoured in the presence of higher particle multiplicity originating from both the jet fragmentation and the underlying event, which could be tested with future measurements of the in-jet multiplicity of Λ+ c-tagged jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The fragmentation of charm quarks into Λ+ c baryons in hadronic collisions exhibits tension with simulations tuned on e+e− data that employ a leading-colour formalism of hadronisation, such as in the Monash tune of PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' This occurs despite their successful description of the fragmentation of charm quarks into D0 mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' However, the inclusion of mechanisms sensitive to the surrounding partonic density that feature colour reconnection beyond the leading-colour approximation results in a better agreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' This result also partially explains the 𝑝T shape of the prompt Λ+ c/D0 cross section ratio [10–14, 38], which shows a peak at low 𝑝T (≈ 3 GeV/𝑐) and is also described within uncertainties by PYTHIA 8 with the CR-BLC Mode 2 tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' The 𝑝T trend of this ratio is driven by the fact that the Λ+ c baryons produced from the fragmenting charm quark carry a significantly lower fraction of the charm-quark transverse momentum than the D0 mesons 6 In-jet Λ+ c production in pp collisions at √𝑠 = 13 TeV ALICE Collaboration produced in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'} +page_content=' References [1] J.' metadata={'source': 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Xiangkun Liu5, Catherine Heymans6, 7, Benjamin +Joachimi8, Marika Asgari9, Maciej Bilicki10, Hendrik Hildebrandt6, Konrad Kuijken11, Tilman Tröster6, Jan Luca van +den Busch8, 12, Angus Wright7, and Ziang Yan7 +1 Shanghai Astronomical Observatory (SHAO), Nandan Road 80, Shanghai 200030, China +2 Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China +3 Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240, China +4 Tsung-Dao Lee Institute, Shanghai, 200240, China +5 South-Western Institute for Astronomy Research, Yunnan University, Kunming, 650500, China +6 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK +7 Ruhr-Universität Bochum, Astronomisches Institut, German Centre for Cosmological Lensing (GCCL), Universitätsstr. 150, +44801, Bochum, Germany +8 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK +9 E.A Milne Centre, University of Hull, Cottingham Road, Hull, HU6 7RX, United Kingdom +10 Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland +11 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands +12 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany +Received January 30, 2023; accepted ? +ABSTRACT +Context. Galaxy shear - cosmic microwave background (CMB) lensing convergence cross-correlations contain additional informa- +tion on cosmology to auto-correlations. While being immune to certain systematic effects, they are affected by the galaxy intrinsic +alignments (IA). This may be responsible for the reported low lensing amplitude of the galaxy shear × CMB convergence cross- +correlations, compared to the standard Planck ΛCDM (cosmological constant and cold dark matter) cosmology prediction. +Aims. In this work, we investigate how IA affects the Kilo-Degree Survey (KiDS) galaxy lensing shear - Planck CMB lensing +convergence cross-correlation and compare it to previous treatments with or without IA taken into consideration. +Methods. More specifically, we compare marginalization over IA parameters and the IA self-calibration (SC) method (with additional +observables defined only from the source galaxies) and prove that SC can efficiently break the degeneracy between the CMB lensing +amplitude Alens and the IA amplitude AIA. We further investigate how different systematics affect the resulting AIA and Alens, and +validate our results with the MICE2 simulation. +Results. We find that by including the SC method to constrain IA, the information loss due to the degeneracy between CMB lensing +and IA is strongly reduced. The best-fit values are Alens = 0.84+0.22 +−0.22 and AIA = 0.60+1.03 +−1.03, while different angular scale cuts can affect +Alens by ∼ 10%. We show that appropriate treatment of the boost factor, cosmic magnification, and photometric redshift modeling is +important for obtaining the correct IA and cosmological results. +Key words. cosmology – weak lensing – CMB lensing – intrinsic alignment – self-calibration +1. Introduction +Weak lensing due to the distortion of light by gravity is a power- +ful probe of the underlying matter distribution and the encoded +secrets of cosmological physics such as dark matter, dark energy, +and the nature of gravity (Refregier 2003; Mandelbaum 2018). +The auto-correlation statistics have been widely used in the anal- +ysis, both for galaxy lensing shear, e.g. “cosmic shear” (Hilde- +brandt et al. 2017; Hamana et al. 2020; Hikage et al. 2019; As- +gari et al. 2021; Secco et al. 2022; Amon et al. 2022), and CMB +lensing convergence (Planck Collaboration et al. 2020c; Omori +et al. 2017). Furthermore, cross-correlations between galaxy +⋆ e-mail: ji.yao@shao.ac.cn +⋆⋆ e-mail: hyshan@shao.ac.cn +⋆⋆⋆ e-mail: zhangpj@sjtu.edu.cn +shear and CMB lensing have been measured extensively (Hand +et al. 2015; Chisari et al. 2015; Liu & Hill 2015; Kirk et al. 2016; +Harnois-Déraps et al. 2016; Singh et al. 2017a; Harnois-Déraps +et al. 2017; Omori et al. 2019; Namikawa et al. 2019; Marques +et al. 2020; Robertson et al. 2021). Cross-correlation statistics +contain highly complementary information to auto-correlations, +both for cosmology and the cross-check of systematics. They +partly reveal the hidden redshift information in CMB lensing +and are more sensitive to structure growth at redshifts between +the epochs probed by galaxy shear and CMB lensing. Cross- +correlations are also immune to additive errors in shear measure- +ment and provide an external diagnosis of multiplicative errors +(Schaan et al. 2017). +Most existing cross-correlation measurements have found a +lower CMB lensing amplitude than the prediction of their as- +Article number, page 1 of 15 +arXiv:2301.13437v1 [astro-ph.CO] 31 Jan 2023 + +A&A proofs: manuscript no. aanda +sumed ΛCDM cosmology (Hand et al. 2015; Liu & Hill 2015; +Kirk et al. 2016; Harnois-Déraps et al. 2016, 2017; Singh et al. +2017a; Marques et al. 2020; Robertson et al. 2021). The ra- +tio, which is normally referred as the CMB lensing amplitude, +Alens ∼ 0.5-0.9, although the deviation from unity is only within +1-2σ. The low lensing amplitude is consistent across many com- +binations of data sets and analysis methods, suggesting the ex- +istence of a common systematic errors or a deviation from the +best-fit Planck cosmology. This might be related to the tension +between galaxy lensing surveys and Planck CMB observation +(Lin & Ishak 2017; Chang et al. 2019; Heymans et al. 2021), +and the Planck internal inconsistencies (Planck Collaboration +et al. 2020a,b). In this paper we focus on the galaxy intrinsic +alignment (IA), which can mimic weak lensing signals (Croft & +Metzler 2000; Catelan et al. 2001; Crittenden et al. 2001; Lee +& Pen 2001; Jing 2002; Hirata & Seljak 2004; Heymans et al. +2004; Bridle & King 2007; Okumura et al. 2009; Joachimi et al. +2013; Kiessling et al. 2015; Blazek et al. 2015; Rong et al. 2015; +Krause et al. 2016; Blazek et al. 2019; Troxel et al. 2018; Chis- +ari et al. 2017; Xia et al. 2017; Samuroff et al. 2019; Yao et al. +2020a; Samuroff et al. 2021; Yao et al. 2020b). Here the CMB +lensing convergence is expected to be anti-correlated with the +intrinsic ellipticities of the foreground galaxy field, resulting in +a dilution of the overall cross-correlation signal (Troxel & Ishak +2014; Chisari et al. 2015; Kirk et al. 2015; Omori et al. 2019; +Robertson et al. 2021). Taking IA into account can alleviate the +tension in Alens, at the expense of a significant loss of lensing +constraining power, because of the degeneracy between the lens- +ing amplitude Alens and the IA amplitude AIA. Therefore, a com- +mon compromise is to fix both the IA model and its amplitude +AIA (Kirk et al. 2016; Harnois-Déraps et al. 2017; Omori et al. +2019) or assume a strong prior (Robertson et al. 2021). +Since IA is already a major limiting factor in the current +cross-correlation analysis, its mitigation will be essential for up- +coming measurements with significantly smaller statistical er- +rors. We utilize the IA self-calibration (SC) method (Zhang +2010a,b; Troxel & Ishak 2012a,b; Yao et al. 2017, 2019), which +is a galaxy-galaxy lensing method but with a different weight- +ing scheme, to mitigate the IA problem in the shear-convergence +cross-correlation. It is based on the fact that the IA-galaxy cor- +relation is insensitive to the redshift order, while it matters for +lensing-galaxy correlation whether the lens is in front of the +source or not. Therefore, we can isolate IA by comparing ex- +tra observables, i.e., the galaxy shear × number density cross- +correlation with a different weighting of the redshift pairs. This +measurement of IA is independent of a physical model of the +IA and requires no data external to the shear data. SC was first +applied to KiDS450/KV450 (Yao et al. 2020a; Pedersen et al. +2020) and DECaLS DR3 (Yao et al. 2020b) and has enabled +significant IA detections. The detected IA signal can then be +applied to remove IA in the lensing shear auto-correlation and +shear-convergence cross-correlation. The IA information is ob- +tained from a shear × number density cross-correlation within +the same photometric redshift (photo-z) bin, more importantly, +with different weighting schemes on the photo-z ordering, which +is usually not used for cosmological parameter constraints. We +find that this removal of IA losses almost no cosmological infor- +mation. +In previous work Yao et al. (2020b), we have demonstrated +the importance and methodology of including certain types of +systematics in the SC lensing-IA separation method, namely +galaxy bias, the covariance between the separated lensing signal +and IA signal, the IA signal drop QIg due to the photo-z selection, +and the scale dependency of the signal drops QGg and QIg. In this +work, we further investigate other sources of systematics, includ- +ing the boost factor (Mandelbaum et al. 2005), photo-z modeling +bias (Yao et al. 2020a), and cosmic magnification (Bartelmann +1995; Bartelmann & Schneider 2001; Yang et al. 2017; Liu et al. +2021). Interestingly, as the survey goes to higher redshift, the +contamination to the SC method from magnification will quickly +increase to a non-negligible level. The cosmic magnification will +change the observed galaxy number density due to the lensing- +magnified flux and lensing-enlarged area, therefore biasing our +SC analysis. We investigate the proper treatments for the above +systematics together with the cosmological study. +This paper is organized as follows. In Sect. 2 we review the +physics of galaxy shear × CMB convergence and how our SC +method works to subtract the IA information. In Sect. 3 we in- +troduce the KiDS-1000 and Planck data used in this work, and +the MICE2 simulation (van den Busch et al. 2020; Fosalba et al. +2015) we use to validate how the SC method is affected by differ- +ent systematics. We show the measurements of the observables +in Sect. 4. The results and summary are shown in Sect. 5 and 6. +2. Methods +We apply our self-calibration method to separate the intrinsic +alignment and the lensing signals and show how the intrinsic +alignment will bias the galaxy shear-CMB convergence corre- +lation. In this section, we review the theory of lensing cross- +correlation and the self-calibration method, with a modification +to account for the contamination from cosmic magnification. +2.1. Galaxy shear × CMB convergence +The gravitational field can distort the shape of the background +source galaxy image and introduce an extra shape that is tan- +gentially aligned to the lens. This gravitational shear γG of the +source galaxy contains integral information of the foreground +overdensity along the line of sight (Bartelmann & Schneider +2001). Similarly, the photons from the CMB are deflected, and +the lensing convergence κ can be reconstructed from the CMB +temperature and polarization observations (Planck Collabora- +tion et al. 2020c). By correlating these two quantities +� +γGκ +� +, +we probe the clustering of the underlying matter field ⟨δδ⟩. In +harmonic space while assuming flat space (Omori et al. 2019; +Marques et al. 2020), we have: +CκgalκCMB(ℓ) = +� χCMB +0 +qgal(χ)qCMB(χ) +χ2 +Pδ +� +k = ℓ + 1/2 +χ +, z +� +dχ. +(1) +Eq. (1) is the galaxy-lensing CMB-lensing cross angular +power spectrum, which probes the matter power spectrum +Pδ(k, z), as well as the background geometry χ(z) if precision +allows. Here z is the redshift, χ is the comoving distance, k is +the wavenumber, ℓ is the angular mode, qgal(χ) and qCMB(χ) are +the lensing efficiency functions for galaxy-lensing and CMB- +lensing, with the analytical forms: +qgal(χl) = 3 +2Ωm +H2 +0 +c2 (1 + zl) +� ∞ +χl +n(χs)(χs − χl)χl +χs +dχs, +(2) +qCMB(χl) = 3 +2Ωm +H2 +0 +c2 (1 + zl)(χs − χl)χl +χs +, +(3) +where χs and χl are the comoving distance to the source and +lens, and the χs in Eq. (3) takes CMB as the source of light +(z ∼ 1100). We note the spacial curvature Ωk = 0 is assumed +Article number, page 2 of 15 + +Yao et al 2022: KiDS shear × Planck lensing and IA removal +so that the comoving angular diameter distances in Eqs. (2) and +(3) are replaced with the comoving radial distances. Here n(χ) +gives the source galaxy distribution as a function of comoving +distance, and it is connected with the galaxy redshift distribution +via n(χ) = n(z)dz/dχ. In this work, we only use one redshift bin +due to the limit of the total S/N on the CMB lensing signal, while +a tomographic example can be found in Harnois-Déraps et al. +(2017). In the future with higher S/N, for example, for CMB-S4 +× LSST, tomography can be used to subtract more cosmological +information. +The shear-convergence cross-correlation function measured +in real space is given by the Hankel transformation: +wGκ(θ) = 1 +2π +� ∞ +0 +dℓℓCκgalκCMB(ℓ)J2(ℓθ), +(4) +where J2(x) is the Bessel function of the first kind and order 2. +The “G” represents the gravitational lensing shear γG, to be sep- +arated from the intrinsic alignment γI in the following section. +Also for the current low S/N reasons, we choose not to in- +vestigate full cosmological constraints in this work. Instead, we +perform a matched-filter fitting, with lensing amplitude Alens that +suits ˆwGκ = AlenswGκ, where ˆwGκ is the measured correlation +function, and wGκ is the theoretical model. +2.2. Intrinsic alignment of galaxies +The observed galaxy shear estimator contains three components: +gravitational shear, an intrinsic alignment term, and random +noise, namely, ˆγ = γG + γI + γN. Both the gravitational shear +and the IA term are related to the underlying matter overdensity +δ and are associated with the large-scale structure. This means +that when we cross-correlate the galaxy shape and the CMB con- +vergence, there will be contributions from both lensing and IA: +⟨ˆγκ⟩ = +� +γGκ +� ++ +� +γIκ +� +. +(5) +Therefore the IA part of the correlation will contaminate the +measurement and lead to a bias in the lensing amplitude Alens +or the cosmological parameters when assuming ⟨ˆγκ⟩ = +� +γGκ +� +. +The IA-convergence correlation function is linked to the IA- +convergence power spectrum +CIκCMB = +� χCMB +0 +n(χ)qCMB(χ) +χ2 +Pδ,γI +� +k = ℓ + 1/2 +χ +, z +� +dχ. +(6) +Here Pδ,γI is the 3D matter-IA power spectrum. The conventional +method is to assume an IA model with some nuisance parame- +ters, which will enter the fitting process. The most widely used +IA model is the non-linear linear tidal alignment model (Cate- +lan et al. 2001; Hirata & Seljak 2004; Bridle & King 2007), ex- +pressed as: +Pδ,γI = −AIA(L, z)C1ρm,0 +D(z) Pδ(k; χ), +(7) +which is proportional to the non-linear matter power spectrum +Pδ, suggesting that the IA is caused by the gravitational tidal +field. AIA is the IA amplitude, which can be redshift(z)- and +luminosity(L)- dependent (Joachimi et al. 2011). Its redshift evo- +lution has been measured recently in simulations (Chisari et al. +2016; Samuroff et al. 2021) and suggestions in observations with +low significance (Johnston et al. 2019; Yao et al. 2020b; Secco +et al. 2022; Tonegawa & Okumura 2022). The other related +quantities include: the mean matter density of the universe at z = +0, expressed as ρm,0 = ρcritΩm,0; C1 = 5 × 10−14(h2Msun/Mpc−3) +the empirical amplitude taken from Brown et al. (2002) and the +normalized linear growth factor D(z). We note that the IA model +in Eq. (7) can be replaced by more complicated models as in +Krause et al. (2016); Blazek et al. (2015, 2019); Fortuna et al. +(2021) for different samples (Yao et al. 2020b; Samuroff et al. +2021; Zjupa et al. 2020). The self-calibration method can intro- +duce new observables to constrain IA with additional constrain- +ing power, and in the future when the signal-to-noise (S/N) al- +lows, it can be extended to constrain more complicated IA mod- +els. +2.3. Self-calibration of intrinsic alignment +The IA self-calibration (SC) method (Zhang 2010b; Yao et al. +2017, 2019, 2020a,b) uses the same galaxy sample as both the +source and the lens, which is different from most galaxy-galaxy +lensing studies. It introduces two observables: the shape-galaxy +correlation in the same redshift bin wγg, and a similar correlation +wγg|S using the pairs where the photo-z of the source galaxy is +lower than the photo-z of the lens galaxy, namely +zP +γ < zP +g +(8) +(this will be denoted as “the SC selection”). +In this work, we extend our methodology to include the im- +pact from cosmic magnification (Bartelmann 1995; Bartelmann +& Schneider 2001; Yang et al. 2017; Liu et al. 2021). Because of +the existence of magnification, the intrinsic galaxy number den- +sity field δg is affected by the foreground lensing convergence +κgal, leading to a lensed galaxy overdensity +δL +g = δg + gmagκgal, +(9) +where the prefactor writes gmag = 2(α − 1) for a complete and +flux-limited sample. It accounts for the increase in galaxy num- +ber density due to lensing-magnified flux (α = −d ln N/d ln F, +where N(F) denotes the galaxy number N that is brighter than +the flux limit F) and the decrease of galaxy number density +due to the lensing-area-enlargement (-2 in gmag). The observed +shape-galaxy correlation is given by +� +ˆγδL +g +� += +� +(γG + γI)(δg + gmagκgal) +� +. +(10) +The two SC observables can be written as: +wγgL +ii (θ) = wGg +ii (θ) + wIg +ii (θ) + gmag +� +wGκgal +ii +(θ) + wIκgal +ii +(θ) +� +, +(11) +wγgL +ii |S(θ) = wGg +ii |S(θ) + wIg +ii |S(θ) + gmag +� +wGκgal +ii +|S(θ) + wIκgal +ii +|S(θ) +� +, +(12) +where the “|S” denotes the SC selection, and i denotes the i-th +redshift bin if tomography is applied. The lensing-galaxy wGg +and the IA-galaxy wIg signal are affected by this SC selection, as +quantified by the Q parameters: +QGg +i (θ) ≡ wGg +ii |S(θ) +wGg +ii (θ) +, +(13) +QIg +i (θ) ≡ wIg +ii |S(θ) +wIg +ii (θ) +. +(14) +For the lensing signal to exist, the redshift of the source, zγ, +needs to be greater than the redshift of the lens, zg: zγ > zg. +Article number, page 3 of 15 + +A&A proofs: manuscript no. aanda +0.4 +0.45 +0.5 +0.55 +0.6 +zP +g +-0.0004 +-0.0002 +0 +0.0002 +0.0004 +wXg(zP|zP +g = 0.5, +) +X=G, lensing, += 1' +X=I, IA, += 1' +X=G, lensing, += 5' +X=I, IA, += 5' +X=G, lensing, += 50' +X=I, IA, += 50' +Fig. 1. A toy model to illustrate the different redshift dependences for +the lensing signal and the IA signal, and why the SC selection Eq. (8) +works. We place many lens galaxies at photo-z zP +g = 0.5 (the grey dotted +line), while allowing the photo-z of the source galaxies zP +γ to change (x- +axis) to evaluate the corresponding lensing correlation function wGg or +IA correlation function wIg at different angular separation θ. The true-z +has a Gaussian scatter of 0.04 (this number is chosen for exhibition, so +that the lensing/IA signals have comparable maximum/minimum val- +ues) around the photo-z, for both source galaxies and lens galaxies. As +the gravitational lensing shear is an optical shape that requires zg < zγ, it +will have a non-symmetric power around zP +g, as the positive solid curves +show. This also demonstrate QGg ≪ 1 according to Eq. (13). As the +IA shape is a dynamical shape, it does not have requirements on the +relative redshifts, leading to a symmetric power around zP +g, as the neg- +ative dashed curves show. This also demonstrate QIg ∼ 1 according to +Eq. (14). These relations hold for signals at different angular separa- +tions (different colors). The different IA models (which could deviate +from Eq. 7 and AIA = 1 being assumed) will only change the rela- +tive amplitudes of the negative signals at different scales, but not the +redshift-dependency around zP +g. We note at such a redshift range, the +magnification signal is much smaller than the IA signal. +The SC photo-z selection zP +γ < zP +g largely reduces the lensing +signal, leading to QGg ≪ 1. The IA signal does not rely on the +ordering along the line-of-sight, with QIg ∼ 1. The lensing-drop +QGg and the IA-drop QIg are dependent on the photo-z quality, +as described in Zhang (2010b); Yao et al. (2017, 2020a,b). If the +photo-z quality is perfect, the SC selection will result in no lens- +ing signal so that QGg approaches 0. For incorrect photo-zs, the +SC selection fails and QGg is ∼ 1. Given a photo-z distribution +nP(zP) and the true-z distribution n(z), the lensing-drop QGg and +IA-drop QIg can be theoretically derived, following Yao et al. +(2020a,b), with more technical details in Appendix A. We also +present a toy model to visualize how the SC selection works in +Fig. 1. +We quantitatively test the terms in Eq. (11), and they gener- +ally follow |wIκgal| < |wGκgal| ≪ |wIg| < |wGg| for z < 0.9 data, +therefore in previous analysis (Zhang 2010b; Yao et al. 2020a,b) +the magnification terms were neglected. For the z ∼ 1 galax- +ies, however, the magnification term wGκgal quickly approaches +wIg and becomes a non-negligible source of contamination to +the SC method. In Fig. 2 we show a theoretical comparison of +the angular power spectra. We can write the SC selection for the +magnification term as wGκgal|S = QGκwGκgal. The drop of the sig- +nal QGκ ∼ QIg ∼ 1 given that these are not z-pair-dependent +correlations, therefore the magnification signal wGκgal will con- +10 +100 +1000 +ℓ +10−10 +10−9 +10−8 +10−7 +C(ℓ) +CGg, bg,eff = 0.88 +CIg, bg,eff = 0.88, AIA = 0.6 +gmagCGκgal, gmag = −0.3 +Fig. 2. A theoretical comparison between the galaxy-shear CGg(ℓ), +galaxy-IA CIg(ℓ) and shear-magnification gmagCGκgal(ℓ) angular power +spectra, with the best-fit of our baseline analysis and the redshift distri- +bution n(z) from KiDS-1000 0.5 < zP < 1.2 shear catalog. The dashed +lines represent negative signals. This figure demonstrates that the mag- +nification contamination is important in the self-calibration method for +the high-z KiDS source sample. +taminate the IA signal wIg due to similar behavior, leaving the +lensing signal wGg unaffected. We note the wIκ term is negligible +in this work. +After measuring the galaxy-galaxy lensing observables +{wγgL, wγgL|S} and the drops of the signals {QGg, QIg} (see +Eq. (13), (14) and Appendix A for more details), the corre- +sponding lensing-galaxy correlation wGg, IA-galaxy correlation +wIg and shear-magnification correlation wGκ can be linearly ob- +tained: +wGg +ii (θ) = QIg +i (θ)wγgL +ii (θ) − wγgL +ii |S(θ) +QIg +i (θ) − QGg +i (θ) +, +(15) +wIg +ii (θ) + wGκgal +ii +(θ) = wγgL +ii |S(θ) − QGg +i (θ)wγgL +ii (θ) +QIg +i (θ) − QGg +i (θ) +. +(16) +In previous work, the IA information was directly extracted +in wIg. However, as shown in Fig. 2 and Eq. 16, for KiDS the +subtracted signal suffers from the contamination from a magni- +fication term wGκ. By constraining the measurements of {wGg, +wIg+wGκgal, wγκCMB} together, including the covariance, will lead +to robust constraints on both the lensing amplitude and the nui- +sance parameters. For the current stage where the S/N for the +measurements are not very high, we choose to ignore the pos- +sible scale-dependent features for the effective galaxy bias bg,eff +and IA amplitude AIA, and assume they are linear and determin- +istic. The parameters {Alens, AIA, bg,eff, gmag} are connected to +the observables following: +wGg(θ) = bg,effwGm +theory(θ), +(17) +wIg(θ) + wGκgal(θ) = bg,effAIAwIm +theory(θ) + gmagwGκgal +theory(θ), +(18) +wγκCMB(θ) = AlenswGκCMB +theory (θ) + AIAwIκCMB +theory(θ), +(19) +where “m” stands for matter, which is the case if one sets the ef- +fective galaxy bias bg,eff = 1. We separate the CMB convergence +and the galaxy convergence (due to magnification) with κCMB +Article number, page 4 of 15 + +Yao et al 2022: KiDS shear × Planck lensing and IA removal +Table 1. The ΛCDM cosmological parameters adopted in this work, +corresponding to the best-fit cosmology from Planck Collaboration +et al. (2020a), and the KiDS-1000 multivariate maximum posterior +(MAP) results from the two-point correlation functions ξ±, the band +powers C(ℓ), and the COSEBIs (Complete Orthogonal Sets of E/B- +Integrals) as in Asgari et al. (2021). +Survey +h0 +Ωbh2 +Ωch2 +ns +σ8 +Planck +0.673 +0.022 +0.120 +0.966 +0.812 +KiDS ξ± +0.711 +0.023 +0.088 +0.928 +0.895 +KiDS C(ℓ) +0.704 +0.022 +0.132 +0.999 +0.723 +KiDS COSEBI +0.727 +0.023 +0.105 +0.949 +0.772 +and κgal. On the LHS of Eq. (17), (18) and (19) are the measure- +ments, while on the RHS the correlations w(θ) are the theoreti- +cal predictions assuming Planck cosmology (Planck Collabora- +tion et al. 2020a), see Table 1. We note the Q values being used +to obtain the LHS are also cosmology dependent, however, the +sensitivity is weak as the cosmological part is mostly canceled +when taking the ratio in Eq. (13) and (14). We tested if the fidu- +cial cosmology is changed to any of the KiDS-1000 cosmolo- +gies in Table 1, the Qs will change by ∼ 1%, similar to Yao et al. +(2020b), and the resulting changes to the fitting parameters {AIA, +bg,eff, gmag, Alens} are negligible. However, considering the RHS, +those four fitting parameters are sensitive to the fiducial cosmol- +ogy used to produce the wtheory values when magnification exists, +which differs from previous analysis (Yao et al. 2020b). The the- +oretical predictions wtheory are calculated with ccl1 (Chisari et al. +2019) and camb2 (Lewis et al. 2000). The effective galaxy bias +bg,eff in this work is used to separate from the true galaxy bias of +this sample, as we will discuss later it can absorb several sources +of systematics. +The theoretical prediction of wGκCMB +theory (θ) is given in Eq. (4), +and wIκgal +theory(θ) is obtained similarly with the Hankel transform +from its power spectrum as in Eq. (6). The wGm +theory, wIm +theory and +wGκgal +theory terms are the Hankel transform from the following angu- +lar power spectra: +CGm(ℓ) = +� zmax +zmin +qgal(χ)n(χ) +χ2 +Pδ +� +k = ℓ + 1/2 +χ +, z +� +dχ, +(20) +CIm(ℓ) = +� zmax +zmin +n(χ)n(χ) +χ2 +Pδ,γI +� +k = ℓ + 1/2 +χ +, z +� +dχ, +(21) +CGκgal(ℓ) = +� zmax +zmin +qgal(χ)qgal(χ) +χ2 +Pδ +� +k = ℓ + 1/2 +χ +, z +� +dχ. +(22) +As discussed in previous work (Yao et al. 2020b), by in- +cluding the effective galaxy bias bg,eff, we can obtain an unbi- +ased estimation of AIA. This information will be propagated into +Eq. (19) to break the degeneracy between AIA and Alens. In this +work, we further extend the fitting to include the impact from +magnification with the nuisance parameter gmag. We will show +later that an unbiased CMB lensing amplitude Alens can be ob- +tained from the simultaneous fitting of Eq. (17), (18) and (19). +3. Data +In this section, we introduce the data we use for the +� +γκCMB� +cross-correlation study. Additionally, we use mock KiDS data, +1 Core Cosmology Library, https://github.com/LSSTDESC/CCL +2 Code for Anisotropies in the Microwave Background, https:// +camb.info/ +based on the MICE2 simulation (see van den Busch et al. (2020) +for details) to quantify the potential bias in the SC method due +to magnification, photo-z modeling, and the boost factor. +3.1. KiDS-1000 shear catalog +We use the fourth data release of the Kilo-Degree Survey that +covers 1006 deg2, known as KiDS-1000 (Kuijken et al. 2019). It +has images from four optical bands ugri and five near-infrared +bands ZYJHKs. The observed galaxies can reach a primary +r−band median limiting 5σ point source magnitude at ∼ 25. The +shear catalog (Giblin et al. 2021) contains ∼ 21 M galaxies and +is divided into five tomographic bins in the range 0.1 < zB < 1.2 +based on the bpz (Benitez 2000) algorithm. The ellipticity dis- +persion σϵ is ∼ 0.27 per component, and the shear multiplicative +bias is generally consistent with 0. +The KiDS data are processed by theli (Erben et al. 2013) +and Astro-WISE (Begeman et al. 2013; de Jong et al. 2015). +Shears are measured using lensfit (Miller et al. 2013), and pho- +tometric redshifts are obtained from PSF-matched photometry +and calibrated using external overlapping spectroscopic surveys +(Hildebrandt et al. 2021). +The application of SC requires not only an accurate redshift +distribution n(z), but also relatively accurate photo-z for each +galaxy, serving for the SC selection (Eq. 8). We discussed in pre- +vious work (Yao et al. 2020a) that the quality of photo-z is very +important for the lensing-IA separation. Therefore in this work, +we choose to combine the three high-z bins, namely bin 3+4+5 +in KiDS-1000 data, as a large bin so that the photo-z error for +an individual galaxy is relatively small compared to the total +bin width. The photo-z and the SOM-calibrated redshift distri- +butions are shown in Fig. 3. We choose to use the high-z bins be- +cause the CMB lensing efficiency Eq. (3) peaks at z ∼ 1 to 2 (see +lower panel of Fig. 3), while the S/N for the cross-correlation is +very low for the two low-z bins of KiDS-1000. +To account for the selection functions for the shape of the +footprint (Mandelbaum et al. 2006) of the overlapped region and +the varying galaxy number density due to observation (Johnston +et al. 2021; Rezaie et al. 2020), we divide the region into 200 +sub-regions with a resolution of Healpix Nside = 512 (∼ 50 +arcmin2 per pixel), and generate random points with 20 times +the number of galaxies of the KiDS-1000 shear catalog in each +sub-region. The pixels within the same sub-region are assigned +the same galaxy numbers. This random catalog is used for the +SC-related galaxy-galaxy lensing calculation, while its potential +defects will not extend to cross-correlations. +3.2. Planck legacy lensing map +We use the CMB lensing map κ(θ) from the Planck data release +(Planck Collaboration et al. 2020c). The CMB lensing map is +reconstructed with the quadratic estimator with the minimum- +variance method combining the temperature map and the polar- +ization map, after foreground removal with the SMICA method +(Planck Collaboration et al. 2020a). It covers fsky = 0.671 of the +whole sky with the maximum multiple ℓ = 4096. +In this work we combine the footprint from the Planck lens- +ing map and the mask of the KiDS-1000 shear catalog, leading +to an overlapped region of ∼ 829 deg2. We include the Planck +Wiener filter (Planck Collaboration et al. 2020c) +ˆκWF +ℓm = +Cφφ,fid +ℓ +Cφφ,fid +ℓ ++ Nφφ +ℓ +ˆκMV +ℓm +(23) +Article number, page 5 of 15 + +A&A proofs: manuscript no. aanda +0 +0.5 +1 +1.5 +2 +2.5 +n(z) +n(z) +nP(zP) +0 +0.5 +1 +1.5 +2 +z +0 +0.2 +0.4 +0.6 +0.8 +1 +lensing efficiency +galaxy lensing +CMB lensing +Fig. 3. The photo-z distribution and the SOM-reconstructed redshift dis- +tribution of the combined galaxy sample in this work. The correspond- +ing galaxy lensing efficiency Eq. (2) and its comparison with CMB lens- +ing efficiency Eq. (3) are shown in the lower panel. +to strengthen the CMB lensing signal at large scales, which will +also lead to a suppression of the power spectrum at small scales, +where the noise dominates (Dong et al. 2021). The Wiener filter +is used both in the CMB lensing κ map and in the theoretical pre- +dictions of Eq. (1) to prevent potential bias. After the application +of the Wiener filter, we use Healpy3 (Górski et al. 2005; Zonca +et al. 2019) to convert the κℓm to the desired κ-map, and rotate +from the galactic coordinates of Planck to the J2000 coordinates +of KiDS with Astropy (Astropy Collaboration et al. 2013). The +two-point correlation functions are calculated with TreeCorr 4 +(Jarvis et al. 2004). +3.3. MICE2 mock catalog +Additionally, we use the MICE2 simulation gold samples (van +den Busch et al. 2020; Fosalba et al. 2015), which highly +mimic the KiDS-1000 shear catalog galaxies, to validate our +SC method, concerning cosmic magnification and photo-z PDF +model bias. MICE2 uses a simulation box width of 3.1 h−1Gpc, +particle mass resolution of 2.9 × 1010 h−1M⊙, and a total particle +number of ∼ 6.9 × 1010. The fiducial cosmology is flat ΛCDM +with Ωm = 0.25, σ8 = 0.8, Ωb = 0.044, ΩΛ = 0.75 and h = 0.7. +The halos are identified with Friends-of-Friends as in Crocce +et al. (2015). The galaxies are populated within the halos with +a mixture of halo abundance matching (HAM) and halo occupa- +tion distribution (HOD) up to z ∼ 1.4 (Carretero et al. 2015). +We note that in the MICE2 simulation that we use for the +KiDS samples, intrinsic alignment is not yet included in the +galaxy shapes (while an IA-included version can be found in +Hoffmann et al. (2022), but for DES). So that we aim to get +AIA = 0 to validate the SC method, while considering system- +atics from cosmic magnification and photo-z model bias, in ad- +3 https://github.com/healpy/healpy +4 https://github.com/rmjarvis/TreeCorr +101 +102 +103 +104 +ℓ +0.2 +0.4 +0.6 +0.8 +1.0 +Q(ℓ) & Q(θ) +QGg(ℓ) +QIg(ℓ) +100 +101 +102 +θ [arcmin] +QGg(θ) +QIg(θ) +Fig. 4. The lensing-drop QGg and the IA-drop QIg as a function of ℓ and +θ by applying the SC selection Eq. (8), see Eq. (13) and (14). These val- +ues are adopted to obtain the separation of wGg and wIg + wGκgal, follow- +ing Eq. (15) and (16). The left panel shows the calculation from power +spectra and the right panel from correlation functions. The right panel +is used to transfer {wγg, wγg|S } to {wGg, wIg} later in Fig. 6. +dition to what has been addressed in Yao et al. (2020b). We use +the galaxy positions (ra, dec), the two noiseless shear compo- +nents (γ1, γ2), and BPZ-measured photo-z zB to calculate the +SC correlations as in Eq. (11) and (12). We test the signal drop +Qs of Eq. (13) and (14) with our photo-z PDF model and with +true-z from simulation (van den Busch et al. 2020). We compare +the results using MICE2 gold samples (which highly mimic the +KiDS-1000 shear catalog galaxies) with magnification (Eq. 9) +and without magnification. For the MICE2 galaxies with mag- +nification, we tested how it will bias the IA measurement, and +proved that when the magnification effect is also included in the +model, IA can be measured in an unbiased way. The validations +will be shown later in our results with some details in Appendix +A. +4. Measurements +We show the estimation of the signal-drops for lensing and IA +due to the SC selection (as in Eqs. 13 and 14), i.e. the lensing- +drop QGg and the IA-drop QIg in Fig. 4. They are responsible for +the lensing-IA separation later in Fig. 6, following Eq. (15) and +(16). We follow the processes in Yao et al. (2020a,b) and adopt +a bi-Gaussian photo-z probability distribution function (PDF) +model with a secondary peak representing the photo-z outlier +problem. We require the PDF model to have the same mean-z as +in Fig. 3, while closest describing the projection from nP(zP) to +n(z). We will also show for the first time how the assumed photo- +z PDF model can affect the results in the next section, with more +details shown in Appendix A. +We calculate the SC correlation function estimator, +wγg(θ) = B(θ) +� +ED wjγ+ +j +(1 + ¯m) � +ED wj +− +� +ER wjγ+ +j +(1 + ¯m) � +ER wj +, +(24) +to obtain the measurements of wγg and wγg|S from the tangential +shear of each galaxy γ+ +j . Here we sum over the ellipticity-density +pairs (� +ED) and the ellipticity-random pairs (� +ER) in an annulus +centered on θ, where the shear weight wj of the j-th galaxy and +the average multiplicative bias ¯m are accounted for. The estima- +tor is binned in angular θ space, with 9 logarithmic bins from 0.5 +Article number, page 6 of 15 + +Yao et al 2022: KiDS shear × Planck lensing and IA removal +100 +101 +102 +θ [arcmin] +1.0 +1.1 +1.2 +1.3 +1.4 +B +Boost +BoostS +Fig. 5. The boost factors for wγgL and wγgL|S are shown in blue and +orange, respectively. The overlapping lines suggest the two signals are +affected by the boost factor in almost the same way. We show the boost +factor is significant at small scales for the SC observables. +to 300 arcmin. We use the averaged multiplicative bias ¯m from +averaging over the three z-bins, weighted by the effective galaxy +number density. This gives ¯m = −0.0036. +We account for the impact of the boost factor (Mandelbaum +et al. 2005; Singh et al. 2017b; Joachimi et al. 2021), which is B +in Eq. (24). It is defined as +B(θ) = +� +ED wj +� +RD wj +, +(25) +which is used to quantify the small-scale bias due to the clus- +tering of lens galaxies and source galaxies (Bernardeau 1998; +Hamana et al. 2002; Yu et al. 2015). We show the measurements +of the boost factor for wγgL and wγgL|S as in Eq. (11) and (12) +in Fig. 5. The fact that the boost factors for wγgL and wγgL|S are +identical suggests this bias can be absorbed by the galaxy bias +bg,eff parameter if magnification is absent (gmag = 0), leading to +an unbiased AIA and Alens. The impact from the boost factor can +potentially break the linear galaxy bias assumption, but later in +Fig.6 we show the linear assumption is fine. The impact of the +boost factor and magnification existing together will be shown +later. +In Fig. 6 we show the SC measurements. In the left panel, the +measured shape-galaxy correlations wγgL are shown in blue: (1) +the boost factor ignored case (B = 1) is shown as blue crosses, +while (2) the boost factor corrected case is shown as blue up- +triangles. With the SC selection Eq. (8), requiring zP +γ < zP +g for +each galaxy pair, the lensing component will drop to QGg ∼ 0.3 +and the IA component will drop to QIg ∼ 0.85 (for more details +on QGg and QIg, see Fig. 4 and Appendix A). Therefore, the se- +lected correlations wγg|S will drop to the orange down-triangles. +Similarly, the boost factor ignored case is shown as crosses. +The separated lensing-galaxy signal wGg and IA-galaxy sig- +nal wIg (which is contaminated by magnification-shear signal +gmagwGκ) are shown in the right panel of Fig. 6. The blue and or- +ange curves are the theoretical predictions with the best-fit {AIA, +bg,eff, gmag}. For the fitting, we cut off the shaded regions at both +large scales and small scales. The small scale cut at θ = 1 ar- +cmin is based on the linear galaxy bias assumption, as including +the θ < 1 arcmin data will make the fitting significantly worse +(increasing the fitting χ2 from 7.5 to 50, with degree-of-freedom +changed from 8 to 10). We note this scale cut could include the +impacts from the 3D non-linear galaxy bias (Fong & Han 2021) +and other small-scale effects such as massive neutrinos or baryon +feedback in the matter power spectrum (Hildebrandt et al. 2017; +Asgari et al. 2021). We emphasize that these systematics will be +absorbed by the effective galaxy bias parameter bg,eff —- with- +out breaking the scale-independent bias assumption —- so that +the IA amplitude will not be affected. As discussed previously +in Yao et al. (2020a,b), the SC method requires significant sep- +aration between wγgL and wγgL|S to accurately get wGg and wIg. +Therefore, we introduce a large-scale cut at θ = 20 arcmin due +to insufficient separation for the left panel of Fig. 6. +Similarly, we measure the ⟨γκ⟩ correlation with the estimator +wγκ(θ) = +� +i j wjγ+ +j κi +(1 + ¯m) � +i j wj +, +(26) +where κi is the CMB lensing convergence in the i-th pixel of +the pixelized map, taking the pixel center for its (ra, dec) co- +ordinates, with nside = 2048 in Healpy. The measured wγκ are +shown in Fig. 7. The tangential shear is shown as blue dots. We +also show the measurements with randomly shuffling galaxy po- +sitions and the shear in red crosses as a null test. We test the +45 deg rotated cross shear for both the above cases and they are +consistent with zero. The theoretical prediction with the best-fit +Alens and AIA are shown as the green curve. If one assumes there +is no IA in the measurements and uses AIA = 0, the theoretical +values for the pure lensing signal are shown in orange. +Note in Fig. 7, because we use the Wiener-filtered κ map +from Planck, both the wγκ measurements and the theoretical pre- +dictions are suppressed at small scales. The Wiener filter can +significantly reduce the impact of the noise of the Planck lens- +ing map and improve the S/N of the measurements. +Together with the measurements in Figs. 6 and 7, we obtain +observables of this work, which are the LHS terms of Eqs. (17), +(18) and (19). We use Jackknife resampling to obtain the co- +variance. 200 Jackknife regions are used, which is much larger +than the length of the data vector (12), based on the analy- +sis of Mandelbaum et al. (2006); Hartlap et al. (2007). The +Jackknife regions are separated using the K-means algorithm +kmeans_radec5. The normalized covariance matrix is shown in +Fig. 8. We find strong anti-correlation between wGg and wIg as +expected (Yao et al. 2020b). Note here in Fig. 8, wIg means the +separated signal in the RHS of Eq. (16), including both the IA +part and the contamination from magnification. There is no sig- +nificant correlation between wγκ and the other two observables. +This covariance will be used in the Monte Carlo Markov Chain +(MCMC) to find the best-fit parameters of {AIA, bg,eff, gmag, +Alens}, while all the other cosmological parameters are fixed to +Planck as in Table 1. +5. Results +5.1. Validation with MICE2 +In this subsection, we apply the IA self-calibration to the MICE2 +mock catalog to test the impact of the systematics and validate +the recovery of the IA signal. The processes of the mock data are +identical to the descriptions in Sec. 4, but only focusing on the +self-calibration part. The measurements are similar to Fig. 6 so +5 https://github.com/esheldon/kmeansradec +Article number, page 7 of 15 + +A&A proofs: manuscript no. aanda +1 +3 +10 +30 +θ [arcmin] +-1 +0 +1 +2 +3 +4 +w(θ) × 104 +wγgL +wγgL|S +1 +3 +10 +30 +θ [arcmin] +-1 +0 +1 +2 +3 +4 +wIg +gmagwGκgal +tot +wGg +wIg (+gmagwGκgal) +Fig. 6. The measurements of SC. The left panel shows the measurement of the two introduced observables wγgL and the one with the SC selection +wγgL|S, while the corresponding 45-deg rotation test is consistent with 0 for both measurements. The significant separation of the two signals shows +that SC is applicable. The right panel shows the separated lensing signal wGg and wIg, where the latter is contaminated by the magnification signal +as shown in Eq. (16). The up- and down-triangles are the results that take the boost factor (Fig. 5) into consideration, while the crosses are the +results that ignore this correction, setting B = 1. The curves are the theoretical value with the best-fit {AIA, bg,eff, gmag} of this work. The blue curve +represents the separated lensing signal as in Eq. (17). The orange curve represents the total contribution of IA and magnification as in Eq. (18). +3 +10 +30 +100 +300 +θ [arcmin] +-1 +0 +1 +2 +3 +wκγ(θ) × 106 +wGκ lensing +w(G+I)κ +best−fit +⟨κγt⟩ +⟨κγshuffle⟩ +Fig. 7. The measurement of the cross-correlation between Planck con- +vergence κ and KiDS-1000 shear γ, based on Eq. (19). The blue dots are +the measurements using tangential shear, with the green curve showing +the best-fit considering both lensing and IA, while the orange curve +shows only the lensing-lensing component. The red crosses show the +null test by randomly shuffling the shear galaxies. The 45-deg rotation +tests for both the blue dots and the red dots are consistent with 0. The +differently shaded regions correspond to our angular scale cuts at 2, 20 +(default), and 40 arcmin. +we choose to skip them. We perform the MCMC calculation us- +ing emcee (Foreman-Mackey et al. 2013). We consider flat priors +in −5 < AIA < 5, 0 < bg,eff < 2 and −3 < gmag < 3. +5.1.1. Impact from magnification +We show how the magnification signal affects the original SC +method (Zhang 2010b; Yao et al. 2020a,b) and the correction +introduced in this work, focusing on the gmag − AIA space. +wGg +wIg +wγκ +wGg +wIg +wγκ +correlation coefficient +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Fig. 8. The normalized covariance matrix (i.e. the correlation coeffi- +cient) used in this work. There exists a strong anti-correlation between +the lensing-galaxy correlation wGg and the IA-galaxy correlation wIg +(including the contamination from wGκgal) as we found in previous work. +The covariance of the 12 data points is calculated from Jackknife re- +sampling with 200 regions. We note the IA information is passed from +1 < θ < 20 [arcmin] for wIg to 20 < θ < 300 [arcmin] for wγκ with the +scale-independent AIA assumption. +In Fig. 9, we show that if magnification is not included in the +modeling, gmag is therefore not constrained. The existing mag- +nification signal will be treated as the IA signal, leading to a +non-vanishing AIA ∼ 0.3, which significantly deviates from the +MICE2 input AIA = 0. When the magnification model is in- +cluded in the analysis, AIA is then consistent with 0. This demon- +strates the importance of including the magnification model in +the SC analysis with high-z data. The results are also summa- +Article number, page 8 of 15 + +Yao et al 2022: KiDS shear × Planck lensing and IA removal +MICE IA +MICE IA+mag +−0.30 +−0.15 +0.00 +0.15 +0.30 +AIA +−0.45 +−0.30 +−0.15 +0.00 +gmag +Fig. 9. The impact of the magnification signal on the IA measurement +in MICE2. The green and blue contours are with and without magni- +fication models, respectively. If the magnification model is used in the +fitting, as in green, the IA amplitude AIA is consistent with 0, which is +the MICE2 input. +rized later in the comparisons in Fig. 11 for MICE2, and in +Fig. 14 for KiDS data. +We note that in the green case of Fig. 9 that considered both +IA and magnification, gmag and AIA strongly degenerate. There- +fore the constraining power in AIA has a significant loss com- +pared with the blue case, which ignores magnification. This de- +generacy can be broken in the future with higher S/N in the ob- +servables. This is because the shape of wIg and wGκ are different +at small scales for correlation functions as in Fig. 6, and on large +scales for power spectra as in Fig. 2. The IA-model-dependency +will be discussed later with other results. Based on the above +analysis, we conclude it is important to include magnification +modeling for SC when using high-z data. +5.1.2. Impact from modeling p(z|zP) +Since the SC selection Eq. (8) plays an important role in the +lensing-IA separation process, it is crucial to understand how the +following aspects affect SC: (1) the quality of the photo-z zP, (2) +the true redshift distribution n(z), and (3) the link between them +p(z|zP). The quality of photo-z and the reconstruction of n(z) has +been studied thoroughly for KiDS data (Kuijken et al. 2019; van +den Busch et al. 2022; Hildebrandt et al. 2021; van den Busch +et al. 2020), we, therefore, trust these results and leave the al- +ternative studies for SC to future works. The uncalibrated PDF +that projects zP → z, on the other hand, has some known prob- +lems, for example when Probability Integral Transform (PIT) is +applied (Newman & Gruen 2022; Hasan et al. 2022). +In this work, we use a bi-Gaussian PDF model to project the +photo-z distribution nP(zP) to the SOM redshift distribution n(z), +which are previously shown in Fig. 3. This modeling ignores the +potential differences for galaxies in the same z-bin (Peng et al. +2022; Xu et al. 2023). However, this is an alternative process, +MICE Qsim+mag +MICE Qmodel +MICE Qmodel+mag +−0.4 +−0.2 +0.0 +0.2 +AIA +−0.4 +0.0 +0.4 +gmag +Fig. 10. The impact from photo-z PDF model bias. The blue case uses +photo-z from the BPZ algorithm and true-z for each galaxy to calcu- +late Eq. A.7 and the resulting QGg and QIg, which are the “sim” cases +in Fig. A.1. This AIA is consistent with 0, which is the MICE2 input. +The green case uses the bi-Gaussian photo-z model for the calculation, +which are the “model” cases in Fig. A.1, while ignoring the magnifica- +tion contribution. This lead to unconstrained gmag and biased AIA. In the +red case, which also uses the photo-z model, but includes the magnifi- +cation model, the resulting AIA is still consistent with 0, with the bias +from photo-z model error absorbed by gmag. +considering the PDF problem for a single galaxy. This analytical +approach is also much faster in calculation than using different +PDFs for different galaxies. +We use Fig. 10 to demonstrate how large this photo-z PDF +modeling bias is with different approaches. We use MICE2 sim- +ulation with galaxy number density affected by magnification. +When the SC calculation uses true-z to calculate the signal drops +QGg and QIg, and the magnification model is also considered, we +find the resulting AIA is consistent with 0, which is the MICE2 +input. The scatter on AIA is ∼ 0.1, thanks to the noiseless shapes +in MICE2. If the Qs are calculated with the assumed photo-z +PDF model, without including the magnification model, then +AIA will be biased towards the negative direction. We proved +with our fiducial analysis that, even if there exists a bias in QGg +due to the assumed photo-z model, as long as the magnification +model is used, this bias will be absorbed by the gmag parameter, +so that the IA amplitude AIA is unbiased (consistent with 0 in +the MICE2 case). The results are also shown later in the com- +parisons in Fig. 11 for MICE2, and in Fig. 14 for KiDS data. +We note that the bias due to photo-z modeling is not an es- +sential problem for SC. In the future, if the photo-z outlier prob- +lem (or the redshift-color degeneracy problem) can be under- +stood better, then a more reliable photo-z model can be used for +our SC study. Alternatively, if the photo-z algorithms can give +unbiased PDFs for each galaxy, this problem can also be directly +solved. +Article number, page 9 of 15 + +A&A proofs: manuscript no. aanda +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +AIA +MICE(mag), Q(sim), w/o mag +MICE(mag), Q(sim), w/ mag +MICE(mag), Q(model), w/o mag +MICE(mag), Q(model), w/ mag +MICE(nomag), Q(model), w/ mag +Fig. 11. We validate our SC method with MICE2 simulation, which +does not have IA implemented; therefore, AIA = 0 is expected. The re- +sults are shown in green, with “MICE(mag)” meaning magnification is +included in the MICE simulation, while “MICE(nomag)” means mag- +nification is not included, “Q(sim)” and “Q(model)” mean if the signal +drops Q values are calculated from true-z from simulation or photo-z +PDF model, and “w/o mag” and “w/ mag” show if the case includes +magnification model in the fitting process. The upper two data are the +results from Fig. 9, showing the impact of the modeling magnification. +The 2nd to the 4th data are the results from Fig. 10, showing the impact +of Q calculation using different PDFs. The 4th data correspond to our +fiducial analysis later for KiDS data, with potential bias ∆AIA < 0.1. +The bottom data is a reference case assuming no magnification effects +in the data, corresponding to our previous work Yao et al. (2020b,a). +5.2. Inference on real data +With the above demonstration that our treatments for magnifica- +tion and photo-z PDF are appropriate, and the resulting bias in +AIA is very small (∆AIA < 0.1 and < 1σ as shown in Fig. 11), +we move on to apply SC to KiDS data and its cross-correlation +with Planck lensing. We show the analysis of the following three +situations: +(1) The case “ignore IA”. We only use the observed wγκ, while +only including Alens in the fit and ignoring the contamination by +IA (by setting AIA = 0). +(2) The case “IA w/o SC”. We only use the observed wγκ, but +consider both Alens and AIA following Eq. (19). +(3) The case “with SC”. We use both wγκ in Fig. 7 and the SC +correlations in Fig. 6. Both the CMB lensing amplitude Alens and +the nuisance parameters {AIA, bg,eff, gmag} will be used in the +analysis, following Eqs. (17), (18) and (19). +The results are shown in Fig. 12. We use flat priors in 0 < +Alens < 2, −5 < AIA < 5, and for the IA self-calibration nuisance +parameters we use 0 < bg,eff < 4, −5 < gmag < 5. +For case (1) “ignore IA”, shown in blue, AIA is unconstrained in +the fitting, giving the best-fit Alens = 0.74+0.18 +−0.17. +For case (2) “IA w/o SC”, when we consider the existence of IA +and apply the IA model as in Eq. (7), but do not use the mea- +surements from SC (Fig. 6 and Eq. 17, 18), there will be a strong +degeneracy between Alens and AIA, as shown in orange. There is +a significant loss of constraining power in the lensing amplitude, +with the best-fit Alens = 0.79+0.43 +−0.46 and AIA = 0.47+3.11 +−3.47. +For case (3) “with SC”, the introduced measurements of wGg +and wIg can not only break the degeneracy between Alens and AIA +(see Eq. 17, 18 and 19), but also bring more constraining power +to AIA, so that the best-fit of Alens will not only be unbiased +(according to the validation using simulation) but also has sig- +−2 +−1 +0 +1 +2 +3 +4 +AIA +0.2 +0.6 +1.0 +1.4 +1.8 +Alens +0.2 +0.6 +1.0 +1.4 +1.8 +Alens +ignore IA +IA w/o SC +with SC +Fig. 12. The constraints on lensing amplitude Alens and the IA ampli- +tude AIA, with three different methods: assume there is no IA in the +measured wκγ (blue), consider the impact of IA with conventional IA +model but do not use SC (orange), use SC to subtract IA information +and constrain together with the CMB lensing cross-correlation (green). +When IA is ignored, AIA is unconstrained. The similar height and width +of Alens PDFs between blue and green prove that by including SC, the +AIA − Alens degeneracy can be efficiently broken so that the constraining +power loss in Alens is very small. +nificantly improved constraining power. The best-fit values are +Alens = 0.84+0.22 +−0.22, AIA = 0.60+1.03 +−1.03, bg,eff = 0.88+0.06 +−0.06, and gmag = +−0.30+1.60 +−1.62. In Fig. 12 we only show AIA and Alens, which are the +focus of this work, while bg,eff and gmag are only related with +the SC observables but not CMB lensing. Also as discussed in +Yao et al. (2020b), the existence of the effective galaxy bias bg,eff +can also absorb some systematics (so it could be a biased bias), +leaving the constraint on AIA unbiased (as shown in Fig. 11). For +example, we tested if magnification is absent, the effect of boost +factor will be purely absorbed by bg,eff, giving unbiased AIA and +Alens. The effective galaxy bias could also absorb the differences +in the assumed fiducial cosmology, with bg,eff ∼ 1.24 with KiDS +COSEBI cosmology, for example. The redshift distribution n(z) +can differ slightly with/without accounting for the lensing weight +(considering the lensing/clustering part in the galaxy-shape cor- +relation), with a ∼ 0.024 difference in the mean-z, which can lead +to ∼ 8% difference in the theoretical lensing signal and ∼ 2% dif- +ference in the theoretical IA signal. Other unaddressed sources +of systematics such as baryonic feedback and massive neutrinos +could have similar effects. We can also see from the validation +using MICE data that although the resulting bg,eff is lower than +the expectation, the AIA result is unbiased. The gmag result also +resides in a reasonable range, considering the KiDS i-band mag- +nitude (Kuijken et al. 2019) and comparing it with Duncan et al. +(2014). The above three cases of IA treatments are also summa- +rized later in Fig. 13 and 14 together with more tests and other +works. +The corresponding best-fit curves are shown in Fig. 2 and 6 +with AIA = 0.60+1.03 +−1.03, bg,eff = 0.88+0.06 +−0.06, and gmag = −0.30+1.60 +−1.62 +. Even though the impact of magnification is comparable to the +IA signal, we can see in both the angular power spectrum and +correlation function that the shapes of IA and magnification are +different. For example, as shown in Fig. 6, the tidal alignment +model wIg and magnification gmagwGκ are comparable at large +Article number, page 10 of 15 + +Yao et al 2022: KiDS shear × Planck lensing and IA removal +0.5 +1 +1.5 +2 +Alens +with SC (Planck) +ignore IA +IA w/o SC +wγκCMB scale > 40 arcmin +wγκCMB scale > 2 arcmin +SC ignore mag +SC ignore boost +with SC (KiDS COSEBI) +Hand+ 2015 Planck +Hand+ 2015 WMAP +Liu+ 2015 +Kirk+ 2016 SPT +Kirk+ 2016 SPT fix-IA +Kirk+ 2016 Planck +Harnois-Deraps+ 2016, CFHT +Harnois-Deraps+ 2016, RCSLenS +Singh+ 2017 +Harnois-Deraps+ 2017, KiDS +Harnois-Deraps+ 2017, Planck +Omori+ 2018 fix-IA +Namikawa+ 2019 +Marques+ 2020 +Robertson+ 2021, Planck +Robertson+ 2021, KiDS +baseline +comparisons +previous w/o IA +previous IA prior +Fig. 13. The comparisons of the constraints on Alens with previous mea- +surements. Our baseline analysis “with SC” is consistent with 1. We +also show some cases where IA is ignored in the analysis and if IA is +considered but the AIA − Alens degeneracy is not broken with SC. These +main results in blue are similar to Fig. 12. We show tests with differ- +ent scale cuts and different treatments to magnification, boost factor, +and different (KiDS) fiducial cosmology in red. We compare with other +works, separated into ignoring IA (orange) and assuming a strong prior +of IA (green). We note that for different work, the different fiducial cos- +mology (the “Planck”, “WMAP”, “KiDS” labels on the y-axis) can lead +to ∼ 10% difference in Alens. +scale, while different at small scale. Therefore, in principle, the +degeneracy between IA and magnification can be broken for fu- +ture data with higher S/N so that the shape/slope information of +the observables can be used. The current degeneracy is due to +the low S/N so that the amplitudes of AIA and gmag degenerate. +Furthermore, if a more complicated IA model is used, for ex- +ample, as in Blazek et al. (2019); Abbott et al. (2022), the small- +scale IA will be different. Based on the study of Shi et al. (2021), +for a wide range of stellar mass, the small-scale IA should have +a higher amplitude (either a direct raise in the amplitude or a +“drop-raise” pattern as we go to smaller scales) than the current +model so that the IA-magnification degeneracy can be broken +further. The appropriate IA model will require studies in many +aspects, and with higher S/N in the measurements, thus we leave +this topic for future work. +We investigate how different choices can change our results. +We first compare the different scale cuts for wκγ. Besides the +baseline analysis of Alens = 0.84+0.22 +−0.22 with θ > 20 arcmin, two +more tests are made with a larger scale cut of θ > 40 arcmin and +a smaller scale cut of θ > 2 arcmin, as shown in Fig. 7, which +give us Alens = 0.97+0.25 +−0.25 and Alens = 0.77+0.21 +−0.22, respectively. The +comparisons are shown in Fig. 13. The large-scale lensing am- +plitude is higher than the small-scale one, which agrees with the +finding in Planck Collaboration et al. (2020c) and other cross- +correlation work (Sun et al. 2022). In this work, we only re- +port this large-scale v.s. small scale difference. However, the cur- +rent S/N of CMB convergence - galaxy shear correlation and the +model assumptions do not allow us to investigate further on this +topic. +-3 +-2 +-1 +0 +1 +2 +3 +4 +AIA +with SC (Planck) +IA w/o SC +SC ignore mag +SC ignore boost +with SC (KiDS COSEBI) +Robertson+ 2021 prior +Asgari+ 2021 C(ℓ) +Asgari+ 2021 COSEBI +Asgari+ 2021 ξ± +DES Y3 Secco+ +HSC Y1 ξ± Hamana+ +HSC Y1 C(ℓ) Hikage+ +this work +KiDS +others +Fig. 14. The comparisons of the constraints on AIA. We show the results +of this work in blue, which contains our fiducial analysis with SC ap- +plied, and the comparisons of (1) without SC, (2) with SC but ignoring +magnification, (3) with SC but ignoring boost factor, and (4) switching +to KiDS fiducial cosmology. We show comparisons with other works +using KiDS-1000 data in orange, and some works using DES or HSC +data in green. +We then compare the different choices in the SC method. We +find that if the magnification model is ignored in the analysis, +the existing magnification signal in the data will be treated as +an IA signal, leading to an over-estimated AIA = 0.81+0.36 +−0.41 and +an over-estimated Alens = 0.87+0.18 +−0.18. On the other hand, we pre- +viously argued that, when magnification is absent, the impact +from the boost factor will be purely absorbed by the effective +galaxy bias bg,eff, leaving AIA and Alens unbiased. Unfortunately, +this does not hold anymore when magnification is present: if the +boost factor is not corrected, all the parameters will be biased +as follows AIA = 1.86+1.01 +−1.05, bg,eff = 0.67+0.06 +−0.06, Alens = 1.00+0.23 +−0.23 +and gmag = 1.55+1.28 +−1.31. We include the comparisons of Alens and +AIA for the above-described cases in Fig. 13 and 14 and empha- +sis the importance of taking magnification and boost factor into +consideration. We also show the impact of the assumed fiducial +cosmology: if the fiducial cosmology is switched from Planck +to KiDS-1000 COSEBI as in Table 1, both Alens and AIA will +change as shown in Fig. 13 (bottom-red) and 14 (bottom-blue). +With the above results in simulation and data, summarized +in Fig. 11, 13 and 14, we show that our measurements on AIA +and Alens are unbiased from magnification, boost factor, and the +assumed photo-z PDF model. These are the new developments +considering the existence of magnification at high redshift z ∼ 1, +beyond the study of Yao et al. (2020b). +Additionally, we compare our analysis with previous works. +The comparisons of Alens are shown in Fig. 13. We find that most +of the previous works ignored the IA contamination (Hand et al. +2015; Liu & Hill 2015; Kirk et al. 2016; Harnois-Déraps et al. +2016; Singh et al. 2017a; Harnois-Déraps et al. 2017; Namikawa +et al. 2019; Marques et al. 2020). For the ones that considered IA, +they either fixed the IA amplitude (Kirk et al. 2016; Omori et al. +2019) or used a strong prior (Robertson et al. 2021) to break the +degeneracy between Alens and AIA, which will otherwise cause +a strong loss in constraining power as we show in Fig. 12. We +are the first to directly achieve the IA amplitude measurement +within the same data and break the lensing-IA degeneracy. Our +Article number, page 11 of 15 + +A&A proofs: manuscript no. aanda +-0.5 +0 +0.5 +1 +1.5 +AIA +Asgari+ 2021 C(ℓ) +SC, C(ℓ) cosmo +Asgari+ 2021 COSEBI +SC, COSEBI cosmo +Asgari+ 2021 ξ± +SC, ξ± cosmo +SC +cosmic shear +Fig. 15. The comparisons of AIA between SC-subtracted results (blue) +and cosmic shear tomography subtracted results (orange) with cosmolo- +gies from different 2-point statistics. The cosmologies are shown in Ta- +ble 1. +baseline analysis is consistent with most of the previous results, +showing the contamination from IA is not significant, mainly due +to the total S/N of CMB lensing - galaxy shear cross-correlation +is only at 3 ∼ 5 σ level at the current stage. However, the correct +treatment for IA will be more and more important in the future +with stage IV cosmic shear surveys and CMB observations. +The comparisons of the AIA constraint with other results us- +ing KiDS-1000 data are shown in Fig. 14, including the prior +assumed in Robertson et al. (2021) and the cosmic shear tomog- +raphy constraint in Asgari et al. (2021). Although the redshift +range is slightly different, the above works have consistent re- +sults on AIA. These comparisons will become more interesting +for the next-stage observations. +As an extended study, we investigate how the choice of fidu- +cial cosmology affects the SC results, namely AIA. In Fig. 14 +we show the results with the fiducial Planck cosmology and the +KiDS-1000 two-point correlation function ξ± best-fit cosmology. +We further compare the results with the KiDS-1000 band power +C(ℓ) cosmology and the COSEBIs cosmology in Fig. 15. The re- +sults from Asgari et al. (2021) (shown in orange) are arranged in +increasing order from bottom to top. We find that when assuming +the same cosmology, the SC results (shown in blue) also follow +the same (weak) trend, meanwhile, they agree very well with the +cosmic shear results. We note the SC results will provide extra +information in constraining IA in cosmic shear in the future. +6. Summary +In this work, we achieved the first application of the self- +calibration (SC) method of intrinsic alignment (IA) of galax- +ies to its cosmological application. We proved that with SC, the +lensing-IA degeneracy could be efficiently broken, i.e., in this +CMB lensing × galaxy shear cross-correlation work, it means +breaking the degeneracy between the lensing amplitude Alens and +the IA amplitude AIA. We showed that for previous treatments, +IA are either ignored or being considered with a strong assumed +prior on AIA. We demonstrated in Fig. 12, 13 and 14 that with +SC to break the degeneracy, the constraining power in both Alens +and AIA is preserved. +We demonstrated that the proper angular scale cuts on wκγ +are important. Our baseline analysis using information from +θ > 20 arcmin gives Alens = 0.84+0.22 +−0.22. If we use informa- +tion only at larger scales with θ > 40 arcmin, the constraint is +Alens = 0.97+0.25 +−0.25. If we include information at much smaller +scales with θ > 2 arcmin, the constraint is Alens = 0.77+0.21 +−0.22. +At the current stage, they do not differ significantly from each +other (even considering they are strongly correlated), as shown +in Fig. 13. However, we note that these differences at differ- +ent scales also exist in other works Planck Collaboration et al. +(2020c) and Sun et al. (2022). We, therefore, emphasize the im- +portance of understanding the possible systematics at different +scales for future studies with higher S/N. +Comparing our CMB lensing amplitude Alens with other +works in Fig. 13, we found consistent results with different treat- +ments of IA throughout almost all the works. We conclude that +IA is not a significant source of systematics for the current stage. +However, it will soon become more important with the stage IV +observations. Nevertheless, we emphasize that the correct treat- +ment to break the lensing-IA degeneracy is very important to +maintain the cosmological constraining power. Our constraint +on the IA amplitude AIA in Fig. 14 is also consistent with the +existing analysis on IA with KiDS-1000 data. We note that the +SC-subtracted IA information can be used as extra constraining +power for any of these analyses. +On the technique side, we further developed the SC method +considering more sources of systematics beyond Yao et al. +(2020b). We showed at z ∼ 1, the impact of galaxy shear × cos- +mic magnification component wGκgal contaminates the separated +IA × galaxy number density signal wIg, and is non-negligible as +shown in Fig. 2 and 6. We use Eq. (16) and (18) to show how the +magnification term enters our observable and how we include +it in the theory as a correction. We show in Fig. 13 and 14 that +the correction of magnification is important when applying SC +to higher redshift data, in order to get the correct constraint on +IA. We also discussed that, with the contamination from mag- +nification, boost factor can no longer be absorbed by the effec- +tive galaxy bias bg,eff, and need to be accounted for correctly, as +shown in Eq. (24), (25) and Fig. 6, 13, 14. +We also validated our analysis with MICE2 simulation, fo- +cusing on two aspects: (1) how good can the magnification +model mitigate the contamination from the magnification-shear +signal; and (2) will the assumed photo-z PDF model (which is +used to calculate the signal drop QGg and QIg) bias the IA mea- +surement. With the strong constraining power from MICE2 with +no shape noise, we can show in Fig. 11 that, when the magnifi- +cation model is included in the analysis, the IA amplitude can be +obtained correctly (consistent within 1σ range of 0, which is the +input of MICE2). Additionally, the bias from the assumed photo- +z model is negligible when the magnification model is used, as +the effective magnification prefactor gmag will absorb the intro- +duced error. We, therefore, emphasize the importance of includ- +ing the magnification model in the SC analysis, especially for fu- +ture high-z surveys like LSST, Euclid, WFIRST, and CSST. We +further notice the contamination from magnification will make +SC no longer an IA-model-independent method, therefore, SC +is more suitable for low-z data when considering alternative IA +models. +Comparing with our first measurements with KV-450 data +(Yao et al. 2020a), a lot of improvements have been added in the +SC method, including: +(1) the covariance, the galaxy bias, the scale-dependency for the +lensing-drop QGg, the IA-drop QIg, and appropriate scale-cuts, +Article number, page 12 of 15 + +Yao et al 2022: KiDS shear × Planck lensing and IA removal +which have been introduced in Yao et al. (2020b); +(2) the boost factor, the cosmic magnification, and the photo-z +PDF modeling, which are introduced in this work; +(3) its first validation using simulation, and its first application +to cosmology in order to break the lensing-IA degeneracy, intro- +duced in this work. +With these improvements, we manage to achieve consistent IA +results between SC and cosmic shear, as shown in Fig. 15, while +previously we got AIA = 2.31+0.42 +−0.42 with the old version of SC +(Yao et al. 2020a) and AIA = 0.981+0.694 +−0.678 for cosmic shear (Hilde- +brandt et al. 2020) with KV-450 data. +Despite SC-obtained AIA is consistent with the MICE input +IA, and when applying to data it is consistent with the KiDS cos- +mic shear results Asgari et al. (2021) and the other CMB lensing +work Robertson et al. (2021), as well as gmag is in reasonable +agreement with (Duncan et al. 2014), our results still suffer from +an unrealisticly low effective galaxy bias bg,eff = 0.88, which +is different from our previous work (Yao et al. 2020b). We dis- +cussed this value may absorb the contribution from (1) fiducial +cosmology, (2) lensing weight in n(z), (3) insufficient modeling +in non-linear galaxy bias, baryonic effects, and massive neutri- +nos, (4) incorrect photo-z v.s. true-z connection as discussed in +Appendix A and (5) possible other sources of systematics. We +emphasize the complication and leave this point for future stud- +ies. +We note that there could still exist other systematics other +than the galaxy bias, such as beyond Limber approximation +(Fang et al. 2020), non-flat ΛCDM (Yu et al. 2021), selection +bias on shear measurement (Li et al. 2021). But they have either +much smaller impacts compared with IA or are strongly reduced +due to our scale cuts. Therefore, they are beyond the scope of +this paper. +Acknowledgements. The authors thank Yu Yu, Hai Yu, Jiaxin Wang for useful +discussions. +This work is supported by National Key R&D Program of China No. +2022YFF0503403. JY acknowledges the support of the National Science +Foundation of China (12203084), the China Postdoctoral Science Foundation +(2021T140451), and the Shanghai Post-doctoral Excellence Program (2021419). +HYS acknowledges the support from CMS-CSST-2021-A01 and CMS-CSST- +2021-B01, NSFC of China under grant 11973070, the Shanghai Committee of +Science and Technology grant No.19ZR1466600 and Key Research Program +of Frontier Sciences, CAS, Grant No. ZDBS-LY-7013. PZ acknowledges the +support of the National Science Foundation of China (11621303, 11433001). +XL acknowledges the support of NSFC of China under Grant No. 11803028, +YNU Grant No. C176220100008, and a grant from the CAS Interdisciplinary +Innovation Team. BJ acknowledges support by STFC Consolidated Grant +ST/V000780/1. MB is supported by the Polish National Science Center through +grants no. 2020/38/E/ST9/00395, 2018/30/E/ST9/00698, 2018/31/G/ST9/03388 +and 2020/39/B/ST9/03494, and by the Polish Ministry of Science and Higher +Education through grant DIR/WK/2018/12. HH is supported by a Heisenberg +grant of the Deutsche Forschungsgemeinschaft (Hi 1495/5-1) as well as +an ERC Consolidator Grant (No. 770935). TT acknowledges support from +the Leverhulme Trust. AW is supported by an European Research Council +Consolidator Grant (No. 770935). ZY acknowledges support from the Max +Planck Society and the Alexander von Humboldt Foundation in the framework +of the Max Planck-Humboldt Research Award endowed by the Federal Ministry +of Education and Research (Germany). The computations in this paper were run +on the π 2.0 cluster supported by the Center for High Performance Computing +at Shanghai Jiao Tong University. +The codes JY produced for this paper were written in Python. JY thanks all its +developers and especially the people behind the following packages: SCIPY +(Jones et al. 2001–), NUMPY (van der Walt et al. 2011), ASTROPY (Astropy +Collaboration et al. 2013) and MATPLOTLIB (Hunter 2007), TreeCorr (Jarvis +et al. 2004), CCL (Chisari et al. 2019), CAMB (Lewis et al. 2000), Healpy +(Górski et al. 2005; Zonca et al. 2019), emcee (Foreman-Mackey et al. 2013), +fitsio6, kmeans_radec7, corner (Foreman-Mackey 2016), ChainConsumer8. The +6 https://github.com/esheldon/fitsio +7 https://github.com/esheldon/kmeansradec +8 https://github.com/Samreay/ChainConsumer +KiDS-1000 results in this paper are based on data products from observations +made with ESO Telescopes at the La Silla Paranal Observatory under pro- +gramme IDs 177.A-3016, 177.A-3017 and 177.A-3018, and on data products +produced by Target/OmegaCEN, INAF-OACN, INAF-OAPD, and the KiDS +production team, on behalf of the KiDS consortium. +Author contributions: All authors contributed to the development and writing +of this paper. The authorship list is given in three groups: the lead authors (JY, +HS, PZ, XL) followed by two alphabetical groups. The first alphabetical group +includes those who are key contributors to both the scientific analysis and the +data products. The second group covers those who have either made a significant +contribution to the data products, or to the scientific analysis. +References +Abbott, T. M. 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(4)) of the angular power spectrum CGg and +CIg. The associated CGg and CGg|S are calculated by: +CGg +ii (ℓ) = +� ∞ +0 +qi(χ)ni(χ) +χ2 +bg,effPδ +� +k = ℓ +χ; χ +� +dχ, +(A.1) +CGg +ii |S(ℓ) = +� ∞ +0 +qi(χ)ni(χ) +χ2 +bg,effPδ +� +k = ℓ +χ; χ +� +ηGg +i (z)dχ. +(A.2) +Similarly, the CIg and CIg|S are given by: +CIg +ii (ℓ) = +� ∞ +0 +ni(χ)ni(χ) +χ2 +bg,effPδ,γI +� +k = ℓ +χ; χ +� +dχ, +(A.3) +CIg +ii |S(ℓ) = +� ∞ +0 +ni(χ)ni(χ) +χ2 +bg,effPδ,γI +� +k = ℓ +χ; χ +� +ηIg +i (z)dχ. +(A.4) +Here ηGg +i (z) = ηGg +i (zL = zg = z) is the function that account +for the effect of the SC selection Eq. (8) in the Limber integral, +similarly for ηIg. They are expressed +ηGg +i (zL, zg) = +2 +� +dzP +G +� +dzP +g +� ∞ +0 dzGWL(zL, zG)S (zP +G, zP +g)K +� +dzP +G +� +dzPg +� ∞ +0 dzGWL(zL, zG)K +, (A.5) +ηIg +i (zL, zg) = +2 +� +dzP +G +� +dzP +g +� ∞ +0 dzGS (zP +G, zP +g)K +� +dzP +G +� +dzPg +� ∞ +0 dzGK +, +(A.6) +as in Yao et al. (2020b), where K is the galaxy-pair redshift dis- +tribution kernel +K(zG, zg, zP +G, zP +g) = p(zG|zP +G)p(zg|zP +g)nP +i (zP +G)nP +i (zP +g), +(A.7) +and S is the SC selection function +S (zP +G, zP +g) = +�1 +for zP +G < zP +g, +0 +otherwise , +(A.8) +which correspond to Eq. 8 in the main text, and the lensing kernel +is +WL(zL, zS ) = +������� +3 +2Ωm +H2 +0 +c2 (1 + zL)χL(1 − χL +χS ) +for zL < zS +0 +otherwise +. +(A.9) +Here zx is the true-z where x can be “G” the source, “L” the +lens, or “g” the galaxy number density. The galaxy photo-z dis- +tribution is nP(zP), and the redshift PDF (probability distribution +function) is p(z|zP). +As shown above, when the galaxy photo-z distribution and +the corresponding true-z distribution are given, as shown in +Fig. 3 in this work, we can follow the above procedure to calcu- +late the lensing-drop QGg and QIg. The results of QGg and QIg for +this work are shown in Fig. 4 for your interest. Generally, given +the tomographic bin width, the better photo-z is, the smaller QGg +will be (it reaches ∼ 0 for perfect photo-z). On the other hand, +non-symmetric photo-z distribution and non-symmetric true-z +distribution will make GIg deviate from 1. For more details on +the Q calculation and its properties, see discussions in Yao et al. +(2020a,b). +101 +102 +103 +104 +ℓ +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Q +gG model +gI model +gG sim +gI sim +100 +101 +102 +θ [arcmin] +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +gG model +gI model +gG sim +gI sim +Fig. A.1. The effect of photo-z modeling with MICE2. By applying the +SC selection Eq. (8) or (A.8), the lensing-drop GGg from photo-z model +(green) is slightly biased compared with the results from true-z (blue), +while the IA-drop GIg from photo-z model (red) is immune to such bias +and agrees with the true-z result (orange). +We note that for the SC calculation, the redshift PDF p(z|zP) +for each galaxy is required. Due to the fact that the PDFs from +photo-z algorithm can be biased due to the color-redshift degen- +eracy in the photometric surveys, calibration is needed (Hilde- +brandt et al. 2017, 2021; Abbott et al. 2022). However, we can +only statistically calibrate the overall redshift distribution n(z) +but not the PDF p(z|zP) for each galaxy. This means in order to +calculate Eq. A.7 we need to assume a photo-z PDF model. We +choose to use a bi-Gaussian model Yao et al. (2020a) +p2G(z|zP) = (1 − fout)pmain(z|zP; ∆1, σ1) + foutpoutlier(z|zP; ∆2, σ2), +(A.10) +with a main Gaussian peak and a Gaussian outlier peak with +different bias ∆i and scatter σi, and an outlier rate fout. +We fit the bi-Gaussian model Eq. (A.10), requiring it to have +same mean redshift ⟨z⟩ with the SOM calibrated n(z) (Asgari +et al. 2021), and minimize the difference between the resulting +model z-distribution +� +nP(zP)p(z|zP)dzP and the SOM n(z). The +best-fit will then be a good description of the photo-z quality and +can be used in Eq. (A.7). The resulting signal drops are shown in +Fig. 4 in the main text. +We validate the bi-Gaussian photo-z model for SC with +MICE2 simulation. We compare with the results that use the +photo-z distribution and true-z distribution in the calculation of +Eq. (A.7). We show in Fig. A.1 that the bi-Gaussian model can +produce the IA-drop QIg measurement very consistent with the +ones with true-z from simulation. However, we find the lensing- +drop QGg from the photo-z model is slightly higher than the true +values from the simulation. This error will be propagated to the +separated lensing signal wGg and the IA+magnification signal +wIg + gmagwGκ according to Eq. (15) and (16). Its impact in AIA +is shown in Fig. 10 and 11. +Article number, page 15 of 15 + diff --git a/9tFQT4oBgHgl3EQf6DZi/content/tmp_files/load_file.txt b/9tFQT4oBgHgl3EQf6DZi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b29613ab15aa9bf93fabf976ac2bc407260b57e --- /dev/null +++ b/9tFQT4oBgHgl3EQf6DZi/content/tmp_files/load_file.txt @@ -0,0 +1,1859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf,len=1858 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda ©ESO 2023 February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2023 KiDS-1000: cross-correlation with Planck cosmic microwave background lensing and intrinsic alignment removal with 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' German Centre for Cosmological Lensing (GCCL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Universitätsstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 150, 44801, Bochum, Germany 8 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK 9 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='A Milne Centre, University of Hull, Cottingham Road, Hull, HU6 7RX, United Kingdom 10 Center for Theoretical Physics, Polish Academy of Sciences, al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Lotników 32/46, 02-668 Warsaw, Poland 11 Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Box 9513, 2300 RA Leiden, the Netherlands 12 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany Received January 30, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' accepted ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Galaxy shear - cosmic microwave background (CMB) lensing convergence cross-correlations contain additional informa- tion on cosmology to auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' While being immune to certain systematic effects, they are affected by the galaxy intrinsic alignments (IA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This may be responsible for the reported low lensing amplitude of the galaxy shear × CMB convergence cross- correlations, compared to the standard Planck ΛCDM (cosmological constant and cold dark matter) cosmology prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we investigate how IA affects the Kilo-Degree Survey (KiDS) galaxy lensing shear - Planck CMB lensing convergence cross-correlation and compare it to previous treatments with or without IA taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' More specifically, we compare marginalization over IA parameters and the IA self-calibration (SC) method (with additional observables defined only from the source galaxies) and prove that SC can efficiently break the degeneracy between the CMB lensing amplitude Alens and the IA amplitude AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We further investigate how different systematics affect the resulting AIA and Alens, and validate our results with the MICE2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We find that by including the SC method to constrain IA, the information loss due to the degeneracy between CMB lensing and IA is strongly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The best-fit values are Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22 and AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='60+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03, while different angular scale cuts can affect Alens by ∼ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show that appropriate treatment of the boost factor, cosmic magnification, and photometric redshift modeling is important for obtaining the correct IA and cosmological results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' cosmology – weak lensing – CMB lensing – intrinsic alignment – self-calibration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Introduction Weak lensing due to the distortion of light by gravity is a power- ful probe of the underlying matter distribution and the encoded secrets of cosmological physics such as dark matter, dark energy, and the nature of gravity (Refregier 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Mandelbaum 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The auto-correlation statistics have been widely used in the anal- ysis, both for galaxy lensing shear, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' “cosmic shear” (Hilde- brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hamana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hikage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' As- gari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Secco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022), and CMB lensing convergence (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Omori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Furthermore, cross-correlations between galaxy ⋆ e-mail: ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='yao@shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='cn ⋆⋆ e-mail: hyshan@shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='cn ⋆⋆⋆ e-mail: zhangpj@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='cn shear and CMB lensing have been measured extensively (Hand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Liu & Hill 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Omori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Namikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Cross-correlation statistics contain highly complementary information to auto-correlations, both for cosmology and the cross-check of systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' They partly reveal the hidden redshift information in CMB lensing and are more sensitive to structure growth at redshifts between the epochs probed by galaxy shear and CMB lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Cross- correlations are also immune to additive errors in shear measure- ment and provide an external diagnosis of multiplicative errors (Schaan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Most existing cross-correlation measurements have found a lower CMB lensing amplitude than the prediction of their as- Article number, page 1 of 15 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='13437v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='CO] 31 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda sumed ΛCDM cosmology (Hand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Liu & Hill 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The ra- tio, which is normally referred as the CMB lensing amplitude, Alens ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9, although the deviation from unity is only within 1-2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The low lensing amplitude is consistent across many com- binations of data sets and analysis methods, suggesting the ex- istence of a common systematic errors or a deviation from the best-fit Planck cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This might be related to the tension between galaxy lensing surveys and Planck CMB observation (Lin & Ishak 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021), and the Planck internal inconsistencies (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this paper we focus on the galaxy intrinsic alignment (IA), which can mimic weak lensing signals (Croft & Metzler 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Catelan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Crittenden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Lee & Pen 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Jing 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hirata & Seljak 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Bridle & King 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Okumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Joachimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Kiessling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Blazek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Blazek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Troxel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Chis- ari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Samuroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Samuroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Here the CMB lensing convergence is expected to be anti-correlated with the intrinsic ellipticities of the foreground galaxy field, resulting in a dilution of the overall cross-correlation signal (Troxel & Ishak 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Omori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Taking IA into account can alleviate the tension in Alens, at the expense of a significant loss of lensing constraining power, because of the degeneracy between the lens- ing amplitude Alens and the IA amplitude AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore, a com- mon compromise is to fix both the IA model and its amplitude AIA (Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Omori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019) or assume a strong prior (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Since IA is already a major limiting factor in the current cross-correlation analysis, its mitigation will be essential for up- coming measurements with significantly smaller statistical er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We utilize the IA self-calibration (SC) method (Zhang 2010a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Troxel & Ishak 2012a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017, 2019), which is a galaxy-galaxy lensing method but with a different weight- ing scheme, to mitigate the IA problem in the shear-convergence cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' It is based on the fact that the IA-galaxy cor- relation is insensitive to the redshift order, while it matters for lensing-galaxy correlation whether the lens is in front of the source or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore, we can isolate IA by comparing ex- tra observables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', the galaxy shear × number density cross- correlation with a different weighting of the redshift pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This measurement of IA is independent of a physical model of the IA and requires no data external to the shear data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' SC was first applied to KiDS450/KV450 (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Pedersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020) and DECaLS DR3 (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b) and has enabled significant IA detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The detected IA signal can then be applied to remove IA in the lensing shear auto-correlation and shear-convergence cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The IA information is ob- tained from a shear × number density cross-correlation within the same photometric redshift (photo-z) bin, more importantly, with different weighting schemes on the photo-z ordering, which is usually not used for cosmological parameter constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We find that this removal of IA losses almost no cosmological infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In previous work Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b), we have demonstrated the importance and methodology of including certain types of systematics in the SC lensing-IA separation method, namely galaxy bias, the covariance between the separated lensing signal and IA signal, the IA signal drop QIg due to the photo-z selection, and the scale dependency of the signal drops QGg and QIg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we further investigate other sources of systematics, includ- ing the boost factor (Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2005), photo-z modeling bias (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a), and cosmic magnification (Bartelmann 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Bartelmann & Schneider 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Interestingly, as the survey goes to higher redshift, the contamination to the SC method from magnification will quickly increase to a non-negligible level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The cosmic magnification will change the observed galaxy number density due to the lensing- magnified flux and lensing-enlarged area, therefore biasing our SC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We investigate the proper treatments for the above systematics together with the cosmological study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2 we review the physics of galaxy shear × CMB convergence and how our SC method works to subtract the IA information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3 we in- troduce the KiDS-1000 and Planck data used in this work, and the MICE2 simulation (van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Fosalba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015) we use to validate how the SC method is affected by differ- ent systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show the measurements of the observables in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The results and summary are shown in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Methods We apply our self-calibration method to separate the intrinsic alignment and the lensing signals and show how the intrinsic alignment will bias the galaxy shear-CMB convergence corre- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this section, we review the theory of lensing cross- correlation and the self-calibration method, with a modification to account for the contamination from cosmic magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Galaxy shear × CMB convergence The gravitational field can distort the shape of the background source galaxy image and introduce an extra shape that is tan- gentially aligned to the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This gravitational shear γG of the source galaxy contains integral information of the foreground overdensity along the line of sight (Bartelmann & Schneider 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Similarly, the photons from the CMB are deflected, and the lensing convergence κ can be reconstructed from the CMB temperature and polarization observations (Planck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' By correlating these two quantities � γGκ � , we probe the clustering of the underlying matter field ⟨δδ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In harmonic space while assuming flat space (Omori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020), we have: CκgalκCMB(ℓ) = � χCMB 0 qgal(χ)qCMB(χ) χ2 Pδ � k = ℓ + 1/2 χ , z � dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (1) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (1) is the galaxy-lensing CMB-lensing cross angular power spectrum, which probes the matter power spectrum Pδ(k, z), as well as the background geometry χ(z) if precision allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Here z is the redshift, χ is the comoving distance, k is the wavenumber, ℓ is the angular mode, qgal(χ) and qCMB(χ) are the lensing efficiency functions for galaxy-lensing and CMB- lensing, with the analytical forms: qgal(χl) = 3 2Ωm H2 0 c2 (1 + zl) � ∞ χl n(χs)(χs − χl)χl χs dχs, (2) qCMB(χl) = 3 2Ωm H2 0 c2 (1 + zl)(χs − χl)χl χs , (3) where χs and χl are the comoving distance to the source and lens, and the χs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (3) takes CMB as the source of light (z ∼ 1100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note the spacial curvature Ωk = 0 is assumed Article number, page 2 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal so that the comoving angular diameter distances in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2) and (3) are replaced with the comoving radial distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Here n(χ) gives the source galaxy distribution as a function of comoving distance, and it is connected with the galaxy redshift distribution via n(χ) = n(z)dz/dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we only use one redshift bin due to the limit of the total S/N on the CMB lensing signal, while a tomographic example can be found in Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In the future with higher S/N, for example, for CMB-S4 × LSST, tomography can be used to subtract more cosmological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The shear-convergence cross-correlation function measured in real space is given by the Hankel transformation: wGκ(θ) = 1 2π � ∞ 0 dℓℓCκgalκCMB(ℓ)J2(ℓθ), (4) where J2(x) is the Bessel function of the first kind and order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The “G” represents the gravitational lensing shear γG, to be sep- arated from the intrinsic alignment γI in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Also for the current low S/N reasons, we choose not to in- vestigate full cosmological constraints in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Instead, we perform a matched-filter fitting, with lensing amplitude Alens that suits ˆwGκ = AlenswGκ, where ˆwGκ is the measured correlation function, and wGκ is the theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Intrinsic alignment of galaxies The observed galaxy shear estimator contains three components: gravitational shear, an intrinsic alignment term, and random noise, namely, ˆγ = γG + γI + γN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Both the gravitational shear and the IA term are related to the underlying matter overdensity δ and are associated with the large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This means that when we cross-correlate the galaxy shape and the CMB con- vergence, there will be contributions from both lensing and IA: ⟨ˆγκ⟩ = � γGκ � + � γIκ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (5) Therefore the IA part of the correlation will contaminate the measurement and lead to a bias in the lensing amplitude Alens or the cosmological parameters when assuming ⟨ˆγκ⟩ = � γGκ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The IA-convergence correlation function is linked to the IA- convergence power spectrum CIκCMB = � χCMB 0 n(χ)qCMB(χ) χ2 Pδ,γI � k = ℓ + 1/2 χ , z � dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (6) Here Pδ,γI is the 3D matter-IA power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The conventional method is to assume an IA model with some nuisance parame- ters, which will enter the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The most widely used IA model is the non-linear linear tidal alignment model (Cate- lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hirata & Seljak 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Bridle & King 2007), ex- pressed as: Pδ,γI = −AIA(L, z)C1ρm,0 D(z) Pδ(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' χ), (7) which is proportional to the non-linear matter power spectrum Pδ, suggesting that the IA is caused by the gravitational tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' AIA is the IA amplitude, which can be redshift(z)- and luminosity(L)- dependent (Joachimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Its redshift evo- lution has been measured recently in simulations (Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Samuroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021) and suggestions in observations with low significance (Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Secco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Tonegawa & Okumura 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The other related quantities include: the mean matter density of the universe at z = 0, expressed as ρm,0 = ρcritΩm,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' C1 = 5 × 10−14(h2Msun/Mpc−3) the empirical amplitude taken from Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2002) and the normalized linear growth factor D(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that the IA model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (7) can be replaced by more complicated models as in Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Blazek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2015, 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Fortuna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021) for different samples (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Samuroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Zjupa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The self-calibration method can intro- duce new observables to constrain IA with additional constrain- ing power, and in the future when the signal-to-noise (S/N) al- lows, it can be extended to constrain more complicated IA mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Self-calibration of intrinsic alignment The IA self-calibration (SC) method (Zhang 2010b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017, 2019, 2020a,b) uses the same galaxy sample as both the source and the lens, which is different from most galaxy-galaxy lensing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' It introduces two observables: the shape-galaxy correlation in the same redshift bin wγg, and a similar correlation wγg|S using the pairs where the photo-z of the source galaxy is lower than the photo-z of the lens galaxy, namely zP γ < zP g (8) (this will be denoted as “the SC selection”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we extend our methodology to include the im- pact from cosmic magnification (Bartelmann 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Bartelmann & Schneider 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Because of the existence of magnification, the intrinsic galaxy number den- sity field δg is affected by the foreground lensing convergence κgal, leading to a lensed galaxy overdensity δL g = δg + gmagκgal, (9) where the prefactor writes gmag = 2(α − 1) for a complete and flux-limited sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' It accounts for the increase in galaxy num- ber density due to lensing-magnified flux (α = −d ln N/d ln F, where N(F) denotes the galaxy number N that is brighter than the flux limit F) and the decrease of galaxy number density due to the lensing-area-enlargement (-2 in gmag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The observed shape-galaxy correlation is given by � ˆγδL g � = � (γG + γI)(δg + gmagκgal) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (10) The two SC observables can be written as: wγgL ii (θ) = wGg ii (θ) + wIg ii (θ) + gmag � wGκgal ii (θ) + wIκgal ii (θ) � , (11) wγgL ii |S(θ) = wGg ii |S(θ) + wIg ii |S(θ) + gmag � wGκgal ii |S(θ) + wIκgal ii |S(θ) � , (12) where the “|S” denotes the SC selection, and i denotes the i-th redshift bin if tomography is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The lensing-galaxy wGg and the IA-galaxy wIg signal are affected by this SC selection, as quantified by the Q parameters: QGg i (θ) ≡ wGg ii |S(θ) wGg ii (θ) , (13) QIg i (θ) ≡ wIg ii |S(θ) wIg ii (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (14) For the lensing signal to exist, the redshift of the source, zγ, needs to be greater than the redshift of the lens, zg: zγ > zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Article number, page 3 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 zP g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0004 wXg(zP|zP g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content="5, ) X=G, lensing, = 1' X=I, IA, = 1' X=G, lensing, = 5' X=I, IA, = 5' X=G, lensing, = 50' X=I, IA, = 50' Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A toy model to illustrate the different redshift dependences for the lensing signal and the IA signal, and why the SC selection Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (8) works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We place many lens galaxies at photo-z zP g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 (the grey dotted line), while allowing the photo-z of the source galaxies zP γ to change (x- axis) to evaluate the corresponding lensing correlation function wGg or IA correlation function wIg at different angular separation θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The true-z has a Gaussian scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='04 (this number is chosen for exhibition, so that the lensing/IA signals have comparable maximum/minimum val- ues) around the photo-z, for both source galaxies and lens galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' As the gravitational lensing shear is an optical shape that requires zg < zγ, it will have a non-symmetric power around zP g, as the positive solid curves show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This also demonstrate QGg ≪ 1 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' As the IA shape is a dynamical shape, it does not have requirements on the relative redshifts, leading to a symmetric power around zP g, as the neg- ative dashed curves show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This also demonstrate QIg ∼ 1 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' These relations hold for signals at different angular separa- tions (different colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The different IA models (which could deviate from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 7 and AIA = 1 being assumed) will only change the rela- tive amplitudes of the negative signals at different scales, but not the redshift-dependency around zP g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note at such a redshift range, the magnification signal is much smaller than the IA signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The SC photo-z selection zP γ < zP g largely reduces the lensing signal, leading to QGg ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The IA signal does not rely on the ordering along the line-of-sight, with QIg ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The lensing-drop QGg and the IA-drop QIg are dependent on the photo-z quality, as described in Zhang (2010b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2017, 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' If the photo-z quality is perfect, the SC selection will result in no lens- ing signal so that QGg approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For incorrect photo-zs, the SC selection fails and QGg is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Given a photo-z distribution nP(zP) and the true-z distribution n(z), the lensing-drop QGg and IA-drop QIg can be theoretically derived, following Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020a,b), with more technical details in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We also present a toy model to visualize how the SC selection works in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We quantitatively test the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (11), and they gener- ally follow |wIκgal| < |wGκgal| ≪ |wIg| < |wGg| for z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9 data, therefore in previous analysis (Zhang 2010b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a,b) the magnification terms were neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For the z ∼ 1 galax- ies, however, the magnification term wGκgal quickly approaches wIg and becomes a non-negligible source of contamination to the SC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2 we show a theoretical comparison of the angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We can write the SC selection for the magnification term as wGκgal|S = QGκwGκgal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The drop of the sig- nal QGκ ∼ QIg ∼ 1 given that these are not z-pair-dependent correlations, therefore the magnification signal wGκgal will con- 10 100 1000 ℓ 10−10 10−9 10−8 10−7 C(ℓ) CGg, bg,eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='88 CIg, bg,eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='88, AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 gmagCGκgal, gmag = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A theoretical comparison between the galaxy-shear CGg(ℓ), galaxy-IA CIg(ℓ) and shear-magnification gmagCGκgal(ℓ) angular power spectra, with the best-fit of our baseline analysis and the redshift distri- bution n(z) from KiDS-1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 < zP < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 shear catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The dashed lines represent negative signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This figure demonstrates that the mag- nification contamination is important in the self-calibration method for the high-z KiDS source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' taminate the IA signal wIg due to similar behavior, leaving the lensing signal wGg unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note the wIκ term is negligible in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' After measuring the galaxy-galaxy lensing observables {wγgL, wγgL|S} and the drops of the signals {QGg, QIg} (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (13), (14) and Appendix A for more details), the corre- sponding lensing-galaxy correlation wGg, IA-galaxy correlation wIg and shear-magnification correlation wGκ can be linearly ob- tained: wGg ii (θ) = QIg i (θ)wγgL ii (θ) − wγgL ii |S(θ) QIg i (θ) − QGg i (θ) , (15) wIg ii (θ) + wGκgal ii (θ) = wγgL ii |S(θ) − QGg i (θ)wγgL ii (θ) QIg i (θ) − QGg i (θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (16) In previous work, the IA information was directly extracted in wIg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 16, for KiDS the subtracted signal suffers from the contamination from a magni- fication term wGκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' By constraining the measurements of {wGg, wIg+wGκgal, wγκCMB} together, including the covariance, will lead to robust constraints on both the lensing amplitude and the nui- sance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For the current stage where the S/N for the measurements are not very high, we choose to ignore the pos- sible scale-dependent features for the effective galaxy bias bg,eff and IA amplitude AIA, and assume they are linear and determin- istic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The parameters {Alens, AIA, bg,eff, gmag} are connected to the observables following: wGg(θ) = bg,effwGm theory(θ), (17) wIg(θ) + wGκgal(θ) = bg,effAIAwIm theory(θ) + gmagwGκgal theory(θ), (18) wγκCMB(θ) = AlenswGκCMB theory (θ) + AIAwIκCMB theory(θ), (19) where “m” stands for matter, which is the case if one sets the ef- fective galaxy bias bg,eff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We separate the CMB convergence and the galaxy convergence (due to magnification) with κCMB Article number, page 4 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The ΛCDM cosmological parameters adopted in this work, corresponding to the best-fit cosmology from Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020a), and the KiDS-1000 multivariate maximum posterior (MAP) results from the two-point correlation functions ξ±, the band powers C(ℓ), and the COSEBIs (Complete Orthogonal Sets of E/B- Integrals) as in Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Survey h0 Ωbh2 Ωch2 ns σ8 Planck 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='812 KiDS ξ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='895 KiDS C(ℓ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='723 KiDS COSEBI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='772 and κgal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' On the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (17), (18) and (19) are the measure- ments, while on the RHS the correlations w(θ) are the theoreti- cal predictions assuming Planck cosmology (Planck Collabora- tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a), see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note the Q values being used to obtain the LHS are also cosmology dependent, however, the sensitivity is weak as the cosmological part is mostly canceled when taking the ratio in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We tested if the fidu- cial cosmology is changed to any of the KiDS-1000 cosmolo- gies in Table 1, the Qs will change by ∼ 1%, similar to Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b), and the resulting changes to the fitting parameters {AIA, bg,eff, gmag, Alens} are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, considering the RHS, those four fitting parameters are sensitive to the fiducial cosmol- ogy used to produce the wtheory values when magnification exists, which differs from previous analysis (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The the- oretical predictions wtheory are calculated with ccl1 (Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019) and camb2 (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The effective galaxy bias bg,eff in this work is used to separate from the true galaxy bias of this sample, as we will discuss later it can absorb several sources of systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The theoretical prediction of wGκCMB theory (θ) is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (4), and wIκgal theory(θ) is obtained similarly with the Hankel transform from its power spectrum as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The wGm theory, wIm theory and wGκgal theory terms are the Hankel transform from the following angu- lar power spectra: CGm(ℓ) = � zmax zmin qgal(χ)n(χ) χ2 Pδ � k = ℓ + 1/2 χ , z � dχ, (20) CIm(ℓ) = � zmax zmin n(χ)n(χ) χ2 Pδ,γI � k = ℓ + 1/2 χ , z � dχ, (21) CGκgal(ℓ) = � zmax zmin qgal(χ)qgal(χ) χ2 Pδ � k = ℓ + 1/2 χ , z � dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (22) As discussed in previous work (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b), by in- cluding the effective galaxy bias bg,eff, we can obtain an unbi- ased estimation of AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This information will be propagated into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (19) to break the degeneracy between AIA and Alens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we further extend the fitting to include the impact from magnification with the nuisance parameter gmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We will show later that an unbiased CMB lensing amplitude Alens can be ob- tained from the simultaneous fitting of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (17), (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Data In this section, we introduce the data we use for the � γκCMB� cross-correlation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Additionally, we use mock KiDS data, 1 Core Cosmology Library, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/LSSTDESC/CCL 2 Code for Anisotropies in the Microwave Background, https:// camb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='info/ based on the MICE2 simulation (see van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020) for details) to quantify the potential bias in the SC method due to magnification, photo-z modeling, and the boost factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' KiDS-1000 shear catalog We use the fourth data release of the Kilo-Degree Survey that covers 1006 deg2, known as KiDS-1000 (Kuijken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' It has images from four optical bands ugri and five near-infrared bands ZYJHKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The observed galaxies can reach a primary r−band median limiting 5σ point source magnitude at ∼ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The shear catalog (Giblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021) contains ∼ 21 M galaxies and is divided into five tomographic bins in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 < zB < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 based on the bpz (Benitez 2000) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The ellipticity dis- persion σϵ is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='27 per component, and the shear multiplicative bias is generally consistent with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The KiDS data are processed by theli (Erben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013) and Astro-WISE (Begeman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' de Jong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Shears are measured using lensfit (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013), and pho- tometric redshifts are obtained from PSF-matched photometry and calibrated using external overlapping spectroscopic surveys (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The application of SC requires not only an accurate redshift distribution n(z), but also relatively accurate photo-z for each galaxy, serving for the SC selection (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We discussed in pre- vious work (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a) that the quality of photo-z is very important for the lensing-IA separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore in this work, we choose to combine the three high-z bins, namely bin 3+4+5 in KiDS-1000 data, as a large bin so that the photo-z error for an individual galaxy is relatively small compared to the total bin width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The photo-z and the SOM-calibrated redshift distri- butions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We choose to use the high-z bins be- cause the CMB lensing efficiency Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (3) peaks at z ∼ 1 to 2 (see lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3), while the S/N for the cross-correlation is very low for the two low-z bins of KiDS-1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' To account for the selection functions for the shape of the footprint (Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2006) of the overlapped region and the varying galaxy number density due to observation (Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Rezaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020), we divide the region into 200 sub-regions with a resolution of Healpix Nside = 512 (∼ 50 arcmin2 per pixel), and generate random points with 20 times the number of galaxies of the KiDS-1000 shear catalog in each sub-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The pixels within the same sub-region are assigned the same galaxy numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This random catalog is used for the SC-related galaxy-galaxy lensing calculation, while its potential defects will not extend to cross-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Planck legacy lensing map We use the CMB lensing map κ(θ) from the Planck data release (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The CMB lensing map is reconstructed with the quadratic estimator with the minimum- variance method combining the temperature map and the polar- ization map, after foreground removal with the SMICA method (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' It covers fsky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='671 of the whole sky with the maximum multiple ℓ = 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work we combine the footprint from the Planck lens- ing map and the mask of the KiDS-1000 shear catalog, leading to an overlapped region of ∼ 829 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We include the Planck Wiener filter (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020c) ˆκWF ℓm = Cφφ,fid ℓ Cφφ,fid ℓ + Nφφ ℓ ˆκMV ℓm (23) Article number, page 5 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 n(z) n(z) nP(zP) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 2 z 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8 1 lensing efficiency galaxy lensing CMB lensing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The photo-z distribution and the SOM-reconstructed redshift dis- tribution of the combined galaxy sample in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The correspond- ing galaxy lensing efficiency Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2) and its comparison with CMB lens- ing efficiency Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (3) are shown in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' to strengthen the CMB lensing signal at large scales, which will also lead to a suppression of the power spectrum at small scales, where the noise dominates (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The Wiener filter is used both in the CMB lensing κ map and in the theoretical pre- dictions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (1) to prevent potential bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' After the application of the Wiener filter, we use Healpy3 (Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019) to convert the κℓm to the desired κ-map, and rotate from the galactic coordinates of Planck to the J2000 coordinates of KiDS with Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The two-point correlation functions are calculated with TreeCorr 4 (Jarvis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' MICE2 mock catalog Additionally, we use the MICE2 simulation gold samples (van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Fosalba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015), which highly mimic the KiDS-1000 shear catalog galaxies, to validate our SC method, concerning cosmic magnification and photo-z PDF model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' MICE2 uses a simulation box width of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 h−1Gpc, particle mass resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9 × 1010 h−1M⊙, and a total particle number of ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9 × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The fiducial cosmology is flat ΛCDM with Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25, σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='044, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='75 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The halos are identified with Friends-of-Friends as in Crocce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The galaxies are populated within the halos with a mixture of halo abundance matching (HAM) and halo occupa- tion distribution (HOD) up to z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 (Carretero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that in the MICE2 simulation that we use for the KiDS samples, intrinsic alignment is not yet included in the galaxy shapes (while an IA-included version can be found in Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2022), but for DES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' So that we aim to get AIA = 0 to validate the SC method, while considering system- atics from cosmic magnification and photo-z model bias, in ad- 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/healpy/healpy 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/rmjarvis/TreeCorr 101 102 103 104 ℓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 Q(ℓ) & Q(θ) QGg(ℓ) QIg(ℓ) 100 101 102 θ [arcmin] QGg(θ) QIg(θ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The lensing-drop QGg and the IA-drop QIg as a function of ℓ and θ by applying the SC selection Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (8), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' These val- ues are adopted to obtain the separation of wGg and wIg + wGκgal, follow- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The left panel shows the calculation from power spectra and the right panel from correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The right panel is used to transfer {wγg, wγg|S } to {wGg, wIg} later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' dition to what has been addressed in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use the galaxy positions (ra, dec), the two noiseless shear compo- nents (γ1, γ2), and BPZ-measured photo-z zB to calculate the SC correlations as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We test the signal drop Qs of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (13) and (14) with our photo-z PDF model and with true-z from simulation (van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We compare the results using MICE2 gold samples (which highly mimic the KiDS-1000 shear catalog galaxies) with magnification (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 9) and without magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For the MICE2 galaxies with mag- nification, we tested how it will bias the IA measurement, and proved that when the magnification effect is also included in the model, IA can be measured in an unbiased way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The validations will be shown later in our results with some details in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Measurements We show the estimation of the signal-drops for lensing and IA due to the SC selection (as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13 and 14), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' the lensing- drop QGg and the IA-drop QIg in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' They are responsible for the lensing-IA separation later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6, following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We follow the processes in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020a,b) and adopt a bi-Gaussian photo-z probability distribution function (PDF) model with a secondary peak representing the photo-z outlier problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We require the PDF model to have the same mean-z as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3, while closest describing the projection from nP(zP) to n(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We will also show for the first time how the assumed photo- z PDF model can affect the results in the next section, with more details shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We calculate the SC correlation function estimator, wγg(θ) = B(θ) � ED wjγ+ j (1 + ¯m) � ED wj − � ER wjγ+ j (1 + ¯m) � ER wj , (24) to obtain the measurements of wγg and wγg|S from the tangential shear of each galaxy γ+ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Here we sum over the ellipticity-density pairs (� ED) and the ellipticity-random pairs (� ER) in an annulus centered on θ, where the shear weight wj of the j-th galaxy and the average multiplicative bias ¯m are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The estima- tor is binned in angular θ space, with 9 logarithmic bins from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 Article number, page 6 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal 100 101 102 θ [arcmin] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 B Boost BoostS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The boost factors for wγgL and wγgL|S are shown in blue and orange, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The overlapping lines suggest the two signals are affected by the boost factor in almost the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show the boost factor is significant at small scales for the SC observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' to 300 arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use the averaged multiplicative bias ¯m from averaging over the three z-bins, weighted by the effective galaxy number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This gives ¯m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We account for the impact of the boost factor (Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Joachimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021), which is B in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' It is defined as B(θ) = � ED wj � RD wj , (25) which is used to quantify the small-scale bias due to the clus- tering of lens galaxies and source galaxies (Bernardeau 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hamana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show the measurements of the boost factor for wγgL and wγgL|S as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (11) and (12) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The fact that the boost factors for wγgL and wγgL|S are identical suggests this bias can be absorbed by the galaxy bias bg,eff parameter if magnification is absent (gmag = 0), leading to an unbiased AIA and Alens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The impact from the boost factor can potentially break the linear galaxy bias assumption, but later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 we show the linear assumption is fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The impact of the boost factor and magnification existing together will be shown later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6 we show the SC measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In the left panel, the measured shape-galaxy correlations wγgL are shown in blue: (1) the boost factor ignored case (B = 1) is shown as blue crosses, while (2) the boost factor corrected case is shown as blue up- triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' With the SC selection Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (8), requiring zP γ < zP g for each galaxy pair, the lensing component will drop to QGg ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3 and the IA component will drop to QIg ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='85 (for more details on QGg and QIg, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4 and Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore, the se- lected correlations wγg|S will drop to the orange down-triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Similarly, the boost factor ignored case is shown as crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The separated lensing-galaxy signal wGg and IA-galaxy sig- nal wIg (which is contaminated by magnification-shear signal gmagwGκ) are shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The blue and or- ange curves are the theoretical predictions with the best-fit {AIA, bg,eff, gmag}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For the fitting, we cut off the shaded regions at both large scales and small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The small scale cut at θ = 1 ar- cmin is based on the linear galaxy bias assumption, as including the θ < 1 arcmin data will make the fitting significantly worse (increasing the fitting χ2 from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 to 50, with degree-of-freedom changed from 8 to 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note this scale cut could include the impacts from the 3D non-linear galaxy bias (Fong & Han 2021) and other small-scale effects such as massive neutrinos or baryon feedback in the matter power spectrum (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We emphasize that these systematics will be absorbed by the effective galaxy bias parameter bg,eff —- with- out breaking the scale-independent bias assumption —- so that the IA amplitude will not be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' As discussed previously in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020a,b), the SC method requires significant sep- aration between wγgL and wγgL|S to accurately get wGg and wIg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore, we introduce a large-scale cut at θ = 20 arcmin due to insufficient separation for the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Similarly, we measure the ⟨γκ⟩ correlation with the estimator wγκ(θ) = � i j wjγ+ j κi (1 + ¯m) � i j wj , (26) where κi is the CMB lensing convergence in the i-th pixel of the pixelized map, taking the pixel center for its (ra, dec) co- ordinates, with nside = 2048 in Healpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The measured wγκ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The tangential shear is shown as blue dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We also show the measurements with randomly shuffling galaxy po- sitions and the shear in red crosses as a null test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We test the 45 deg rotated cross shear for both the above cases and they are consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The theoretical prediction with the best-fit Alens and AIA are shown as the green curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' If one assumes there is no IA in the measurements and uses AIA = 0, the theoretical values for the pure lensing signal are shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Note in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 7, because we use the Wiener-filtered κ map from Planck, both the wγκ measurements and the theoretical pre- dictions are suppressed at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The Wiener filter can significantly reduce the impact of the noise of the Planck lens- ing map and improve the S/N of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Together with the measurements in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6 and 7, we obtain observables of this work, which are the LHS terms of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (17), (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use Jackknife resampling to obtain the co- variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 200 Jackknife regions are used, which is much larger than the length of the data vector (12), based on the analy- sis of Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hartlap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The Jackknife regions are separated using the K-means algorithm kmeans_radec5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The normalized covariance matrix is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We find strong anti-correlation between wGg and wIg as expected (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Note here in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 8, wIg means the separated signal in the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (16), including both the IA part and the contamination from magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' There is no sig- nificant correlation between wγκ and the other two observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This covariance will be used in the Monte Carlo Markov Chain (MCMC) to find the best-fit parameters of {AIA, bg,eff, gmag, Alens}, while all the other cosmological parameters are fixed to Planck as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Validation with MICE2 In this subsection, we apply the IA self-calibration to the MICE2 mock catalog to test the impact of the systematics and validate the recovery of the IA signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The processes of the mock data are identical to the descriptions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4, but only focusing on the self-calibration part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The measurements are similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6 so 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/esheldon/kmeansradec Article number, page 7 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda 1 3 10 30 θ [arcmin] 1 0 1 2 3 4 w(θ) × 104 wγgL wγgL|S 1 3 10 30 θ [arcmin] 1 0 1 2 3 4 wIg gmagwGκgal tot wGg wIg (+gmagwGκgal) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The measurements of SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The left panel shows the measurement of the two introduced observables wγgL and the one with the SC selection wγgL|S, while the corresponding 45-deg rotation test is consistent with 0 for both measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The significant separation of the two signals shows that SC is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The right panel shows the separated lensing signal wGg and wIg, where the latter is contaminated by the magnification signal as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The up- and down-triangles are the results that take the boost factor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5) into consideration, while the crosses are the results that ignore this correction, setting B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The curves are the theoretical value with the best-fit {AIA, bg,eff, gmag} of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The blue curve represents the separated lensing signal as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The orange curve represents the total contribution of IA and magnification as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3 10 30 100 300 θ [arcmin] 1 0 1 2 3 wκγ(θ) × 106 wGκ lensing w(G+I)κ best−fit ⟨κγt⟩ ⟨κγshuffle⟩ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The measurement of the cross-correlation between Planck con- vergence κ and KiDS-1000 shear γ, based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The blue dots are the measurements using tangential shear, with the green curve showing the best-fit considering both lensing and IA, while the orange curve shows only the lensing-lensing component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The red crosses show the null test by randomly shuffling the shear galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The 45-deg rotation tests for both the blue dots and the red dots are consistent with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The differently shaded regions correspond to our angular scale cuts at 2, 20 (default), and 40 arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' we choose to skip them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We perform the MCMC calculation us- ing emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We consider flat priors in −5 < AIA < 5, 0 < bg,eff < 2 and −3 < gmag < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Impact from magnification We show how the magnification signal affects the original SC method (Zhang 2010b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a,b) and the correction introduced in this work, focusing on the gmag − AIA space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' wGg wIg wγκ wGg wIg wγκ correlation coefficient −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='00 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The normalized covariance matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' the correlation coeffi- cient) used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' There exists a strong anti-correlation between the lensing-galaxy correlation wGg and the IA-galaxy correlation wIg (including the contamination from wGκgal) as we found in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The covariance of the 12 data points is calculated from Jackknife re- sampling with 200 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note the IA information is passed from 1 < θ < 20 [arcmin] for wIg to 20 < θ < 300 [arcmin] for wγκ with the scale-independent AIA assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 9, we show that if magnification is not included in the modeling, gmag is therefore not constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The existing mag- nification signal will be treated as the IA signal, leading to a non-vanishing AIA ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3, which significantly deviates from the MICE2 input AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' When the magnification model is in- cluded in the analysis, AIA is then consistent with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This demon- strates the importance of including the magnification model in the SC analysis with high-z data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The results are also summa- Article number, page 8 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal MICE IA MICE IA+mag −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='30 AIA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='00 gmag Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The impact of the magnification signal on the IA measurement in MICE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The green and blue contours are with and without magni- fication models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' If the magnification model is used in the fitting, as in green, the IA amplitude AIA is consistent with 0, which is the MICE2 input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' rized later in the comparisons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11 for MICE2, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 14 for KiDS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that in the green case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 9 that considered both IA and magnification, gmag and AIA strongly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' There- fore the constraining power in AIA has a significant loss com- pared with the blue case, which ignores magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This de- generacy can be broken in the future with higher S/N in the ob- servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This is because the shape of wIg and wGκ are different at small scales for correlation functions as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6, and on large scales for power spectra as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The IA-model-dependency will be discussed later with other results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Based on the above analysis, we conclude it is important to include magnification modeling for SC when using high-z data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Impact from modeling p(z|zP) Since the SC selection Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (8) plays an important role in the lensing-IA separation process, it is crucial to understand how the following aspects affect SC: (1) the quality of the photo-z zP, (2) the true redshift distribution n(z), and (3) the link between them p(z|zP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The quality of photo-z and the reconstruction of n(z) has been studied thoroughly for KiDS data (Kuijken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' van den Busch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020), we, therefore, trust these results and leave the al- ternative studies for SC to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The uncalibrated PDF that projects zP → z, on the other hand, has some known prob- lems, for example when Probability Integral Transform (PIT) is applied (Newman & Gruen 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we use a bi-Gaussian PDF model to project the photo-z distribution nP(zP) to the SOM redshift distribution n(z), which are previously shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This modeling ignores the potential differences for galaxies in the same z-bin (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, this is an alternative process, MICE Qsim+mag MICE Qmodel MICE Qmodel+mag −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 AIA −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 gmag Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The impact from photo-z PDF model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The blue case uses photo-z from the BPZ algorithm and true-z for each galaxy to calcu- late Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7 and the resulting QGg and QIg, which are the “sim” cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This AIA is consistent with 0, which is the MICE2 input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The green case uses the bi-Gaussian photo-z model for the calculation, which are the “model” cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1, while ignoring the magnifica- tion contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This lead to unconstrained gmag and biased AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In the red case, which also uses the photo-z model, but includes the magnifi- cation model, the resulting AIA is still consistent with 0, with the bias from photo-z model error absorbed by gmag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' considering the PDF problem for a single galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This analytical approach is also much faster in calculation than using different PDFs for different galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 10 to demonstrate how large this photo-z PDF modeling bias is with different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use MICE2 sim- ulation with galaxy number density affected by magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' When the SC calculation uses true-z to calculate the signal drops QGg and QIg, and the magnification model is also considered, we find the resulting AIA is consistent with 0, which is the MICE2 input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The scatter on AIA is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1, thanks to the noiseless shapes in MICE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' If the Qs are calculated with the assumed photo-z PDF model, without including the magnification model, then AIA will be biased towards the negative direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We proved with our fiducial analysis that, even if there exists a bias in QGg due to the assumed photo-z model, as long as the magnification model is used, this bias will be absorbed by the gmag parameter, so that the IA amplitude AIA is unbiased (consistent with 0 in the MICE2 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The results are also shown later in the com- parisons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11 for MICE2, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 14 for KiDS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that the bias due to photo-z modeling is not an es- sential problem for SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In the future, if the photo-z outlier prob- lem (or the redshift-color degeneracy problem) can be under- stood better, then a more reliable photo-z model can be used for our SC study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Alternatively, if the photo-z algorithms can give unbiased PDFs for each galaxy, this problem can also be directly solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Article number, page 9 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3 AIA MICE(mag), Q(sim), w/o mag MICE(mag), Q(sim), w/ mag MICE(mag), Q(model), w/o mag MICE(mag), Q(model), w/ mag MICE(nomag), Q(model), w/ mag Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We validate our SC method with MICE2 simulation, which does not have IA implemented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' therefore, AIA = 0 is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The re- sults are shown in green, with “MICE(mag)” meaning magnification is included in the MICE simulation, while “MICE(nomag)” means mag- nification is not included, “Q(sim)” and “Q(model)” mean if the signal drops Q values are calculated from true-z from simulation or photo-z PDF model, and “w/o mag” and “w/ mag” show if the case includes magnification model in the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The upper two data are the results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 9, showing the impact of the modeling magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The 2nd to the 4th data are the results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 10, showing the impact of Q calculation using different PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The 4th data correspond to our fiducial analysis later for KiDS data, with potential bias ∆AIA < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The bottom data is a reference case assuming no magnification effects in the data, corresponding to our previous work Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Inference on real data With the above demonstration that our treatments for magnifica- tion and photo-z PDF are appropriate, and the resulting bias in AIA is very small (∆AIA < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 and < 1σ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11), we move on to apply SC to KiDS data and its cross-correlation with Planck lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show the analysis of the following three situations: (1) The case “ignore IA”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We only use the observed wγκ, while only including Alens in the fit and ignoring the contamination by IA (by setting AIA = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2) The case “IA w/o SC”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We only use the observed wγκ, but consider both Alens and AIA following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (3) The case “with SC”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use both wγκ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 7 and the SC correlations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Both the CMB lensing amplitude Alens and the nuisance parameters {AIA, bg,eff, gmag} will be used in the analysis, following Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (17), (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use flat priors in 0 < Alens < 2, −5 < AIA < 5, and for the IA self-calibration nuisance parameters we use 0 < bg,eff < 4, −5 < gmag < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For case (1) “ignore IA”, shown in blue, AIA is unconstrained in the fitting, giving the best-fit Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For case (2) “IA w/o SC”, when we consider the existence of IA and apply the IA model as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (7), but do not use the mea- surements from SC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 17, 18), there will be a strong degeneracy between Alens and AIA, as shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' There is a significant loss of constraining power in the lensing amplitude, with the best-fit Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='46 and AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='47+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='11 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For case (3) “with SC”, the introduced measurements of wGg and wIg can not only break the degeneracy between Alens and AIA (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 17, 18 and 19), but also bring more constraining power to AIA, so that the best-fit of Alens will not only be unbiased (according to the validation using simulation) but also has sig- −2 −1 0 1 2 3 4 AIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8 Alens 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8 Alens ignore IA IA w/o SC with SC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The constraints on lensing amplitude Alens and the IA ampli- tude AIA, with three different methods: assume there is no IA in the measured wκγ (blue), consider the impact of IA with conventional IA model but do not use SC (orange), use SC to subtract IA information and constrain together with the CMB lensing cross-correlation (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' When IA is ignored, AIA is unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The similar height and width of Alens PDFs between blue and green prove that by including SC, the AIA − Alens degeneracy can be efficiently broken so that the constraining power loss in Alens is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' nificantly improved constraining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The best-fit values are Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22, AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='60+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03, bg,eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='06, and gmag = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='30+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='60 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 12 we only show AIA and Alens, which are the focus of this work, while bg,eff and gmag are only related with the SC observables but not CMB lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Also as discussed in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b), the existence of the effective galaxy bias bg,eff can also absorb some systematics (so it could be a biased bias), leaving the constraint on AIA unbiased (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For example, we tested if magnification is absent, the effect of boost factor will be purely absorbed by bg,eff, giving unbiased AIA and Alens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The effective galaxy bias could also absorb the differences in the assumed fiducial cosmology, with bg,eff ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='24 with KiDS COSEBI cosmology, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The redshift distribution n(z) can differ slightly with/without accounting for the lensing weight (considering the lensing/clustering part in the galaxy-shape cor- relation), with a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='024 difference in the mean-z, which can lead to ∼ 8% difference in the theoretical lensing signal and ∼ 2% dif- ference in the theoretical IA signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Other unaddressed sources of systematics such as baryonic feedback and massive neutrinos could have similar effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We can also see from the validation using MICE data that although the resulting bg,eff is lower than the expectation, the AIA result is unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The gmag result also resides in a reasonable range, considering the KiDS i-band mag- nitude (Kuijken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019) and comparing it with Duncan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The above three cases of IA treatments are also summa- rized later in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13 and 14 together with more tests and other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The corresponding best-fit curves are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2 and 6 with AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='60+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03, bg,eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='06, and gmag = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='30+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='60 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='62 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Even though the impact of magnification is comparable to the IA signal, we can see in both the angular power spectrum and correlation function that the shapes of IA and magnification are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6, the tidal alignment model wIg and magnification gmagwGκ are comparable at large Article number, page 10 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 2 Alens with SC (Planck) ignore IA IA w/o SC wγκCMB scale > 40 arcmin wγκCMB scale > 2 arcmin SC ignore mag SC ignore boost with SC (KiDS COSEBI) Hand+ 2015 Planck Hand+ 2015 WMAP Liu+ 2015 Kirk+ 2016 SPT Kirk+ 2016 SPT fix-IA Kirk+ 2016 Planck Harnois-Deraps+ 2016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' CFHT Harnois-Deraps+ 2016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' RCSLenS Singh+ 2017 Harnois-Deraps+ 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' KiDS Harnois-Deraps+ 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Planck Omori+ 2018 fix-IA Namikawa+ 2019 Marques+ 2020 Robertson+ 2021,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Planck Robertson+ 2021,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' KiDS baseline comparisons previous w/o IA previous IA prior Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The comparisons of the constraints on Alens with previous mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Our baseline analysis “with SC” is consistent with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We also show some cases where IA is ignored in the analysis and if IA is considered but the AIA − Alens degeneracy is not broken with SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' These main results in blue are similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show tests with differ- ent scale cuts and different treatments to magnification, boost factor, and different (KiDS) fiducial cosmology in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We compare with other works, separated into ignoring IA (orange) and assuming a strong prior of IA (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that for different work, the different fiducial cos- mology (the “Planck”, “WMAP”, “KiDS” labels on the y-axis) can lead to ∼ 10% difference in Alens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' scale, while different at small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore, in principle, the degeneracy between IA and magnification can be broken for fu- ture data with higher S/N so that the shape/slope information of the observables can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The current degeneracy is due to the low S/N so that the amplitudes of AIA and gmag degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Furthermore, if a more complicated IA model is used, for ex- ample, as in Blazek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2022), the small- scale IA will be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Based on the study of Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021), for a wide range of stellar mass, the small-scale IA should have a higher amplitude (either a direct raise in the amplitude or a “drop-raise” pattern as we go to smaller scales) than the current model so that the IA-magnification degeneracy can be broken further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The appropriate IA model will require studies in many aspects, and with higher S/N in the measurements, thus we leave this topic for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We investigate how different choices can change our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We first compare the different scale cuts for wκγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Besides the baseline analysis of Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22 with θ > 20 arcmin, two more tests are made with a larger scale cut of θ > 40 arcmin and a smaller scale cut of θ > 2 arcmin, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 7, which give us Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25 and Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The comparisons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The large-scale lensing am- plitude is higher than the small-scale one, which agrees with the finding in Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020c) and other cross- correlation work (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In this work, we only re- port this large-scale v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' small scale difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, the cur- rent S/N of CMB convergence - galaxy shear correlation and the model assumptions do not allow us to investigate further on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3 2 1 0 1 2 3 4 AIA with SC (Planck) IA w/o SC SC ignore mag SC ignore boost with SC (KiDS COSEBI) Robertson+ 2021 prior Asgari+ 2021 C(ℓ) Asgari+ 2021 COSEBI Asgari+ 2021 ξ± DES Y3 Secco+ HSC Y1 ξ± Hamana+ HSC Y1 C(ℓ) Hikage+ this work KiDS others Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The comparisons of the constraints on AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show the results of this work in blue, which contains our fiducial analysis with SC ap- plied, and the comparisons of (1) without SC, (2) with SC but ignoring magnification, (3) with SC but ignoring boost factor, and (4) switching to KiDS fiducial cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show comparisons with other works using KiDS-1000 data in orange, and some works using DES or HSC data in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We then compare the different choices in the SC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We find that if the magnification model is ignored in the analysis, the existing magnification signal in the data will be treated as an IA signal, leading to an over-estimated AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='41 and an over-estimated Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' On the other hand, we pre- viously argued that, when magnification is absent, the impact from the boost factor will be purely absorbed by the effective galaxy bias bg,eff, leaving AIA and Alens unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Unfortunately, this does not hold anymore when magnification is present: if the boost factor is not corrected, all the parameters will be biased as follows AIA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='86+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='01 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='05, bg,eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='06, Alens = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='23 and gmag = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='55+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='28 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We include the comparisons of Alens and AIA for the above-described cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13 and 14 and empha- sis the importance of taking magnification and boost factor into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We also show the impact of the assumed fiducial cosmology: if the fiducial cosmology is switched from Planck to KiDS-1000 COSEBI as in Table 1, both Alens and AIA will change as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13 (bottom-red) and 14 (bottom-blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' With the above results in simulation and data, summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11, 13 and 14, we show that our measurements on AIA and Alens are unbiased from magnification, boost factor, and the assumed photo-z PDF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' These are the new developments considering the existence of magnification at high redshift z ∼ 1, beyond the study of Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Additionally, we compare our analysis with previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The comparisons of Alens are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We find that most of the previous works ignored the IA contamination (Hand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Liu & Hill 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Harnois-Déraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Namikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For the ones that considered IA, they either fixed the IA amplitude (Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Omori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019) or used a strong prior (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021) to break the degeneracy between Alens and AIA, which will otherwise cause a strong loss in constraining power as we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We are the first to directly achieve the IA amplitude measurement within the same data and break the lensing-IA degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Our Article number, page 11 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' aanda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 AIA Asgari+ 2021 C(ℓ) SC, C(ℓ) cosmo Asgari+ 2021 COSEBI SC, COSEBI cosmo Asgari+ 2021 ξ± SC, ξ± cosmo SC cosmic shear Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The comparisons of AIA between SC-subtracted results (blue) and cosmic shear tomography subtracted results (orange) with cosmolo- gies from different 2-point statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The cosmologies are shown in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' baseline analysis is consistent with most of the previous results, showing the contamination from IA is not significant, mainly due to the total S/N of CMB lensing - galaxy shear cross-correlation is only at 3 ∼ 5 σ level at the current stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, the correct treatment for IA will be more and more important in the future with stage IV cosmic shear surveys and CMB observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The comparisons of the AIA constraint with other results us- ing KiDS-1000 data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 14, including the prior assumed in Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021) and the cosmic shear tomog- raphy constraint in Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Although the redshift range is slightly different, the above works have consistent re- sults on AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' These comparisons will become more interesting for the next-stage observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' As an extended study, we investigate how the choice of fidu- cial cosmology affects the SC results, namely AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 14 we show the results with the fiducial Planck cosmology and the KiDS-1000 two-point correlation function ξ± best-fit cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We further compare the results with the KiDS-1000 band power C(ℓ) cosmology and the COSEBIs cosmology in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The re- sults from Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021) (shown in orange) are arranged in increasing order from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We find that when assuming the same cosmology, the SC results (shown in blue) also follow the same (weak) trend, meanwhile, they agree very well with the cosmic shear results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note the SC results will provide extra information in constraining IA in cosmic shear in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Summary In this work, we achieved the first application of the self- calibration (SC) method of intrinsic alignment (IA) of galax- ies to its cosmological application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We proved that with SC, the lensing-IA degeneracy could be efficiently broken, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', in this CMB lensing × galaxy shear cross-correlation work, it means breaking the degeneracy between the lensing amplitude Alens and the IA amplitude AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We showed that for previous treatments, IA are either ignored or being considered with a strong assumed prior on AIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 12, 13 and 14 that with SC to break the degeneracy, the constraining power in both Alens and AIA is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We demonstrated that the proper angular scale cuts on wκγ are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Our baseline analysis using information from θ > 20 arcmin gives Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' If we use informa- tion only at larger scales with θ > 40 arcmin, the constraint is Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' If we include information at much smaller scales with θ > 2 arcmin, the constraint is Alens = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' At the current stage, they do not differ significantly from each other (even considering they are strongly correlated), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, we note that these differences at differ- ent scales also exist in other works Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020c) and Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We, therefore, emphasize the im- portance of understanding the possible systematics at different scales for future studies with higher S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Comparing our CMB lensing amplitude Alens with other works in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13, we found consistent results with different treat- ments of IA throughout almost all the works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We conclude that IA is not a significant source of systematics for the current stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, it will soon become more important with the stage IV observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Nevertheless, we emphasize that the correct treat- ment to break the lensing-IA degeneracy is very important to maintain the cosmological constraining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Our constraint on the IA amplitude AIA in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 14 is also consistent with the existing analysis on IA with KiDS-1000 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that the SC-subtracted IA information can be used as extra constraining power for any of these analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' On the technique side, we further developed the SC method considering more sources of systematics beyond Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We showed at z ∼ 1, the impact of galaxy shear × cos- mic magnification component wGκgal contaminates the separated IA × galaxy number density signal wIg, and is non-negligible as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (16) and (18) to show how the magnification term enters our observable and how we include it in the theory as a correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 13 and 14 that the correction of magnification is important when applying SC to higher redshift data, in order to get the correct constraint on IA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We also discussed that, with the contamination from mag- nification, boost factor can no longer be absorbed by the effec- tive galaxy bias bg,eff, and need to be accounted for correctly, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (24), (25) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 6, 13, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We also validated our analysis with MICE2 simulation, fo- cusing on two aspects: (1) how good can the magnification model mitigate the contamination from the magnification-shear signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' and (2) will the assumed photo-z PDF model (which is used to calculate the signal drop QGg and QIg) bias the IA mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' With the strong constraining power from MICE2 with no shape noise, we can show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11 that, when the magnifi- cation model is included in the analysis, the IA amplitude can be obtained correctly (consistent within 1σ range of 0, which is the input of MICE2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Additionally, the bias from the assumed photo- z model is negligible when the magnification model is used, as the effective magnification prefactor gmag will absorb the intro- duced error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We, therefore, emphasize the importance of includ- ing the magnification model in the SC analysis, especially for fu- ture high-z surveys like LSST, Euclid, WFIRST, and CSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We further notice the contamination from magnification will make SC no longer an IA-model-independent method, therefore, SC is more suitable for low-z data when considering alternative IA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Comparing with our first measurements with KV-450 data (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a), a lot of improvements have been added in the SC method, including: (1) the covariance, the galaxy bias, the scale-dependency for the lensing-drop QGg, the IA-drop QIg, and appropriate scale-cuts, Article number, page 12 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal which have been introduced in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2) the boost factor, the cosmic magnification, and the photo-z PDF modeling, which are introduced in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (3) its first validation using simulation, and its first application to cosmology in order to break the lensing-IA degeneracy, intro- duced in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' With these improvements, we manage to achieve consistent IA results between SC and cosmic shear, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 15, while previously we got AIA = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='42 with the old version of SC (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020a) and AIA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='981+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='694 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='678 for cosmic shear (Hilde- brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020) with KV-450 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Despite SC-obtained AIA is consistent with the MICE input IA, and when applying to data it is consistent with the KiDS cos- mic shear results Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021) and the other CMB lensing work Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2021), as well as gmag is in reasonable agreement with (Duncan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2014), our results still suffer from an unrealisticly low effective galaxy bias bg,eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='88, which is different from our previous work (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We dis- cussed this value may absorb the contribution from (1) fiducial cosmology, (2) lensing weight in n(z), (3) insufficient modeling in non-linear galaxy bias, baryonic effects, and massive neutri- nos, (4) incorrect photo-z v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' true-z connection as discussed in Appendix A and (5) possible other sources of systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We emphasize the complication and leave this point for future stud- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that there could still exist other systematics other than the galaxy bias, such as beyond Limber approximation (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020), non-flat ΛCDM (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021), selection bias on shear measurement (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' But they have either much smaller impacts compared with IA or are strongly reduced due to our scale cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Therefore, they are beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The authors thank Yu Yu, Hai Yu, Jiaxin Wang for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This work is supported by National Key R&D Program of China No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022YFF0503403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' JY acknowledges the support of the National Science Foundation of China (12203084), the China Postdoctoral Science Foundation (2021T140451), and the Shanghai Post-doctoral Excellence Program (2021419).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' HYS acknowledges the support from CMS-CSST-2021-A01 and CMS-CSST- 2021-B01, NSFC of China under grant 11973070, the Shanghai Committee of Science and Technology grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='19ZR1466600 and Key Research Program of Frontier Sciences, CAS, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' ZDBS-LY-7013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' PZ acknowledges the support of the National Science Foundation of China (11621303, 11433001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' XL acknowledges the support of NSFC of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 11803028, YNU Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' C176220100008, and a grant from the CAS Interdisciplinary Innovation Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' BJ acknowledges support by STFC Consolidated Grant ST/V000780/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' MB is supported by the Polish National Science Center through grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020/38/E/ST9/00395, 2018/30/E/ST9/00698, 2018/31/G/ST9/03388 and 2020/39/B/ST9/03494, and by the Polish Ministry of Science and Higher Education through grant DIR/WK/2018/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' HH is supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft (Hi 1495/5-1) as well as an ERC Consolidator Grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 770935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' TT acknowledges support from the Leverhulme Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' AW is supported by an European Research Council Consolidator Grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 770935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' ZY acknowledges support from the Max Planck Society and the Alexander von Humboldt Foundation in the framework of the Max Planck-Humboldt Research Award endowed by the Federal Ministry of Education and Research (Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The computations in this paper were run on the π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The codes JY produced for this paper were written in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' JY thanks all its developers and especially the people behind the following packages: SCIPY (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2001–), NUMPY (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2011), ASTROPY (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013) and MATPLOTLIB (Hunter 2007), TreeCorr (Jarvis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2004), CCL (Chisari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019), CAMB (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2000), Healpy (Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Zonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019), emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2013), fitsio6, kmeans_radec7, corner (Foreman-Mackey 2016), ChainConsumer8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/esheldon/fitsio 7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/esheldon/kmeansradec 8 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='com/Samreay/ChainConsumer KiDS-1000 results in this paper are based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under pro- gramme IDs 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='A-3016, 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='A-3017 and 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='A-3018, and on data products produced by Target/OmegaCEN, INAF-OACN, INAF-OAPD, and the KiDS production team, on behalf of the KiDS consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Author contributions: All authors contributed to the development and writing of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The authorship list is given in three groups: the lead authors (JY, HS, PZ, XL) followed by two alphabetical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The first alphabetical group includes those who are key contributors to both the scientific analysis and the data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The second group covers those who have either made a significant contribution to the data products, or to the scientific analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' References Abbott, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015, A&A, 582, A62 Dong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021, ApJ, 923, 153 Duncan, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Troxel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019, MNRAS, 489, 5453 Samuroff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Mandelbaum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', & Blazek, J.' metadata={'source': 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Jullo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020b, ApJ, 904, 135 Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', & Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='03298 Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', & Cui, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2015, ApJ, 803, 46 Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2010a, MNRAS, 406, L95 Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2010b, ApJ, 720, 1090 Zjupa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Schäfer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', & Hahn, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2020, arXiv e-prints, arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='07951 Zonca, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Singer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', Lenz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2019, Journal of Open Source Software, 4, 1298 Article number, page 14 of 15 Yao et al 2022: KiDS shear × Planck lensing and IA removal Appendix A: The signal drop Q We keep the main text of this paper focused on the physics while keeping the details of the SC method, more specifically the cal- culation for the lensing-drop QGg and the IA-drop QIg in this ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The Qs are calculated through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (13) and (14), while the correlation functions being used are just the Hankel trans- form (similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (4)) of the angular power spectrum CGg and CIg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The associated CGg and CGg|S are calculated by: CGg ii (ℓ) = � ∞ 0 qi(χ)ni(χ) χ2 bg,effPδ � k = ℓ χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' χ � dχ, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1) CGg ii |S(ℓ) = � ∞ 0 qi(χ)ni(χ) χ2 bg,effPδ � k = ℓ χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' χ � ηGg i (z)dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='2) Similarly, the CIg and CIg|S are given by: CIg ii (ℓ) = � ∞ 0 ni(χ)ni(χ) χ2 bg,effPδ,γI � k = ℓ χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' χ � dχ, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='3) CIg ii |S(ℓ) = � ∞ 0 ni(χ)ni(χ) χ2 bg,effPδ,γI � k = ℓ χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' χ � ηIg i (z)dχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4) Here ηGg i (z) = ηGg i (zL = zg = z) is the function that account for the effect of the SC selection Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (8) in the Limber integral, similarly for ηIg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' They are expressed ηGg i (zL, zg) = 2 � dzP G � dzP g � ∞ 0 dzGWL(zL, zG)S (zP G, zP g)K � dzP G � dzPg � ∞ 0 dzGWL(zL, zG)K , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5) ηIg i (zL, zg) = 2 � dzP G � dzP g � ∞ 0 dzGS (zP G, zP g)K � dzP G � dzPg � ∞ 0 dzGK , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6) as in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020b), where K is the galaxy-pair redshift dis- tribution kernel K(zG, zg, zP G, zP g) = p(zG|zP G)p(zg|zP g)nP i (zP G)nP i (zP g), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7) and S is the SC selection function S (zP G, zP g) = �1 for zP G < zP g, 0 otherwise , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8) which correspond to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 8 in the main text, and the lensing kernel is WL(zL, zS ) = ������� 3 2Ωm H2 0 c2 (1 + zL)χL(1 − χL χS ) for zL < zS 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9) Here zx is the true-z where x can be “G” the source, “L” the lens, or “g” the galaxy number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The galaxy photo-z dis- tribution is nP(zP), and the redshift PDF (probability distribution function) is p(z|zP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' As shown above, when the galaxy photo-z distribution and the corresponding true-z distribution are given, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 3 in this work, we can follow the above procedure to calcu- late the lensing-drop QGg and QIg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The results of QGg and QIg for this work are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4 for your interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Generally, given the tomographic bin width, the better photo-z is, the smaller QGg will be (it reaches ∼ 0 for perfect photo-z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' On the other hand, non-symmetric photo-z distribution and non-symmetric true-z distribution will make GIg deviate from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' For more details on the Q calculation and its properties, see discussions in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 101 102 103 104 ℓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 Q gG model gI model gG sim gI sim 100 101 102 θ [arcmin] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='0 gG model gI model gG sim gI sim Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The effect of photo-z modeling with MICE2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' By applying the SC selection Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (8) or (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='8), the lensing-drop GGg from photo-z model (green) is slightly biased compared with the results from true-z (blue), while the IA-drop GIg from photo-z model (red) is immune to such bias and agrees with the true-z result (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We note that for the SC calculation, the redshift PDF p(z|zP) for each galaxy is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Due to the fact that the PDFs from photo-z algorithm can be biased due to the color-redshift degen- eracy in the photometric surveys, calibration is needed (Hilde- brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2017, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, we can only statistically calibrate the overall redshift distribution n(z) but not the PDF p(z|zP) for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This means in order to calculate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7 we need to assume a photo-z PDF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We choose to use a bi-Gaussian model Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (2020a) p2G(z|zP) = (1 − fout)pmain(z|zP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' ∆1, σ1) + foutpoutlier(z|zP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' ∆2, σ2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='10) with a main Gaussian peak and a Gaussian outlier peak with different bias ∆i and scatter σi, and an outlier rate fout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We fit the bi-Gaussian model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='10), requiring it to have same mean redshift ⟨z⟩ with the SOM calibrated n(z) (Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 2021), and minimize the difference between the resulting model z-distribution � nP(zP)p(z|zP)dzP and the SOM n(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The best-fit will then be a good description of the photo-z quality and can be used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' The resulting signal drops are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 4 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We validate the bi-Gaussian photo-z model for SC with MICE2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We compare with the results that use the photo-z distribution and true-z distribution in the calculation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content='1 that the bi-Gaussian model can produce the IA-drop QIg measurement very consistent with the ones with true-z from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' However, we find the lensing- drop QGg from the photo-z model is slightly higher than the true values from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' This error will be propagated to the separated lensing signal wGg and the IA+magnification signal wIg + gmagwGκ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' (15) and (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Its impact in AIA is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' 10 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} +page_content=' Article number, page 15 of 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQf6DZi/content/2301.13437v1.pdf'} diff --git a/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf b/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..50fdbcdf24bff217b1ce4d42fc94eabd8fb88526 --- /dev/null +++ b/A9E4T4oBgHgl3EQfEwz_/content/2301.04881v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6cbc06bdaf6ea9aa78225d67b87e2c84728685d4eafe5e01f56752ab89906747 +size 216122 diff --git a/A9E4T4oBgHgl3EQfEwz_/vector_store/index.pkl b/A9E4T4oBgHgl3EQfEwz_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d91dad1c2240288f3847b7a961e81d043899cd13 --- /dev/null +++ b/A9E4T4oBgHgl3EQfEwz_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:37ec59e87fd253e1f7e078900ab615a6e4d9a817467b707c983ed1c67422bb75 +size 98306 diff --git a/AdE0T4oBgHgl3EQfPgCp/content/tmp_files/2301.02180v1.pdf.txt b/AdE0T4oBgHgl3EQfPgCp/content/tmp_files/2301.02180v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..58cb6f4a1a3b72f4770e724c40e9ccfc31c89673 --- /dev/null +++ b/AdE0T4oBgHgl3EQfPgCp/content/tmp_files/2301.02180v1.pdf.txt @@ -0,0 +1,1237 @@ +arXiv:2301.02180v1 [math.DS] 5 Jan 2023 +Existence of robust non-uniformly hyperbolic +endomorphism in homotopy classes +Victor Janeiro +victorgjaneiro@gmail.com, ICEx-UFMG, Belo Horizonte-MG, Brazil. +Abstract +We extend the results of [1] by showing that any homothety in 핋2 is homo- +topic to a non-uniformly hyperbolic ergodic area preserving map, provided that +its degree is at least 52. We also address other small topological degree cases not +considered in the previous article. This proves the existence of a 1 open set +of non-uniformly hyperbolic systems, that intersects essentially every homotopy +classes in 핋2, where the Lyapunov exponents vary continuously. +1 +Introduction +We study conservative maps of the two-torus 핋2 from the point of view of smooth +ergodic theory. We are interested in the Lyapunov exponents of these systems, in +particular, in extending the results obtained in [1] to the homothety case and some +cases with lower topological degree, which were not included in the previous results. +With this in mind, some familiarity with the results of [1] is desirable. +For a differentiable covering map 푓 ∶ 핋2 → 핋2 and a pair (푥, 푣) ∈ 푇핋2, the number +̃(푥, 푣) = lim sup +푛→∞ +log ‖퐷푥푓 푛(푣)‖ +푛 +is the Lyapunov exponent of 푓 at (푥, 푣). See [2] for background in Smooth Ergodic +Theory. Due to Oseledet’s Theorem [3] , there is a full area set 푀0 on 핋2 where the +previous limit exists for every 푣, and there exists a measurable bundle 퐸− defined on +푀0 such that for 푥 ∈ 푀0, 푣 ≠ 0 ∈ 퐸−(푥): +(푥, 푣) ∶= lim +푛→∞ +log ‖퐷푥푓 푛(푣)‖ +푛 += lim +푛→∞ +log 푚(퐷푥푓 푛) +푛 +∶= −(푥), +while for 푣 ∈ ℝ2 ⧵ 퐸−(푥): +(푥, 푣) = lim +푛→∞ +log ‖퐷푥푓 푛‖ +푛 +∶= +(푥), +Moreover, if 휇 denotes the Lebesgue (Haar) measure on 핋2, then: +∫ (+(푥) + −(푥))푑휇(푥) = ∫ log | det 퐷푥푓 |푑휇(푥) > 0, +(1) +1 + +so +(푥) > 0 almost everywhere. At last, we say that 푓 is non-uniformly hyperbolic +(NUH) if −(푥) < 0 < +(푥) almost everywhere. +Non uniformly hyperbolic systems provide a generalization of the classical Anosov +surface maps [4]. Here, we will only be concerned with the non-invertible case in an +attempt to aid the understanding of their statistical properties, which is still under +development. For the general ergodic theory of endomorphisms, the reader is directed +to [5]. +Any map 푓 ∶ 핋2 → 핋2 is homotopic to a linear endomorphism 퐸 ∶ 핋2 → 핋2, +induced by an integer matrix that we denote by the same letter. In [1], it is established +the existence of a 1 open set of non-uniformly hyperbolic systems that intersects +every homotopy class that does not contain a homothety, provided that the degree is +not too small. The authors then conjecture that the same is true for homotheties. In +this article, we prove this conjecture, provided that the degree is at least 52. There are +other low topological degree cases not covered by Andersson, Carrasco and Saghin, +which we also address here. +Let End푟 +휇(핋2) be the set of 푟 local diffeomorphisms of 핋2 preserving the Lebesgue +measure 휇, that are not invertible. For 푓 ∈ End푟 +휇(핋2), (푥, 푣) ∈ 푇 1핋2 define: +퐼(푥, 푣; 푓 푛) = +∑ +푦∈푓 −푛(푥) +log ‖(퐷푦푓 푛)−1푣‖ +det(퐷푦푓 푛) +, +and +퐶(푓 ) = ∑ +푛∈ℕ +1 +푛 +inf +(푥,푣)∈푇 1핋2 퐼(푥, 푣; 푓 푛). +Define the set + ∶= {푓 ∈ End푟 +휇(핋2) ∶ 퐶(푓 ) > 0}, +which is open in the 1-topology. On Subsection 2.3 of the main reference [1], it is +proved: +Theorem 1. If 푓 ∈  , then 푓 is non-uniformly hyperbolic. +Our main results are: +Theorem A. For 퐸 = 푘 ⋅퐼푑 ∈ 푀2×2(ℤ), with |푘| ≥ 5, the intersection [퐸]∩ is non-empty +and in fact contains maps that are real analytically homotopic to E. +Theorem B. If 퐸 ∈ 푀2×2(ℤ) is not a homothety and 푑푒푡(퐸) > 4, the intersection [퐸] ∩  +is non-empty and in fact contains maps that are real analytically homotopic to E. +Our Theorem B is equivalent to the Theorem A of [1] but includes three cases +which are not proved there. The main difficulty for our results is that, in the case of +2 + +a homothety, the induced projective action is trivial; non-triviality of this projective +action is a central piece in the method of Andersson et al. +Finally, by inspection on the proofs of Theorems B and C of [1], we can see that it +works for all cases included here. Hence, defining: +퐶det(푓 ) ∶= sup +푛∈ℕ +1 +푛 inf +푥∈핋2 log(det(퐷푥푓 푛)) > 0, +and the open set: +1 ∶= +{ +푓 ∈ End푟 +휇(핋2) ∶ 퐶(푓 ) > −1 +2퐶det(푓 ) +} +, +we have from Theorems A and B that if a linear endomorphism 퐸 satisfies the condi- +tions of either of the Theorems, then [퐸] ∩ 1 ≠ ∅. Therefore, by Theorem 퐵 of [1], we +have conituity of the maps 1 ∋ 푓 ↦ ∫핋2 ±(푓 )푑휇 in the 1 topology. +From Theorem C of [1], we conclude that for any linear endomorphism E as in +Theorem A or B. If ±1 is not an eigenvalue of 퐸, then [퐸] ∩  contains stably ergodic +endomorphisms. In fact, it contains stably Bernoulli endomorphisms and, in particular, +maps that are mixing of all orders. +Acknowledgements +The results presented here were conjectured by Martin Andersson, Pablo D. Carrasco +and Radu Saghin in [1], I thank Pablo D. Carrasco, who is also my MSc advisor, for the +suggestion of the problem and for the hours of conversations on the subject that were +crucial to this article. +This work has been supported by the Brazillian research agencies CAPES and +CNPq. +2 +Preliminary +In order to prove Theorems A and B, we require a result on the computation of the +numbers 퐼(푥, 푣; 푓 푛) which the proof can be found in [1]: +Proposition 2.1. For any 푛 ∈ ℕ, it holds: +퐼(푥, 푣; 푓 푛) = +푛−1 +∑ +푖=0 +∑ +푦∈푓 −푖(푥) +퐼(푦, 퐹 −푖 +푦 푣; 푓 ) +det(퐷푦푓 푖) , +(2) +where 퐹 −푖 +푦 푣 = +(퐷푦푓 푖)−1푣 +‖(퐷푦푓 푖)−푖푣‖. +3 + +2.1 +Shears +For fixed points 푧1, 푧2, 푧3, 푧4 ∈ 핋1, in this order, take the closed intervals 퐼1 = [푧1, 푧2], +퐼3 = [푧3, 푧4], and the open intervals 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1). +Definition 2.1. We define the horizontal and vertical critical regions in 핋2 as ℎ = (퐼1 ∪ +퐼3) × 핋1, 푣 = 핋1 × (퐼1 ∪ 퐼3) and its complements ℎ = 핋2 ⧵ ℎ , 푣 = 핋2 ⧵ 푣 are respectively +the horizontal and vertical good region. +We then divide the good regions into + +ℎ = 퐼4 × 핋1, − +ℎ = 퐼2 × 핋1, + +푣 = 핋1 × 퐼4 and +− +푣 = 핋1 × 퐼2. +For fixed numbers 0 < 푎 < 푏, we take 푠 ∶ 핋1 → ℝ as an analytic map satisfying +the following conditions: +1. If 푧 ∈ 퐼4, then 푎 < 푠′(푧) < 푏; +2. If 푧 ∈ 퐼2, then −푏 < 푠′(푧) < −푎; +3. If 푧 ∈ 퐼1 ∪ 퐼3, then |푠′(푧)| < 푏. +Consider the two families of conservative diffeomorphisms of the torus given by: +ℎ푡(푥1, 푥2) = (푥1, 푥2 + 푡푠(푥1)), 푣푟(푥1, 푥2) = (푥1 + 푟푠(푥2), 푥2), +푡, 푟 ∈ ℝ. +Note that: +퐷(푥1,푥2)ℎ푡 = ( +1 +0 +푡푠′(푥1) +1) , +퐷(푥1,푥2)푣푟 = ( +1 +푟푠′(푥2) +0 +1 +) . +In order to simplify the computations we will consider the maximum norm on 푇핋2 +as ‖(푢1, 푢2)‖ = max{|푢1|, |푢2|}, and all the computations from now on are performed +using this norm. This way, we get, for every 푥 ∈ 핋2: +‖퐷푥ℎ푡‖ < 푏푡 + 1, and ‖퐷푥푣푟‖ < 푏푡 + 1. +Definition 2.2. Given 훼 > 0, the corresponding horizontal cone is Δℎ +훼 = {(푢1, 푢2) ∈ ℝ2 ∶ +|푢2| ≤ 훼|푢1|}, while the corresponding vertical cone is its complement Δ푣 +훼 = ℝ2 ⧵ Δℎ +훼, +Lemma 2.1. For 훼 > 1, let Δℎ +훼 and Δ푣 +훼 be the corresponding horizontal and vertical cones. +Then, for every 푡, 푟 > 2훼 +푎 , and, for every unit vector 푢 ∈ 푇푥핋2, the following holds: +1. If 푢 ∈ Δ푣 +훼, and: +(a) 푥 ∈ 푣, then +• (퐷푥푣푟)−1푢 ∈ Δℎ +훼 (퐷푥푣−1 +푟 Δ푣 +훼 ⊂ Δℎ +훼); +• ‖(퐷푥푣푟)−1푢‖ > 푎푟−훼 +훼 += 푟 +푎− 훼 +푟 +훼 ; +4 + +(b) 푥 ∈ 푣, then ‖(퐷푥푣푟)−1푢‖ > 1 +훼 . +2. If 푢 = ±(1, 푢2) ∈ Δℎ +훼, then: +(a) either for every 푥 ∈ + +푣 ( if 푢2 ≤ 0) or for every 푥 ∈ − +푣 (if 푢2 ≥ 0) it holds: +• (퐷푥푣푟)−1푢 ∈ Δℎ +훼; +• ‖(퐷푥푣푟)−1푢‖ > 1; +(b) for all other 푥, we have ‖(퐷푥푣푟)−1푢‖ > +1 +푏푟+1. +3. If 푢 ∈ Δℎ +훼, and: +(a) 푥 ∈ ℎ, then +• (퐷푥ℎ푡)−1푢 ∈ Δ푣 +훼 (퐷푥ℎ−1 +푡 Δℎ +훼 ⊂ Δ푣 +훼); +• ‖(퐷푥ℎ푡)−1푢‖ > 푎푡−훼 +훼 += 푡 +푎− 훼 +푡 +훼 ; +(b) 푥 ∈ ℎ, then ‖(퐷푥ℎ푡)−1푢‖ > 1 +훼 . +4. If 푢 = ±(푢1, 1) ∈ Δ푣 +훼, then: +(a) either for every 푥 ∈ + +ℎ ( if 푢1 ≤ 0) or for every 푥 ∈ − +ℎ (if 푢1 ≥ 0) it holds: +• (퐷푥ℎ푡)−1푢 ∈ Δ푣 +훼; +• ‖(퐷푥ℎ푡)−1푢‖ > 1; +(b) for all other 푥, we have ‖(퐷푥ℎ푡)−1푢‖ > +1 +푏푡+1. +Proof. We prove items 1 and 2, the case for ℎ푡 is analogous. Let 푥 = (푥1, 푥2) ∈ 푣, and +푢± = (1, ±훼) then: +(퐷푥푣푟)−1푢± = ( +1 +−푟푠′(푥2) +0 +1 +) ( +1 +±훼) = ( +1 ∓ 푟푠′(푥2)훼 +±훼 +) , +also since 푥 ∈ 푣, 푎 < |푠′(푥2)| < 푏, we also have 훼 > 1 and 푟 > 2훼 +푎 , hence: +|1 ∓ 푟푠′(푥2)훼| ≥ 푟훼푎 − 1 > 2훼2 − 1 > 훼 > 1, +which shows that (퐷푥푣푟)−1Δ푣 +훼 ⊂ Δℎ +훼. Also, ‖(퐷푥푣푟)−1푢‖ = |1 ∓ 푟푠′(푥2)훼| > 푟푎훼 − 1. Now, +noticing that the minimal expansion of vectors in Δ푣 +훼 occurs on either of (1, ±훼), we +have for every unit vector 푢 ∈ Δ푣 +훼: +‖(퐷푥푣푟)−1푢‖ ≥ ‖(퐷푥푣푟)−1(1, ±훼)‖ +‖(1, ±훼)‖ +> 푟훼 − 1 +훼 +. +For part 2 (a), we have for x ∈ + +푣 푠′(푥2) > 푎 > 0, and for 푥 ∈ − +푣, 푠′(푥2) < −푎 < 0, +thus, by simple calculations analogous to the last one, we get the results. Finally, for +(b) we just use 푚((퐷푥푣푟)−1) = +1 +‖퐷푥푣푟‖ > +1 +푏푟+1 for every 푥 ∈ 핋2. +5 + +3 +Endomorphisms and Shears: Proof of Theorem A +Fix 퐸 = 푘 ⋅ 퐼푑, for some 푘 ∈ ℕ (we shall make the entire argument on 푘 ∈ ℕ for the +sake of simplicity of notation, we emphasize that the entire argument works for 푘 ∈ ℤ +by replacing 푘 for |푘| when necessary). Fix a 훿 < +1 +4푘 and define the critical and good +regions as in Def. 2.1 for points 푧1, 푧2, 푧3, 푧4 ∈ 핋1 such that: +• 퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 2훿; +• The translation of 퐼1 by a multiple of 1 +푘 does not intersect 퐼3. +• 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1 +푘 [ +푘−1 +2 ], where [푝] +denotes the floor of 푝. +It is obtained directly from the definitions that: +Proposition 3.1. For every 푥 = (푥1, 푥2) ∈ 핋2, 퐸−1(푥) has 푘2 points given by: +퐸−1(푥1, 푥2) = +{ +( +푥1 + 푖 +푘 +, 푥2 + 푗 +푘 +) ∶ 푖, 푗 = 0, ⋯ , 푘 − 1 +} +. +At least 푘 [ +푘−1 +2 ] are inside each of + +푣, − +푣, + +ℎ and − +ℎ, and at most 푘 of them are inside +each of 푣, ℎ. +From now on, in this section, we fix any 훼 > 1 and the corresponding cones as in +Def. 2.2. We consider the analytic maps: +푓(푡,푟) = 퐸◦푣푟◦ℎ푡, +which we shall denote only by 푓 = 푓(푡,푟). Clearly 푓 is an area preserving endomorphism +isotopic to E. We observe that, given 푥 ∈ 핋2 and 푦 ∈ 푓 −1(푥), we have: +(퐷푦푓 )−1 = (퐷푦ℎ푡)−1(퐷ℎ푡(푦)푣푟)−1퐸−1. +The goal is for (퐷ℎ푡(푦)푣푟)−1 to take vectors in the vertical cone and expand them +in the horizontal direction and then (퐷푦ℎ푡)−1 takes its images and expands them in +the vertical direction, resulting in (퐷푦푓 )−1 expanding in the vertical direction for most +points in 푓 −1(푥). Thus, in order to keep track of this derivative, we must localize the +points 푦 ∈ 푓 −1(푥) in regard to which of ℎ or ℎ they belong, and {ℎ푡(푦) ∶ 푦 ∈ 푓 −1(푥)} = +(퐸◦푣푟)−1(푥) regarding which of 푣 or 푣 they belong. +Lemma 3.1. For every 푥 ∈ 핋2, we have: +1. (푣푟◦퐸)−1(푥) has 푘2 points of which at least 푘 [ +푘−1 +2 ] of them are in each one of + +푣 and +− +푣 and at most 푘 of them are in 푣; +6 + +2. 푓 −1(푥) has 푘2 points of which at least 푘 [ +푘−1 +2 ] of them are in each one of + +ℎ and − +ℎ +and at most 푘 of them are in ℎ. +Proof. +1. It is a direct consequence of Prop. 3.1 along with the fact that the regions ++ +푣, − +푣 and 푣 are invariant under 푣푟. +2. Notice that in each row of pre-images by E of a point 푥 = (푥1, 푥2) given by +{ +( +푥1+푖 +푘 , 푥2+푗0 +푘 ) ∶ 푖 = 0, ⋯ , 푘 − 1 +} +for a fixed 푗0 ∈ {0, ⋯ , 푘 − 1}, 푣−1 +푟 +is a rotation +by −푟푠 ( +푥2+푗0 +푘 ) in the circle 핋1 × +{ 푥2+푗0 +푘 +} +. Hence, at least [ +푘−1 +2 ] of the 푘 points of +this row are inside each one of + +ℎ and − +ℎ, and at most 1 is in ℎ. +As this is also true for all the 푘 rows of pre-images by E, we get at least 푘 [ +푘−1 +2 ] +pre-images by 퐸◦푣푟 are inside each one of + +ℎ and − +ℎ, and at most 푘 pre-images +by 퐸◦푣푟 are inside ℎ. Finally, since these sets are invariant under ℎ푡, we get the +desired result. +Remark 3.1. Even knowing which regions is a point 푦 ∈ (퐸◦푣푟)−1(푥), we cannot de- +termine the region which ℎ−1 +푡 (푦) is inside, as 푡 is varying. That is, there may be points +푦 ∈ 푓 −1(푥) that are in ℎ such that ℎ푡(푦) ∈ 푣 and vice-versa. +Definition 3.1. In order to keep track of the vectors, define: +• For 푢 = (푢1, 푢2) ∈ ℝ2 with 푢2 ≠ 0: +∗ (푢) = +{ +−sgn ( +푢1 +푢2) , if 푢1 ≠ 0, +−sgn(푢2), +if 푢1 = 0. +Notice that ∗ (푢) = ∗ (퐸−1푢), for every 푢 ∈ ℝ2. +• For 푥 ∈ 핋2, 푦 ∈ 푓 −1(푥) and 푢 ∈ ℝ2, let (푤1, 푤2) = (퐷ℎ푡(푦)푣푟)−1퐸−1푢: +∗푦 (푢) = +⎧⎪⎪ +⎨⎪⎪⎩ +−sgn ( +푤1 +푤2) , if 푤1, 푤2 ≠ 0, +−sgn(푤2), +if 푤2 ≠ 0, 푤1 = 0, +−sgn(푤1), +if 푤1 ≠ 0, 푤2 = 0. +In view of item 4 of Lemma 2.1, even though (퐷ℎ푡(푦)푣푟)−1 may not send a vector +푢 ∈ Δ푣 +훼 to the horizontal cone if ℎ푡(푦) ∈ 푣, we can still end up having expansion in the +vertical direction, depending on whether 푦 ∈  +∗푦(푢) +ℎ +or not. In this regard, from Lemma +3.1, there are 푘 points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) are in 푣, and these points (ℎ푡(푦)) are +all in the same circle 핋1 × +{ 푥2+푗0 +푘 +} +, hence the derivative (퐷ℎ푡(푦)푣푟)−1 is the same for those +points. We get: +7 + +Proposition 3.2. For every 푢 ∈ ℝ2, 푥 ∈ 핋2, then the sign ∗푦 (푢) = sg ( +푤1 +푤2) is the same for +all points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) ∈ 푣, where ∗푦 (푢) is as in Definition 3.1. +Definition 3.2. For a fixed 푥 ∈ 핋2 and: +• 푢 ∈ Δ푣 +훼, define: +⎧⎪⎪⎪⎪ +⎨⎪⎪⎪⎪⎩ +퐴 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ ℎ, ℎ푡(푦) ∈ 푣}. +퐵 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈  +∗푦(푢) +ℎ +, ℎ푡(푦) ∈ 푣}, +푣 = 퐴 ∪ 퐵, +ℎ = 푓 −1(푥) ⧵ 푣. +• 푢 ∈ Δℎ +훼, define: +⎧⎪⎪⎪⎪ +⎨⎪⎪⎪⎪⎩ +퐶 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ ℎ, ℎ푡(푦) ∈ ∗(푢) +푣 }. +퐷 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈  +∗푦(푢) +ℎ +, ℎ푡(푦) ∈ 푣 ∪ −∗(푢) +푣 +}, +푣 = 퐶 ∪ 퐷, +ℎ = 푓 −1(푥) ⧵ 푣. +A direct consequence of Lemma 3.1 and Prop. 3.2, having Remark. 3.1 in mind, is +the following: +Lemma 3.2. For a fixed (푥, 푢) ∈ 푇핋2, 푓 −1(푥) has 푘2 points, of which: +1. For 푢 ∈ Δ푣 +훼, at most 2푘 − 1 − [ +푘−1 +2 ] of them are in ℎ and at least (푘 − 1)2 + [ +푘−1 +2 ] +are inside 푣, because: +• At least (푘 − 1)2 are in A and, +• at least [ +푘−1 +2 ] are in B. +2. For 푢 ∈ Δℎ +훼, at most 푘2 − [ +푘−1 +2 ] (푘 + [ +푘−1 +2 ]) are in ℎ and at least [ +푘−1 +2 ] (푘 + [ +푘−1 +2 ]) +are in 푣, because: +• At least (푘 − 1) [ +푘−1 +2 ] are in C and, +• at least [ +푘−1 +2 ] (1 + [ +푘−1 +2 ]) are in D. +Knowing that for every unit vector 푢 ∈ ℝ2 we have ‖퐸−1푢‖ = 1 +푘 (maximum norm), +from Lemma 2.1 we get: +Lemma 3.3. For 푡, 푟 > 2훼 +푎 and for fixed 푥 ∈ 핋2, it holds: +1. If 푢 ∈ Δ푣 +훼, then for all 푦 ∈ 푣 we have (퐷푦푓 )−1푢 ∈ Δ푣 +훼; +2. If 푢 ∈ Δ푣 +훼 is a unit vector, then: +‖(퐷푦푓 )−1푢‖ > +⎧⎪⎪⎪ +⎨⎪⎪⎪⎩ +( +푎− 훼 +푡 +훼 ) ( +푎− 훼 +푟 +훼 ) +푡푟 +푘 , 푦 ∈ 퐴, +1 +훼푘, +푦 ∈ 퐵, +1 +(푏푡+1)훼푘, +푦 ∈ ℎ; +8 + +3. If 푢 ∈ Δℎ +훼, then for all 푦 ∈ 푣 we have (퐷푦푓 )−1푢 ∈ Δ푣 +훼; +4. If 푢 ∈ Δℎ +훼 is a unit vector, then: +‖(퐷푦푓 )−1푢‖ > +⎧⎪⎪⎪ +⎨⎪⎪⎪⎩ +( +푎− 훼 +푡 +훼 ) +푡 +푘, +푦 ∈ 퐶, +1 +(푏푟+1)푘, +푦 ∈ 퐷, +1 +(푏푡+1)(푏푟+1)푘, 푦 ∈ ℎ. +3.1 +Non-uniform hyperbolicity +For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ denote by +퐷푓 −푛(푥, 푢) = {(푦, 푤) ∈ 푇핋2 ∶ 푓 푛(푦) = 푥, 퐷푦푓 푛푤 = 푢}. +For any non-zero tangent vector (푥, 푢) and 푛 ≥ 0, define: +푛 = {(푧, 푤) ∈ 퐷푓 −푛(푥, 푢) ∶ 푤 ∈ Δ푣 +훼}, +푛 = 퐷푓 −푛(푥, 푢) ⧵ 푛, +푔푛 = #푛, +푏푛 = #푛 = 푘2푛 − 푔푛. +From Lemmas 3.2, 3.3 one deduces: +Lemma 3.4. Let (푥, 푢) ∈ 푇핋2. +1. If 푢 ∈ Δ푣 +훼, then at least (푘 − 1)2 + [ +푘−1 +2 ] of its pre-images under 퐷푓 are also in Δ푣 +훼; +2. If 푢 ∈ Δℎ +훼, then at least [ +푘−1 +2 ] (푘 + [ +푘−1 +2 ]) of its pre-images under 퐷푓 are in Δ푣 +훼. +By the lemma above, we get: +푔푛+1 ≥ ((푘 − 1)2 + [ +푘 − 1 +2 +]) 푔푛 + [ +푘 − 1 +2 +] (푘 + [ +푘 − 1 +2 +]) 푏푛 += ((푘 − 1)2 − [ +푘 − 1 +2 +] (푘 − 1 + [ +푘 − 1 +2 +])) 푔푛 + [ +푘 − 1 +2 +] (푘 + [ +푘 − 1 +2 +]) 푘2푛, +hence: +푔푛+1 +푘2(푛+1) ≥ 1 +푘2 ((푘 − 1)2 − [ +푘 − 1 +2 +] (푘 − 1 + [ +푘 − 1 +2 +])) +푔푛 +푘2푛 ++ 1 +푘2 [ +푘 − 1 +2 +] (푘 + [ +푘 − 1 +2 +]) . +9 + +Denoting by 푎푛 = 푔푛 +푘2푛 and +푐 = 1 +푘2 ((푘 − 1)2 − [ +푘 − 1 +2 +] (푘 − 1 + [ +푘 − 1 +2 +])) , +푒 = 1 +푘2 [ +푘 − 1 +2 +] (푘 + [ +푘 − 1 +2 +]) , +the inequality above becomes: +푎푛+1 ≥ 푐 ⋅ 푎푛 + 푒. +Lemma 3.5. For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0 it holds: +푎푛 ≥ +푒 +1 − 푐 (1 − 푐푛) += +[ +푘−1 +2 ] (푘 + [ +푘−1 +2 ]) +2푘 − 1 + [ +푘−1 +2 ] (푘 − 1 + [ +푘−1 +2 ]) +(1 − 푐푛) +In particular, +lim inf 푎푛 ≥ +[ +푘−1 +2 ] (푘 + [ +푘−1 +2 ]) +2푘 − 1 + [ +푘−1 +2 ] (푘 − 1 + [ +푘−1 +2 ]) +∶= 퐿(푘), +uniformly in (푥, 푢) ∈ 핋2. +From now on we shall denote by 퐿(푘) = +[ 푘−1 +2 ](푘+[ 푘−1 +2 ]) +2푘−1+[ 푘−1 +2 ](푘−1+[ 푘−1 +2 ]). As another direct con- +sequence of Lemmas 3.2 and 3.3 we have the following: +Lemma 3.6. If 푟, 푡 > 2훼 +푎 , then for all (푥, 푢) ∈ 푇핋2 we have: +1. If 푢 ∈ Δ푣 +훼, then: +퐼(푥, 푢; 푓) ≥(푘 − 1)2 +푘2 +log 푟 + ( +푘2 − 4푘 + 2 + [ +푘−1 +2 ] +푘2 +) log 푡 ++ log ( +1 +훼푘 ((푎 − 훼 +푡 ) (푎 − 훼 +푟 )) +(푘−1)2 +푘2 +(푏 + 1 +푡 ) +− 1 +푘2(2푘−1−[ 푘−1 +2 ]) +) . +2. If 푢 ∈ Δℎ +훼, then: +퐼(푥, 푢; 푓) ≥ − ( +푘2 − (푘 − 1) [ +푘−1 +2 ] +푘2 +) log 푟 − ( +푘2 − [ +푘−1 +2 ] (2푘 − 1 + [ +푘−1 +2 ]) +푘2 +) log 푡 ++ log ( +1 +푘 ( +1 +훼 (푎 − 훼 +푡 )) +푘−1 +푘2 [ 푘−1 +2 ]−1 +(푏 + 1 +푡 ) +1 +푘2[ 푘−1 +2 ](푘+[ 푘−1 +2 ])−1 +) . +10 + +Now, to calculate (푓 ), we use Prop. 2.1 to compute: +퐼(푥, 푢; 푓 푛) = +푛−1 +∑ +푖=0 +∑ +푦∈푓 −푖(푥) +퐼(푦, (퐷푦푓 푖)−1푢; 푓) +푘2푖 +∶= +푛−1 +∑ +푖=0 +퐽푖, +and, if 푡, 푟 > 2훼 +푎 , for each 푖 we obtain: +퐽푖 = 1 +푘2푖 +∑ +푦∈푓 −1(푥) +퐼(푦, (퐷푦푓 푖)−1푢; 푓 ) = 1 +푘2푖 +∑ +(푦,푤)∈푖 +퐼(푦, 푤; 푓) + 1 +푘2푖 +∑ +(푦,푤)∈푖 +퐼(푦, 푤; 푓) +≥ 푎푖푉(푡, 푟, 푘) + (1 − 푎푖)퐻(푡, 푟, 푘), +where V and H are the right side of the inequalities obtained in Lemma 3.6 for 푢 ∈ Δ푣 +훼 +and 푢 ∈ Δℎ +훼 respectively. It follows from Lemma 3.5, with 퐿(푘) as above and 푐푘 = [ +푘−1 +2 ], +to simplify the notation, that: +lim +푖→∞ 퐽푖 ≥ 퐿(푘)푉(푡, 푟, 푘) + (1 − 퐿(푘))퐻(푡, 푟, 푘) += 퐶(푡, 푟, 푘) + 1 +푘2 (퐿(푘) ((푘 − 1) (2푘 − 푐푘) + 1) − (푘2 − (푘 − 1)푐푘)) log 푟 + +1 +푘2 (퐿(푘) (2(푘 − 1)2 − 푐푘 (2(푘 − 1) + 푐푘)) − (푘2 − 푐푘 (2푘 − 1 + 푐푘))) log 푡 +, +where +퐶(푡, 푟, 푘) = 퐿(푘)퐶1(푡, 푟, 푘) + (1 − 퐿(푘))퐶2(푡, 푟, 푘), +with +퐶1(푡, 푟, 푘) = log ( +1 +훼푘 ((푎 − 훼 +푡 ) (푎 − 훼 +푟 )) +(푘−1)2 +푘2 +(푏 + 1 +푡 ) +− 1 +푘2(2푘−1−[ 푘−1 +2 ]) +) +퐶2(푡, 푟, 푘) = log ( +1 +푘 ( +1 +훼 (푎 − 훼 +푡 )) +푘−1 +푘2 [ 푘−1 +2 ]−1 +(푏 + 1 +푡 ) +1 +푘2[ 푘−1 +2 ](푘+[ 푘−1 +2 ])−1 +) , +as in Lemma 3.6. From this, we get that for any 푘, 퐶(푡, 푟, 푘) is growing as 푡 and 푟 grow, +then for 푡, 푟 > 2훼 +푎 , 퐶(푡, 푟, 푘) > 퐶 is uniformly bounded from below by some constant 퐶. +Now, in order to get lim +푖→∞ 퐽푖 > 0, we can either make 푡 or 푟 large, depending on +whether the constant (which depends on 푘) multiplying log 푡 or log 푟 is positive or +negative. However, for both of them, we only get positivity of the constant if 푘 ≥ 5. +Thus, for 푘 ≥ 5, since all the bounds above are uniform for all non-zero tangent +vectors (푥, 푢), we obtain that for 푡 (or 푟) sufficiently large, for all 푖 greater than some 푖0, +and for all nonzero tangent vectors (푥, 푢), 퐽푖(푥, 푢) > 푁 > 0 for some constant 푁. Hence, +there exists some 푛0 such that +1 +푛0 +퐼(푥, 푢; 푓 푛0) = 1 +푛0 +푛0−1 +∑ +푖=0 +퐽푖(푥, 푢) > 푁 +2 > 0, +11 + +for all nonzero tangent vectors (푥, 푢). Therefore, (푓 ) > 0 which by Theorem 1 con- +cludes the proof of Theorem A. +We finish this section by including some examples for a better visualization that +for a fixed 푘 ∈ ℕ, the bounds obtained in this section are quite simple. For that, we fix +푘 = 5, we get 퐿(5) = 2 +3, the limitations of our last calculations become: +lim +푖→∞ 퐽푖 ≥ 퐶(푡, 푟, 5) + 5 log 푟 + 5 log 푡, +with +퐶(푡, 푟, 5) = log ( +1 +5 +훼 +17 +25 +푎2/3 (푎 − 훼 +푡 ) +1 +5 +(푎 − 훼 +푟 ) +32 +75 +(푏 + 1 +푡 ) +− 18 +25 +) +Thus, taking the map 푠 ∶ 핋1 → ℝ as 푠(푢) = sin(2휋푢), 훿 = 1 +20, 푎 = 2휋 sin( 휋 +10), 푏 = 2휋, +and 훼 = 1.1, we get that for every 푡, 푟 ⪆ 2푎 +훼 ≈ 1.77 the number 퐶(푡, 푟, 5)+5 log 푟 +5 log 푡 +is positive. Thus, the maps 푓(푡, 푟) = 퐸◦푣푟◦ℎ푡 satisfy the results of Theorem A. +4 +Proof of Theorem B +For 푘 ⋅ 퐼푑 ≠ 퐸 ∈ 푀2×2(ℤ), let 휏1(퐸) be the greatest common divisor of the entries of +E, 휏2(퐸) = det(퐸)/휏1(퐸), so that 푑 = 휏1 ⋅ 휏2 coincides with the topological degree of the +induced endomorphism 퐸 ∶ 핋2 → 핋2. +We want to make a slight change in the argument used in [1] so that for every +푥 ∈ 핋2, 푓 −1(푥) has at most one point in the critical zone. This solves the cases where +the pair (휏1, 휏2) is (2, 4), (3, 3) or (4, 4). For the remaining four cases (1, 2), (1, 3), (1, 4) +and (2, 2), even with this improvement in the argument, the proportion we obtain for +vectors in the good region (which in these cases is the optimum one for the argument +presented here) is still insufficient to obtain expansion in the vertical direction, given +the small amount of pre-images. +The numbers 휏1, 휏2 are the elementary divisors of E and, as in Section 2.4 of [1], +there exists 푃 ∈ 퐺퐿2(ℤ) such that the matrix 퐺 = 푃−1 ⋅ 퐸 ⋅ 푃 satisfies: +퐺−1(ℤ) = +{ +( +푖 +휏2푗 +휏1) ∶ 푖, 푗 ∈ ℤ +} +Moreover, as E is not a homothety, by another change of coordinates if necessary +we may assume that E does not have (0, 1) as an eigenvector. +With this in mind, we assume that ℙ퐸 does not fix [(0, 1)] and that 퐸−1ℤ2 = 1 +휏2ℤ× 1 +휏1ℤ. +So there exists an 훼 > 휏2 > 1 such that if Δℎ +훼 and Δ푣 +훼 are the corresponding horizontal +and vertical cones as in Def. 2.2, then 퐸−1Δ푣훼 ⊂ 퐼푛푡(Δℎ +훼). From now on, we fix such +훼 > 휏2. +Let 퐿 < max +{ +1 +4휏2, 휏−1 +2 −훼−1 +2 +} +, choose points 푧1, 푧2, 푧2, 푧4 ∈ 핋1, in this order, such that: +12 + +• 퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 퐿; +• the translation of 퐼1 by a multiple of 1/휏2 does not intersect 퐼3; +• 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1 +휏2 [ +휏2−1 +2 ], +and define the critical and good regions ℎ, ℎ and ± +ℎ as in Def. 2.1. As an immediate +consequence of the definition we get: +Proposition 4.1. For every 푥 ∈ 핋2, 퐸−1(푥) has 푑 points of which at least 1 +휏2 [ +휏2−1 +2 ] are +inside each of + +ℎ and − +ℎ, and at most 휏1 of them are inside of ℎ. +In order to have at most one pre-image of each point in the critical zone of the shear +ℎ푡(푥1, 푥2) = (푥1, 푥2+푡푠(푥1) defined as before, we define the conservative diffeomorphism +of the torus 푣(푥1, 푥2) = (푥1 + ̃푠(푥2), 푥2), with ̃푠 ∶ 핋1 → ℝ an analytic map which we shall +impose restrictions later. We then study the family: +푓푡 = 퐸◦푣◦ℎ푡, +of area preserving endomorphism of the torus isotopic to E. We shall denote 푓 = 푓푡 to +simplify the notation. +Given 푥 ∈ 핋2, the set 푓 −1(푥) = ℎ−1 +푡 ◦푣−1◦퐸−1(푥) is composed by d points, and given +푦 ∈ 푓 −1(푥), we have (퐷푦푓 )−1 = (퐷푦ℎ푡)−1◦(퐷ℎ푡(푦)푣)−1◦퐸−1. +In order to define 푣 in a way that only one pre-image of 푥 by 푓 remains in the +critical zone, we notice that 퐸−1(푥) is composed by 푑 points which, by the change of +coordinates made initially, are aligned in a lattice of height 휏1 and length 휏2. We also +notice that the map ℎ−1 +푡 keeps the vertical lines invariant. Therefore, the map 푣−1 needs +to act in a way that it moves points on a vertical line enough so that only one remains +in the critical zone, and, also, it cannot move them so much that we have new points +entering the critical zone. +In this way, we took the analytic map ̃푠 ∶ 핋1 → ℝ satisfying: +1. If 퐿 is the size of the intervals 퐼1, 퐼3 then |̃푠(푢)| < 1 +휏2 − 퐿, for all 푢 ∈ 핋1. +2. For all 푢 ∈ 핋1, we have that |||̃푠 (푢 + 푗 +휏1)||| > 퐿 for all 푗 ∈ {0, 1, ⋯ , 휏1 − 1} except at +most one index. +3. |̃푠′(푢)| < (2훼)−1, for all 푢 ∈ 핋1, where 훼 is the size of the cones fixed in the previous +subsection. +Notice that conditions 2 and 3 are not mutually exclusives thanks to the conditions +for 훼 and 퐿 imposed in the previous subsection. Now, conditions 1 and 2 give us: +13 + +Lemma 4.1. For all 푥 ∈ 핋2, 푓 −1(푥) is composed by 푑 points of which at most one is inside +ℎ. At least 푑 − 1 of the pre-images are inside  of which at least 휏1 [ +휏2−1 +2 ] are inside each +of + +ℎ and − +ℎ. +Proof. In the case where 퐸−1(푥) has no points in the critical zone, due to condition 1 +together with the fact that ℎ푡 preserves vertical lines, the map ℎ−1 +푡 ◦푣−1 does not take +any of those points to the critical zone. +In the case where 퐸−1(푥) has a point in the critical zone, it implies that we have +exactly 휏1 points there. Due to condition 2, only one of those points is able to remain +there, and due to condition 1, none of the other points is getting inside. +For the minimum amount of points in each of + +ℎ and − +ℎ, we notice that, by Prop. +4.1, 퐸−1(푥) already has at least 휏1 [ +휏2−1 +2 ] points inside each one, and, due to condition 1, +those points must remain there. +At last, condition 3 gives us the next lemma, required for the whole construction +to work: +Lemma 4.2. There exists 훽 > 훼 such that for all 푦 ∈ 핋2, (퐷푦푣)−1◦퐸−1Δ푣 +훽 ⊂ Δℎ +훽, where Δ푣 +훽 +and Δℎ +훽 are the corresponding vertical and horizontal cones of size 훽 as in Def. 2.2. +Proof. For 푦 = (푦1, 푦2), 퐷푦푣 = ( +1 +̃푠′(푦2) +0 +1 +). Then, due to condition 3, for all 휆 ∈ ℝ, +퐷푦푣 ⋅ 휆푒2 = 휆(̃푠′(푦2), 1) ∈ Δ푣 +2훼. Since, by the definition of 훼, we have 퐸−1 ⋅ 휆푒2 ∈ 푖푛푡(Δℎ +훼), +we conclude that for all 푦 ∈ 핋2, ℙ((퐷푦푣)−1◦퐸−1)⋅[푒2] is uniformly away from [푒2], hence +there exists such 훽 as we wanted. +Remark 4.1. Items 3 and 4 of Lemma 2.1 also works in this cases for Δ푣 +훽 and Δℎ +훽. +We give the correspondent to Lemma 3.3 for this case, as a consequence of items +3 and 4 of Lemma 2.1, Remark 4.1 and Lemma 4.2 . From now on, we fix 훽 > 훼 as in +Lemma 4.2 and let: +푒푣 = inf +{ +‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δ푣 +훽 +} +, +푒ℎ = inf +{ +‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δℎ +훽 +} +. +Lemma 4.3. For 푡 > 2훽 +푎 it holds: +1. if 푦 ∈ ℎ then (퐷푦푓 )−1Δ푣 +훽 ⊂ Δ푣 +훽, it is strictly invariant. +2. if 푢 ∈ Δ푣 +훽 is a unit vector, then +‖(퐷푦푓 )−1푢‖ > +{ +푒푣(푎−훽/푡)) +훽 +푡, 푦 ∈ ℎ, +푒푣 +훽 , +푦 ∈ ℎ. +14 + +3. if 푢 ∈ Δℎ +훽, and (퐷ℎ푡(푦)푣)−1◦퐸−1 ⋅ 푢 = (푤1, 푤2) let ∗푦 (푢) be as in Def. 3.1. Then if +푦 ∈  +∗푦(푢) +ℎ +we have (퐷푦푓 )−1(푢) ∈ Δ푣 +훽. +4. if 푢 ∈ Δℎ +훽 is a unit vector, then +‖(퐷푦푓 )−1푢‖ > +{ +푒ℎ, +푦 ∈  +∗푦(푢) +ℎ +, +푒ℎ +푏+ 1 +푡 푡−1, 푦 ∉  +∗푦(푢) +ℎ +. +We notice that, analogously to the homothety case, we have the problem that ∗푦 (푢) +depends on 푦 ∈ 푓 −1(푥), therefore even though we have at least 휏1 [ +휏2−1 +2 ] points in each +of ± +ℎ, there could be a vector 푢 ∈ ℝ2 such that for all 푦 ∈ + +ℎ, ∗푦 (푢) = − and vice-versa. +However, we can see that this is not the case: +Proposition 4.2. For every 푥 ∈ 핋2, 푢 ∈ ℝ2, there are at least 휏2 [ +휏2−1 +2 ] points 푦 ∈ 푓 −1(푥) +such that 푦 ∈  +∗푦(푢) +ℎ +, where ∗푦 (푢) is as in Def. 3.1 changing 푣푟 for 푣. +Proof. By the same argument used in Prop. 3.2, we can see that ∗푦 (푢) is constant for +points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) lies in the same horizontal line. There are exactly +휏2 pre-images 푦′ such that ℎ푡(푦) and ℎ푡(푦′) are in the same horizontal line, hence at +least [ +휏2−1 +2 ] of these lies in  +∗푦(푢) +ℎ +. As 푣−1◦퐸−1(푥) has 휏1 different vertical lines, we get the +result. +4.1 +Non-uniform hyperbolicity +We end up having calculations completely mirrored in those made in Subsection 3.1, +and for that reason we will skip the details. For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ, +we define the sets 퐷푓 −푛(푥, 푢), 푛, 푛, and the numbers 푔푛, 푏푛 = 푑푛 − 푔푛 as before. From +Lemmas 4.1, 4.3 and Prop. 4.2 we deduce: +Lemma 4.4. Let (푥, 푢) ∈ 푇핋2. +1. If 푢 ∈ Δ푣 +훽, then at least 푑 − 1 of its pre-images under 퐷푓 are also in Δ푣 +훽. +2. If 푢 ∈ Δℎ +훽, then at least 휏1 [ +휏2−1 +2 ] of its pre-images under 퐷푓 are in Δℎ +훽. +For that, we get for all 푛 ∈ ℕ: +푔푛+1 ≥ (푑 − 1 − 휏1 [ +휏2 − 1 +2 +]) 푔푛 + 휏1 [ +휏2 − 1 +2 +] 푑푛, +hence, putting 푎푛 = 푔푛 +푑푛 : +푎푛+1 ≥ ( +푑 − 1 +푑 +− 1 +휏2 [ +휏2 − 1 +2 +]) 푎푛 + 1 +휏2 [ +휏2 − 1 +2 +] . +Thus, we get: +15 + +Lemma 4.5. For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0, it holds: +lim inf 푎푛 ≥ 1 +휏2 [ +휏2 − 1 +2 +] +푑 +1 + 휏1 [ +휏2−1 +2 ] +∶= 퐿(휏1, 휏2). +Remark 4.2. This is where we are able to verify that this argument will work for the cases +(휏1, 휏2) as (2, 4), (3, 3) and (4, 4), where we have 퐿(휏1, 휏2) as 2/3, 3/4 and 4/5, respectively. +And it won’t work for the other cases (1, 2), (1, 3), (1, 4) and (2, 2) where we will get 퐿(휏1, 휏2) +as 0, 1/2, 1/2 and 0, respectively. As we will see, for the rest of the argument to work, we +need this lower bound strictly greater than 1/2. +As another consequence of Lemmas 4.1, 4.3 and Prop. 4.2, we get: +Lemma 4.6. If 푡 > 2훽 +푎 , then for all (푥, 푢) ∈ 푇핋2, it holds: +1. If 푢 ∈ Δ푣 +훽, then: +퐼(푥, 푢; 푓) ≥ 푑 − 1 +푑 +log 푡 + log ( +푒푣 +훽 (푎 − 훽 +푡 ) +푑−1 +푑 +) . +2. If 푢 ∈ Δℎ +훽, then: +퐼(푥, 푢; 푓) ≥ − (1 − 1 +휏2 [ +휏2 − 1 +2 +]) log 푡 + log (푒ℎ (푏 + 1 +푡 ) +−(1− 1 +휏2[ +휏2−1 +2 ]) +) . +Again, by Prop. 2.1, we have: +퐼(푥, 푢; 푓 푛) = +푛−1 +∑ +푖=0 +∑ +푦∈푓 −푖(푥) +퐼(푦, (퐷푦푓 푖)−1푢; 푓) +푘2푖 +∶= +푛−1 +∑ +푖=0 +퐽푖, +we compute, for 푡 > 2훽 +푎 , for all 푖 ≥ 0: +퐽푖 = 1 +푑 +∑ +(푦,푤)∈푖 +퐼(푦, 푤; 푓) + 1 +푑 +∑ +(푦,푤)∈푖 +퐼(푦, 푤; 푓) +≥ 푎푖푉(푡, 휏1, 휏2) + (1 − 푎푖)퐻(푡, 휏1, 휏2), +where 푎푖 is as in Lemma 4.5, 푉 and 퐻 are the right side of the inequalities obtained in +Lemma 4.6 for 푢 ∈ Δ푣 +훽 and 푢 ∈ Δℎ +훽 respectively. It follows: +lim +푖→∞ 퐽푖 ≥ 퐿(휏1, 휏2)푉(푡, 휏1, 휏2) + (1 − 퐿(휏1, 휏2))퐻(푡, 휏1, 휏2) += (휏1 − 2 +휏2) [ +휏2−1 +2 ] − 1 +1 + 휏1 [ +휏2−1 +2 ] +log 푡 + 퐶(푡, 휏1, 휏2), +16 + +where: +퐶(푡, 휏1, 휏2) =퐿(휏1, 휏2) log ( +푒푣 +훽 (푎 − 훽 +푡 ) +푑−1 +푑 +) ++ (1 − 퐿(휏1, 휏2)) log (푒ℎ (푏 + 1 +푡 ) +−(1− 1 +휏2[ +휏2−1 +2 ]) +) > 퐶, +for all 푡 > 2훽 +푎 , that is, 퐶(푡, 휏1, 휏2) is uniformly bounded from below by some constant C. +Since 푑 = 휏1 ⋅ 휏2 > 4, the constant multiplying log 푡 is positive. Therefore, since all +the bounds above are uniform for all non-zero tangent vectors (푥, 푢), as in the homo- +thety case we obtain that for 푡 sufficiently large, for all 푛 greater than some 푛0, and for +all nonzero tangent vectors (푥, 푢): +1 +푛퐼(푥, 푢; 푓 푛) = 1 +푛 +푛−1 +∑ +푖=0 +퐽푖(푥, 푢) > 0, +hence, (푓 ) > 0 which by Theorem 1 concludes the proof of Theorem B. +References +[1] M. Andersson, P. D. Carrasco, and R. Saghin, “Non-uniformly hyperbolic endo- +morphisms,” 2022. +[2] L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory. Graduate Studies +in Mathematics, American Mathematical Society, 2013. +[3] V. I. Oseledets, “A multiplicative ergodic theorem. characteristic ljapunov, expo- +nents of dynamical systems,” Trudy Moskovskogo Matematicheskogo Obshchestva, +vol. 19, pp. 179–210, 1968. +[4] D. V. Anosov, “Geodesic flows on closed riemannian manifolds of negative curva- +ture,” Trudy Mat. Inst. Steklov, vol. 90, pp. 3–210, 1967. +[5] M. Qian, J.-S. Xie, and S. Zhu, Smooth Ergodic Theory for Endomorphisms, vol. 1978 +of Lecture Notes in Mathematics. 01 2009. +17 + diff --git a/AdE0T4oBgHgl3EQfPgCp/content/tmp_files/load_file.txt b/AdE0T4oBgHgl3EQfPgCp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65c89aa87c49c5673b4a3d2905a750a994efc6db --- /dev/null +++ b/AdE0T4oBgHgl3EQfPgCp/content/tmp_files/load_file.txt @@ -0,0 +1,429 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf,len=428 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='02180v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='DS] 5 Jan 2023 Existence of robust non-uniformly hyperbolic endomorphism in homotopy classes Victor Janeiro victorgjaneiro@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='com, ICEx-UFMG, Belo Horizonte-MG, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Abstract We extend the results of [1] by showing that any homothety in 핋2 is homo- topic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We also address other small topological degree cases not considered in the previous article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' This proves the existence of a \ue22f1 open set of non-uniformly hyperbolic systems, that intersects essentially every homotopy classes in 핋2, where the Lyapunov exponents vary continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 1 Introduction We study conservative maps of the two-torus 핋2 from the point of view of smooth ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We are interested in the Lyapunov exponents of these systems, in particular, in extending the results obtained in [1] to the homothety case and some cases with lower topological degree, which were not included in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' With this in mind, some familiarity with the results of [1] is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For a differentiable covering map 푓 ∶ 핋2 → 핋2 and a pair (푥, 푣) ∈ 푇핋2, the number ̃\ue244(푥, 푣) = lim sup 푛→∞ log ‖퐷푥푓 푛(푣)‖ 푛 is the Lyapunov exponent of 푓 at (푥, 푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' See [2] for background in Smooth Ergodic Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Due to Oseledet’s Theorem [3] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' there is a full area set 푀0 on 핋2 where the previous limit exists for every 푣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' and there exists a measurable bundle 퐸− defined on 푀0 such that for 푥 ∈ 푀0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푣 ≠ 0 ∈ 퐸−(푥): \ue244(푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푣) ∶= lim 푛→∞ log ‖퐷푥푓 푛(푣)‖ 푛 = lim 푛→∞ log 푚(퐷푥푓 푛) 푛 ∶= \ue244−(푥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' while for 푣 ∈ ℝ2 ⧵ 퐸−(푥): \ue244(푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푣) = lim 푛→∞ log ‖퐷푥푓 푛‖ 푛 ∶= \ue244+(푥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' if 휇 denotes the Lebesgue (Haar) measure on 핋2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' then: ∫ (\ue244+(푥) + \ue244−(푥))푑휇(푥) = ∫ log | det 퐷푥푓 |푑휇(푥) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' (1) 1 so \ue244+(푥) > 0 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' At last, we say that 푓 is non-uniformly hyperbolic (NUH) if \ue244−(푥) < 0 < \ue244+(푥) almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Non uniformly hyperbolic systems provide a generalization of the classical Anosov surface maps [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Here, we will only be concerned with the non-invertible case in an attempt to aid the understanding of their statistical properties, which is still under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For the general ergodic theory of endomorphisms, the reader is directed to [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Any map 푓 ∶ 핋2 → 핋2 is homotopic to a linear endomorphism 퐸 ∶ 핋2 → 핋2, induced by an integer matrix that we denote by the same letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In [1], it is established the existence of a \ue22f1 open set of non-uniformly hyperbolic systems that intersects every homotopy class that does not contain a homothety, provided that the degree is not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' The authors then conjecture that the same is true for homotheties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In this article, we prove this conjecture, provided that the degree is at least 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' There are other low topological degree cases not covered by Andersson, Carrasco and Saghin, which we also address here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Let End푟 휇(핋2) be the set of \ue22f푟 local diffeomorphisms of 핋2 preserving the Lebesgue measure 휇, that are not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푓 ∈ End푟 휇(핋2), (푥, 푣) ∈ 푇 1핋2 define: 퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛) = ∑ 푦∈푓 −푛(푥) log ‖(퐷푦푓 푛)−1푣‖ det(퐷푦푓 푛) , and 퐶\ue244(푓 ) = ∑ 푛∈ℕ 1 푛 inf (푥,푣)∈푇 1핋2 퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Define the set \ue241 ∶= {푓 ∈ End푟 휇(핋2) ∶ 퐶\ue244(푓 ) > 0}, which is open in the \ue22f1-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' On Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3 of the main reference [1], it is proved: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푓 ∈ \ue241 , then 푓 is non-uniformly hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Our main results are: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 퐸 = 푘 ⋅퐼푑 ∈ 푀2×2(ℤ), with |푘| ≥ 5, the intersection [퐸]∩\ue241 is non-empty and in fact contains maps that are real analytically homotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 퐸 ∈ 푀2×2(ℤ) is not a homothety and 푑푒푡(퐸) > 4, the intersection [퐸] ∩ \ue241 is non-empty and in fact contains maps that are real analytically homotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Our Theorem B is equivalent to the Theorem A of [1] but includes three cases which are not proved there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' The main difficulty for our results is that, in the case of 2 a homothety, the induced projective action is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' non-triviality of this projective action is a central piece in the method of Andersson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Finally, by inspection on the proofs of Theorems B and C of [1], we can see that it works for all cases included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Hence, defining: 퐶det(푓 ) ∶= sup 푛∈ℕ 1 푛 inf 푥∈핋2 log(det(퐷푥푓 푛)) > 0, and the open set: \ue2411 ∶= { 푓 ∈ End푟 휇(핋2) ∶ 퐶\ue244(푓 ) > −1 2퐶det(푓 ) } , we have from Theorems A and B that if a linear endomorphism 퐸 satisfies the condi- tions of either of the Theorems, then [퐸] ∩ \ue2411 ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Therefore, by Theorem 퐵 of [1], we have conituity of the maps \ue2411 ∋ 푓 ↦ ∫핋2 \ue244±(푓 )푑휇 in the \ue22f1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From Theorem C of [1], we conclude that for any linear endomorphism E as in Theorem A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If ±1 is not an eigenvalue of 퐸, then [퐸] ∩ \ue241 contains stably ergodic endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In fact, it contains stably Bernoulli endomorphisms and, in particular, maps that are mixing of all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Acknowledgements The results presented here were conjectured by Martin Andersson, Pablo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Carrasco and Radu Saghin in [1], I thank Pablo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Carrasco, who is also my MSc advisor, for the suggestion of the problem and for the hours of conversations on the subject that were crucial to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' This work has been supported by the Brazillian research agencies CAPES and CNPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2 Preliminary In order to prove Theorems A and B, we require a result on the computation of the numbers 퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛) which the proof can be found in [1]: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For any 푛 ∈ ℕ, it holds: 퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛) = 푛−1 ∑ 푖=0 ∑ 푦∈푓 −푖(푥) 퐼(푦, 퐹 −푖 푦 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 ) det(퐷푦푓 푖) , (2) where 퐹 −푖 푦 푣 = (퐷푦푓 푖)−1푣 ‖(퐷푦푓 푖)−푖푣‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 Shears For fixed points 푧1, 푧2, 푧3, 푧4 ∈ 핋1, in this order, take the closed intervals 퐼1 = [푧1, 푧2], 퐼3 = [푧3, 푧4], and the open intervals 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We define the horizontal and vertical critical regions in 핋2 as \ue22fℎ = (퐼1 ∪ 퐼3) × 핋1, \ue22f푣 = 핋1 × (퐼1 ∪ 퐼3) and its complements \ue233ℎ = 핋2 ⧵ \ue22fℎ , \ue233푣 = 핋2 ⧵ \ue22f푣 are respectively the horizontal and vertical good region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We then divide the good regions into \ue233+ ℎ = 퐼4 × 핋1, \ue233− ℎ = 퐼2 × 핋1, \ue233+ 푣 = 핋1 × 퐼4 and \ue233− 푣 = 핋1 × 퐼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For fixed numbers 0 < 푎 < 푏, we take 푠 ∶ 핋1 → ℝ as an analytic map satisfying the following conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푧 ∈ 퐼4, then 푎 < 푠′(푧) < 푏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푧 ∈ 퐼2, then −푏 < 푠′(푧) < −푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푧 ∈ 퐼1 ∪ 퐼3, then |푠′(푧)| < 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Consider the two families of conservative diffeomorphisms of the torus given by: ℎ푡(푥1, 푥2) = (푥1, 푥2 + 푡푠(푥1)), 푣푟(푥1, 푥2) = (푥1 + 푟푠(푥2), 푥2), 푡, 푟 ∈ ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Note that: 퐷(푥1,푥2)ℎ푡 = ( 1 0 푡푠′(푥1) 1) , 퐷(푥1,푥2)푣푟 = ( 1 푟푠′(푥2) 0 1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In order to simplify the computations we will consider the maximum norm on 푇핋2 as ‖(푢1, 푢2)‖ = max{|푢1|, |푢2|}, and all the computations from now on are performed using this norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' This way, we get, for every 푥 ∈ 핋2: ‖퐷푥ℎ푡‖ < 푏푡 + 1, and ‖퐷푥푣푟‖ < 푏푡 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Given 훼 > 0, the corresponding horizontal cone is Δℎ 훼 = {(푢1, 푢2) ∈ ℝ2 ∶ |푢2| ≤ 훼|푢1|}, while the corresponding vertical cone is its complement Δ푣 훼 = ℝ2 ⧵ Δℎ 훼, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 훼 > 1, let Δℎ 훼 and Δ푣 훼 be the corresponding horizontal and vertical cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Then, for every 푡, 푟 > 2훼 푎 , and, for every unit vector 푢 ∈ 푇푥핋2, the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훼, and: (a) 푥 ∈ \ue233푣, then (퐷푥푣푟)−1푢 ∈ Δℎ 훼 (퐷푥푣−1 푟 Δ푣 훼 ⊂ Δℎ 훼);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' ‖(퐷푥푣푟)−1푢‖ > 푎푟−훼 훼 = 푟 푎− 훼 푟 훼 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4 (b) 푥 ∈ \ue22f푣, then ‖(퐷푥푣푟)−1푢‖ > 1 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 = ±(1, 푢2) ∈ Δℎ 훼, then: (a) either for every 푥 ∈ \ue233+ 푣 ( if 푢2 ≤ 0) or for every 푥 ∈ \ue233− 푣 (if 푢2 ≥ 0) it holds: (퐷푥푣푟)−1푢 ∈ Δℎ 훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' ‖(퐷푥푣푟)−1푢‖ > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' (b) for all other 푥, we have ‖(퐷푥푣푟)−1푢‖ > 1 푏푟+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훼, and: (a) 푥 ∈ \ue233ℎ, then (퐷푥ℎ푡)−1푢 ∈ Δ푣 훼 (퐷푥ℎ−1 푡 Δℎ 훼 ⊂ Δ푣 훼);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' ‖(퐷푥ℎ푡)−1푢‖ > 푎푡−훼 훼 = 푡 푎− 훼 푡 훼 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' (b) 푥 ∈ \ue22fℎ, then ‖(퐷푥ℎ푡)−1푢‖ > 1 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 = ±(푢1, 1) ∈ Δ푣 훼, then: (a) either for every 푥 ∈ \ue233+ ℎ ( if 푢1 ≤ 0) or for every 푥 ∈ \ue233− ℎ (if 푢1 ≥ 0) it holds: (퐷푥ℎ푡)−1푢 ∈ Δ푣 훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' ‖(퐷푥ℎ푡)−1푢‖ > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' (b) for all other 푥, we have ‖(퐷푥ℎ푡)−1푢‖ > 1 푏푡+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We prove items 1 and 2, the case for ℎ푡 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Let 푥 = (푥1, 푥2) ∈ \ue233푣, and 푢± = (1, ±훼) then: (퐷푥푣푟)−1푢± = ( 1 −푟푠′(푥2) 0 1 ) ( 1 ±훼) = ( 1 ∓ 푟푠′(푥2)훼 ±훼 ) , also since 푥 ∈ \ue233푣, 푎 < |푠′(푥2)| < 푏, we also have 훼 > 1 and 푟 > 2훼 푎 , hence: |1 ∓ 푟푠′(푥2)훼| ≥ 푟훼푎 − 1 > 2훼2 − 1 > 훼 > 1, which shows that (퐷푥푣푟)−1Δ푣 훼 ⊂ Δℎ 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Also, ‖(퐷푥푣푟)−1푢‖ = |1 ∓ 푟푠′(푥2)훼| > 푟푎훼 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Now, noticing that the minimal expansion of vectors in Δ푣 훼 occurs on either of (1, ±훼), we have for every unit vector 푢 ∈ Δ푣 훼: ‖(퐷푥푣푟)−1푢‖ ≥ ‖(퐷푥푣푟)−1(1, ±훼)‖ ‖(1, ±훼)‖ > 푟훼 − 1 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For part 2 (a), we have for x ∈ \ue233+ 푣 푠′(푥2) > 푎 > 0, and for 푥 ∈ \ue233− 푣, 푠′(푥2) < −푎 < 0, thus, by simple calculations analogous to the last one, we get the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Finally, for (b) we just use 푚((퐷푥푣푟)−1) = 1 ‖퐷푥푣푟‖ > 1 푏푟+1 for every 푥 ∈ 핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 5 3 Endomorphisms and Shears: Proof of Theorem A Fix 퐸 = 푘 ⋅ 퐼푑, for some 푘 ∈ ℕ (we shall make the entire argument on 푘 ∈ ℕ for the sake of simplicity of notation, we emphasize that the entire argument works for 푘 ∈ ℤ by replacing 푘 for |푘| when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Fix a 훿 < 1 4푘 and define the critical and good regions as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 for points 푧1, 푧2, 푧3, 푧4 ∈ 핋1 such that: 퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 2훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' The translation of 퐼1 by a multiple of 1 푘 does not intersect 퐼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1 푘 [ 푘−1 2 ], where [푝] denotes the floor of 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' It is obtained directly from the definitions that: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every 푥 = (푥1, 푥2) ∈ 핋2, 퐸−1(푥) has 푘2 points given by: 퐸−1(푥1, 푥2) = { ( 푥1 + 푖 푘 , 푥2 + 푗 푘 ) ∶ 푖, 푗 = 0, ⋯ , 푘 − 1 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' At least 푘 [ 푘−1 2 ] are inside each of \ue233+ 푣, \ue233− 푣, \ue233+ ℎ and \ue233− ℎ, and at most 푘 of them are inside each of \ue22f푣, \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From now on, in this section, we fix any 훼 > 1 and the corresponding cones as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We consider the analytic maps: 푓(푡,푟) = 퐸◦푣푟◦ℎ푡, which we shall denote only by 푓 = 푓(푡,푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Clearly 푓 is an area preserving endomorphism isotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We observe that, given 푥 ∈ 핋2 and 푦 ∈ 푓 −1(푥), we have: (퐷푦푓 )−1 = (퐷푦ℎ푡)−1(퐷ℎ푡(푦)푣푟)−1퐸−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' The goal is for (퐷ℎ푡(푦)푣푟)−1 to take vectors in the vertical cone and expand them in the horizontal direction and then (퐷푦ℎ푡)−1 takes its images and expands them in the vertical direction, resulting in (퐷푦푓 )−1 expanding in the vertical direction for most points in 푓 −1(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Thus, in order to keep track of this derivative, we must localize the points 푦 ∈ 푓 −1(푥) in regard to which of \ue233ℎ or \ue22fℎ they belong, and {ℎ푡(푦) ∶ 푦 ∈ 푓 −1(푥)} = (퐸◦푣푟)−1(푥) regarding which of \ue233푣 or \ue22f푣 they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every 푥 ∈ 핋2, we have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' (푣푟◦퐸)−1(푥) has 푘2 points of which at least 푘 [ 푘−1 2 ] of them are in each one of \ue233+ 푣 and \ue233− 푣 and at most 푘 of them are in \ue22f푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 −1(푥) has 푘2 points of which at least 푘 [ 푘−1 2 ] of them are in each one of \ue233+ ℎ and \ue233− ℎ and at most 푘 of them are in \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' It is a direct consequence of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 along with the fact that the regions \ue233+ 푣, \ue233− 푣 and \ue22f푣 are invariant under 푣푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Notice that in each row of pre-images by E of a point 푥 = (푥1, 푥2) given by { ( 푥1+푖 푘 , 푥2+푗0 푘 ) ∶ 푖 = 0, ⋯ , 푘 − 1 } for a fixed 푗0 ∈ {0, ⋯ , 푘 − 1}, 푣−1 푟 is a rotation by −푟푠 ( 푥2+푗0 푘 ) in the circle 핋1 × { 푥2+푗0 푘 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Hence, at least [ 푘−1 2 ] of the 푘 points of this row are inside each one of \ue233+ ℎ and \ue233− ℎ, and at most 1 is in \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' As this is also true for all the 푘 rows of pre-images by E, we get at least 푘 [ 푘−1 2 ] pre-images by 퐸◦푣푟 are inside each one of \ue233+ ℎ and \ue233− ℎ, and at most 푘 pre-images by 퐸◦푣푟 are inside \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Finally, since these sets are invariant under ℎ푡, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Even knowing which regions is a point 푦 ∈ (퐸◦푣푟)−1(푥), we cannot de- termine the region which ℎ−1 푡 (푦) is inside, as 푡 is varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' That is, there may be points 푦 ∈ 푓 −1(푥) that are in \ue233ℎ such that ℎ푡(푦) ∈ \ue22f푣 and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In order to keep track of the vectors, define: For 푢 = (푢1, 푢2) ∈ ℝ2 with 푢2 ≠ 0: ∗ (푢) = { −sgn ( 푢1 푢2) , if 푢1 ≠ 0, −sgn(푢2), if 푢1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Notice that ∗ (푢) = ∗ (퐸−1푢), for every 푢 ∈ ℝ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푥 ∈ 핋2, 푦 ∈ 푓 −1(푥) and 푢 ∈ ℝ2, let (푤1, 푤2) = (퐷ℎ푡(푦)푣푟)−1퐸−1푢: ∗푦 (푢) = ⎧⎪⎪ ⎨⎪⎪⎩ −sgn ( 푤1 푤2) , if 푤1, 푤2 ≠ 0, −sgn(푤2), if 푤2 ≠ 0, 푤1 = 0, −sgn(푤1), if 푤1 ≠ 0, 푤2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In view of item 4 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, even though (퐷ℎ푡(푦)푣푟)−1 may not send a vector 푢 ∈ Δ푣 훼 to the horizontal cone if ℎ푡(푦) ∈ \ue22f푣, we can still end up having expansion in the vertical direction, depending on whether 푦 ∈ \ue233 ∗푦(푢) ℎ or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In this regard, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, there are 푘 points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) are in \ue22f푣, and these points (ℎ푡(푦)) are all in the same circle 핋1 × { 푥2+푗0 푘 } , hence the derivative (퐷ℎ푡(푦)푣푟)−1 is the same for those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We get: 7 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every 푢 ∈ ℝ2, 푥 ∈ 핋2, then the sign ∗푦 (푢) = sg ( 푤1 푤2) is the same for all points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) ∈ \ue22f푣, where ∗푦 (푢) is as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For a fixed 푥 ∈ 핋2 and: 푢 ∈ Δ푣 훼, define: ⎧⎪⎪⎪⎪ ⎨⎪⎪⎪⎪⎩ 퐴 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233ℎ, ℎ푡(푦) ∈ \ue233푣}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 퐵 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233 ∗푦(푢) ℎ , ℎ푡(푦) ∈ \ue22f푣}, \ue242푣 = 퐴 ∪ 퐵, \ue242ℎ = 푓 −1(푥) ⧵ \ue242푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푢 ∈ Δℎ 훼, define: ⎧⎪⎪⎪⎪ ⎨⎪⎪⎪⎪⎩ 퐶 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233ℎ, ℎ푡(푦) ∈ \ue233∗(푢) 푣 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 퐷 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233 ∗푦(푢) ℎ , ℎ푡(푦) ∈ \ue22f푣 ∪ \ue233−∗(푢) 푣 }, \ue234푣 = 퐶 ∪ 퐷, \ue234ℎ = 푓 −1(푥) ⧵ \ue234푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' A direct consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2, having Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 in mind, is the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For a fixed (푥, 푢) ∈ 푇핋2, 푓 −1(푥) has 푘2 points, of which: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푢 ∈ Δ푣 훼, at most 2푘 − 1 − [ 푘−1 2 ] of them are in \ue242ℎ and at least (푘 − 1)2 + [ 푘−1 2 ] are inside \ue242푣, because: At least (푘 − 1)2 are in A and, at least [ 푘−1 2 ] are in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푢 ∈ Δℎ 훼, at most 푘2 − [ 푘−1 2 ] (푘 + [ 푘−1 2 ]) are in \ue234ℎ and at least [ 푘−1 2 ] (푘 + [ 푘−1 2 ]) are in \ue234푣, because: At least (푘 − 1) [ 푘−1 2 ] are in C and, at least [ 푘−1 2 ] (1 + [ 푘−1 2 ]) are in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Knowing that for every unit vector 푢 ∈ ℝ2 we have ‖퐸−1푢‖ = 1 푘 (maximum norm), from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 we get: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푡, 푟 > 2훼 푎 and for fixed 푥 ∈ 핋2, it holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훼, then for all 푦 ∈ \ue242푣 we have (퐷푦푓 )−1푢 ∈ Δ푣 훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훼 is a unit vector, then: ‖(퐷푦푓 )−1푢‖ > ⎧⎪⎪⎪ ⎨⎪⎪⎪⎩ ( 푎− 훼 푡 훼 ) ( 푎− 훼 푟 훼 ) 푡푟 푘 , 푦 ∈ 퐴, 1 훼푘, 푦 ∈ 퐵, 1 (푏푡+1)훼푘, 푦 ∈ \ue242ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훼, then for all 푦 ∈ \ue234푣 we have (퐷푦푓 )−1푢 ∈ Δ푣 훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훼 is a unit vector, then: ‖(퐷푦푓 )−1푢‖ > ⎧⎪⎪⎪ ⎨⎪⎪⎪⎩ ( 푎− 훼 푡 훼 ) 푡 푘, 푦 ∈ 퐶, 1 (푏푟+1)푘, 푦 ∈ 퐷, 1 (푏푡+1)(푏푟+1)푘, 푦 ∈ \ue234ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 Non-uniform hyperbolicity For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ denote by 퐷푓 −푛(푥, 푢) = {(푦, 푤) ∈ 푇핋2 ∶ 푓 푛(푦) = 푥, 퐷푦푓 푛푤 = 푢}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For any non-zero tangent vector (푥, 푢) and 푛 ≥ 0, define: \ue233푛 = {(푧, 푤) ∈ 퐷푓 −푛(푥, 푢) ∶ 푤 ∈ Δ푣 훼}, \ue22e푛 = 퐷푓 −푛(푥, 푢) ⧵ \ue233푛, 푔푛 = #\ue233푛, 푏푛 = #\ue22e푛 = 푘2푛 − 푔푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3 one deduces: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Let (푥, 푢) ∈ 푇핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훼, then at least (푘 − 1)2 + [ 푘−1 2 ] of its pre-images under 퐷푓 are also in Δ푣 훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훼, then at least [ 푘−1 2 ] (푘 + [ 푘−1 2 ]) of its pre-images under 퐷푓 are in Δ푣 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' By the lemma above, we get: 푔푛+1 ≥ ((푘 − 1)2 + [ 푘 − 1 2 ]) 푔푛 + [ 푘 − 1 2 ] (푘 + [ 푘 − 1 2 ]) 푏푛 = ((푘 − 1)2 − [ 푘 − 1 2 ] (푘 − 1 + [ 푘 − 1 2 ])) 푔푛 + [ 푘 − 1 2 ] (푘 + [ 푘 − 1 2 ]) 푘2푛, hence: 푔푛+1 푘2(푛+1) ≥ 1 푘2 ((푘 − 1)2 − [ 푘 − 1 2 ] (푘 − 1 + [ 푘 − 1 2 ])) 푔푛 푘2푛 + 1 푘2 [ 푘 − 1 2 ] (푘 + [ 푘 − 1 2 ]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 9 Denoting by 푎푛 = 푔푛 푘2푛 and 푐 = 1 푘2 ((푘 − 1)2 − [ 푘 − 1 2 ] (푘 − 1 + [ 푘 − 1 2 ])) , 푒 = 1 푘2 [ 푘 − 1 2 ] (푘 + [ 푘 − 1 2 ]) , the inequality above becomes: 푎푛+1 ≥ 푐 ⋅ 푎푛 + 푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0 it holds: 푎푛 ≥ 푒 1 − 푐 (1 − 푐푛) = [ 푘−1 2 ] (푘 + [ 푘−1 2 ]) 2푘 − 1 + [ 푘−1 2 ] (푘 − 1 + [ 푘−1 2 ]) (1 − 푐푛) In particular, lim inf 푎푛 ≥ [ 푘−1 2 ] (푘 + [ 푘−1 2 ]) 2푘 − 1 + [ 푘−1 2 ] (푘 − 1 + [ 푘−1 2 ]) ∶= 퐿(푘), uniformly in (푥, 푢) ∈ 핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From now on we shall denote by 퐿(푘) = [ 푘−1 2 ](푘+[ 푘−1 2 ]) 2푘−1+[ 푘−1 2 ](푘−1+[ 푘−1 2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' As another direct con- sequence of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3 we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푟, 푡 > 2훼 푎 , then for all (푥, 푢) ∈ 푇핋2 we have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훼, then: 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) ≥(푘 − 1)2 푘2 log 푟 + ( 푘2 − 4푘 + 2 + [ 푘−1 2 ] 푘2 ) log 푡 + log ( 1 훼푘 ((푎 − 훼 푡 ) (푎 − 훼 푟 )) (푘−1)2 푘2 (푏 + 1 푡 ) − 1 푘2(2푘−1−[ 푘−1 2 ]) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훼, then: 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) ≥ − ( 푘2 − (푘 − 1) [ 푘−1 2 ] 푘2 ) log 푟 − ( 푘2 − [ 푘−1 2 ] (2푘 − 1 + [ 푘−1 2 ]) 푘2 ) log 푡 + log ( 1 푘 ( 1 훼 (푎 − 훼 푡 )) 푘−1 푘2 [ 푘−1 2 ]−1 (푏 + 1 푡 ) 1 푘2[ 푘−1 2 ](푘+[ 푘−1 2 ])−1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 10 Now, to calculate \ue22f\ue244(푓 ), we use Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 to compute: 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛) = 푛−1 ∑ 푖=0 ∑ 푦∈푓 −푖(푥) 퐼(푦, (퐷푦푓 푖)−1푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) 푘2푖 ∶= 푛−1 ∑ 푖=0 퐽푖, and, if 푡, 푟 > 2훼 푎 , for each 푖 we obtain: 퐽푖 = 1 푘2푖 ∑ 푦∈푓 −1(푥) 퐼(푦, (퐷푦푓 푖)−1푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 ) = 1 푘2푖 ∑ (푦,푤)∈\ue233푖 퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) + 1 푘2푖 ∑ (푦,푤)∈\ue22e푖 퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) ≥ 푎푖푉(푡, 푟, 푘) + (1 − 푎푖)퐻(푡, 푟, 푘), where V and H are the right side of the inequalities obtained in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='6 for 푢 ∈ Δ푣 훼 and 푢 ∈ Δℎ 훼 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' with 퐿(푘) as above and 푐푘 = [ 푘−1 2 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' to simplify the notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' that: lim 푖→∞ 퐽푖 ≥ 퐿(푘)푉(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) + (1 − 퐿(푘))퐻(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) = 퐶(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) + 1 푘2 (퐿(푘) ((푘 − 1) (2푘 − 푐푘) + 1) − (푘2 − (푘 − 1)푐푘)) log 푟 + 1 푘2 (퐿(푘) (2(푘 − 1)2 − 푐푘 (2(푘 − 1) + 푐푘)) − (푘2 − 푐푘 (2푘 − 1 + 푐푘))) log 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' where 퐶(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) = 퐿(푘)퐶1(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) + (1 − 퐿(푘))퐶2(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' with 퐶1(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) = log ( 1 훼푘 ((푎 − 훼 푡 ) (푎 − 훼 푟 )) (푘−1)2 푘2 (푏 + 1 푡 ) − 1 푘2(2푘−1−[ 푘−1 2 ]) ) 퐶2(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푘) = log ( 1 푘 ( 1 훼 (푎 − 훼 푡 )) 푘−1 푘2 [ 푘−1 2 ]−1 (푏 + 1 푡 ) 1 푘2[ 푘−1 2 ](푘+[ 푘−1 2 ])−1 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From this, we get that for any 푘, 퐶(푡, 푟, 푘) is growing as 푡 and 푟 grow, then for 푡, 푟 > 2훼 푎 , 퐶(푡, 푟, 푘) > 퐶 is uniformly bounded from below by some constant 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Now, in order to get lim 푖→∞ 퐽푖 > 0, we can either make 푡 or 푟 large, depending on whether the constant (which depends on 푘) multiplying log 푡 or log 푟 is positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' However, for both of them, we only get positivity of the constant if 푘 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Thus, for 푘 ≥ 5, since all the bounds above are uniform for all non-zero tangent vectors (푥, 푢), we obtain that for 푡 (or 푟) sufficiently large, for all 푖 greater than some 푖0, and for all nonzero tangent vectors (푥, 푢), 퐽푖(푥, 푢) > 푁 > 0 for some constant 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Hence, there exists some 푛0 such that 1 푛0 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛0) = 1 푛0 푛0−1 ∑ 푖=0 퐽푖(푥, 푢) > 푁 2 > 0, 11 for all nonzero tangent vectors (푥, 푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Therefore, \ue22f\ue244(푓 ) > 0 which by Theorem 1 con- cludes the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We finish this section by including some examples for a better visualization that for a fixed 푘 ∈ ℕ, the bounds obtained in this section are quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For that, we fix 푘 = 5, we get 퐿(5) = 2 3, the limitations of our last calculations become: lim 푖→∞ 퐽푖 ≥ 퐶(푡, 푟, 5) + 5 log 푟 + 5 log 푡, with 퐶(푡, 푟, 5) = log ( 1 5 훼 17 25 푎2/3 (푎 − 훼 푡 ) 1 5 (푎 − 훼 푟 ) 32 75 (푏 + 1 푡 ) − 18 25 ) Thus, taking the map 푠 ∶ 핋1 → ℝ as 푠(푢) = sin(2휋푢), 훿 = 1 20, 푎 = 2휋 sin( 휋 10), 푏 = 2휋, and 훼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, we get that for every 푡, 푟 ⪆ 2푎 훼 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='77 the number 퐶(푡, 푟, 5)+5 log 푟 +5 log 푡 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Thus, the maps 푓(푡, 푟) = 퐸◦푣푟◦ℎ푡 satisfy the results of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4 Proof of Theorem B For 푘 ⋅ 퐼푑 ≠ 퐸 ∈ 푀2×2(ℤ), let 휏1(퐸) be the greatest common divisor of the entries of E, 휏2(퐸) = det(퐸)/휏1(퐸), so that 푑 = 휏1 ⋅ 휏2 coincides with the topological degree of the induced endomorphism 퐸 ∶ 핋2 → 핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We want to make a slight change in the argument used in [1] so that for every 푥 ∈ 핋2, 푓 −1(푥) has at most one point in the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' This solves the cases where the pair (휏1, 휏2) is (2, 4), (3, 3) or (4, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For the remaining four cases (1, 2), (1, 3), (1, 4) and (2, 2), even with this improvement in the argument, the proportion we obtain for vectors in the good region (which in these cases is the optimum one for the argument presented here) is still insufficient to obtain expansion in the vertical direction, given the small amount of pre-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' The numbers 휏1, 휏2 are the elementary divisors of E and, as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='4 of [1], there exists 푃 ∈ 퐺퐿2(ℤ) such that the matrix 퐺 = 푃−1 ⋅ 퐸 ⋅ 푃 satisfies: 퐺−1(ℤ) = { ( 푖 휏2푗 휏1) ∶ 푖, 푗 ∈ ℤ } Moreover, as E is not a homothety, by another change of coordinates if necessary we may assume that E does not have (0, 1) as an eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' With this in mind, we assume that ℙ퐸 does not fix [(0, 1)] and that 퐸−1ℤ2 = 1 휏2ℤ× 1 휏1ℤ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' So there exists an 훼 > 휏2 > 1 such that if Δℎ 훼 and Δ푣 훼 are the corresponding horizontal and vertical cones as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2, then 퐸−1Δ푣훼 ⊂ 퐼푛푡(Δℎ 훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From now on, we fix such 훼 > 휏2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Let 퐿 < max { 1 4휏2, 휏−1 2 −훼−1 2 } , choose points 푧1, 푧2, 푧2, 푧4 ∈ 핋1, in this order, such that: 12 퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 퐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' the translation of 퐼1 by a multiple of 1/휏2 does not intersect 퐼3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1 휏2 [ 휏2−1 2 ], and define the critical and good regions \ue22fℎ, \ue233ℎ and \ue233± ℎ as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' As an immediate consequence of the definition we get: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every 푥 ∈ 핋2, 퐸−1(푥) has 푑 points of which at least 1 휏2 [ 휏2−1 2 ] are inside each of \ue233+ ℎ and \ue233− ℎ, and at most 휏1 of them are inside of \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In order to have at most one pre-image of each point in the critical zone of the shear ℎ푡(푥1, 푥2) = (푥1, 푥2+푡푠(푥1) defined as before, we define the conservative diffeomorphism of the torus 푣(푥1, 푥2) = (푥1 + ̃푠(푥2), 푥2), with ̃푠 ∶ 핋1 → ℝ an analytic map which we shall impose restrictions later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We then study the family: 푓푡 = 퐸◦푣◦ℎ푡, of area preserving endomorphism of the torus isotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We shall denote 푓 = 푓푡 to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Given 푥 ∈ 핋2, the set 푓 −1(푥) = ℎ−1 푡 ◦푣−1◦퐸−1(푥) is composed by d points, and given 푦 ∈ 푓 −1(푥), we have (퐷푦푓 )−1 = (퐷푦ℎ푡)−1◦(퐷ℎ푡(푦)푣)−1◦퐸−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In order to define 푣 in a way that only one pre-image of 푥 by 푓 remains in the critical zone, we notice that 퐸−1(푥) is composed by 푑 points which, by the change of coordinates made initially, are aligned in a lattice of height 휏1 and length 휏2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We also notice that the map ℎ−1 푡 keeps the vertical lines invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Therefore, the map 푣−1 needs to act in a way that it moves points on a vertical line enough so that only one remains in the critical zone, and, also, it cannot move them so much that we have new points entering the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In this way, we took the analytic map ̃푠 ∶ 핋1 → ℝ satisfying: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 퐿 is the size of the intervals 퐼1, 퐼3 then |̃푠(푢)| < 1 휏2 − 퐿, for all 푢 ∈ 핋1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For all 푢 ∈ 핋1, we have that |||̃푠 (푢 + 푗 휏1)||| > 퐿 for all 푗 ∈ {0, 1, ⋯ , 휏1 − 1} except at most one index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' |̃푠′(푢)| < (2훼)−1, for all 푢 ∈ 핋1, where 훼 is the size of the cones fixed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Notice that conditions 2 and 3 are not mutually exclusives thanks to the conditions for 훼 and 퐿 imposed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Now, conditions 1 and 2 give us: 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For all 푥 ∈ 핋2, 푓 −1(푥) is composed by 푑 points of which at most one is inside \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' At least 푑 − 1 of the pre-images are inside \ue233 of which at least 휏1 [ 휏2−1 2 ] are inside each of \ue233+ ℎ and \ue233− ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In the case where 퐸−1(푥) has no points in the critical zone, due to condition 1 together with the fact that ℎ푡 preserves vertical lines, the map ℎ−1 푡 ◦푣−1 does not take any of those points to the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' In the case where 퐸−1(푥) has a point in the critical zone, it implies that we have exactly 휏1 points there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Due to condition 2, only one of those points is able to remain there, and due to condition 1, none of the other points is getting inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For the minimum amount of points in each of \ue233+ ℎ and \ue233− ℎ, we notice that, by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, 퐸−1(푥) already has at least 휏1 [ 휏2−1 2 ] points inside each one, and, due to condition 1, those points must remain there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' At last, condition 3 gives us the next lemma, required for the whole construction to work: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' There exists 훽 > 훼 such that for all 푦 ∈ 핋2, (퐷푦푣)−1◦퐸−1Δ푣 훽 ⊂ Δℎ 훽, where Δ푣 훽 and Δℎ 훽 are the corresponding vertical and horizontal cones of size 훽 as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푦 = (푦1, 푦2), 퐷푦푣 = ( 1 ̃푠′(푦2) 0 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Then, due to condition 3, for all 휆 ∈ ℝ, 퐷푦푣 ⋅ 휆푒2 = 휆(̃푠′(푦2), 1) ∈ Δ푣 2훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Since, by the definition of 훼, we have 퐸−1 ⋅ 휆푒2 ∈ 푖푛푡(Δℎ 훼), we conclude that for all 푦 ∈ 핋2, ℙ((퐷푦푣)−1◦퐸−1)⋅[푒2] is uniformly away from [푒2], hence there exists such 훽 as we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Items 3 and 4 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 also works in this cases for Δ푣 훽 and Δℎ 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We give the correspondent to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3 for this case, as a consequence of items 3 and 4 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From now on, we fix 훽 > 훼 as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2 and let: 푒푣 = inf { ‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δ푣 훽 } , 푒ℎ = inf { ‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δℎ 훽 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For 푡 > 2훽 푎 it holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' if 푦 ∈ \ue233ℎ then (퐷푦푓 )−1Δ푣 훽 ⊂ Δ푣 훽, it is strictly invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' if 푢 ∈ Δ푣 훽 is a unit vector, then ‖(퐷푦푓 )−1푢‖ > { 푒푣(푎−훽/푡)) 훽 푡, 푦 ∈ \ue233ℎ, 푒푣 훽 , 푦 ∈ \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' if 푢 ∈ Δℎ 훽, and (퐷ℎ푡(푦)푣)−1◦퐸−1 ⋅ 푢 = (푤1, 푤2) let ∗푦 (푢) be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Then if 푦 ∈ \ue233 ∗푦(푢) ℎ we have (퐷푦푓 )−1(푢) ∈ Δ푣 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' if 푢 ∈ Δℎ 훽 is a unit vector, then ‖(퐷푦푓 )−1푢‖ > { 푒ℎ, 푦 ∈ \ue233 ∗푦(푢) ℎ , 푒ℎ 푏+ 1 푡 푡−1, 푦 ∉ \ue233 ∗푦(푢) ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' We notice that, analogously to the homothety case, we have the problem that ∗푦 (푢) depends on 푦 ∈ 푓 −1(푥), therefore even though we have at least 휏1 [ 휏2−1 2 ] points in each of \ue233± ℎ, there could be a vector 푢 ∈ ℝ2 such that for all 푦 ∈ \ue233+ ℎ, ∗푦 (푢) = − and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' However, we can see that this is not the case: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every 푥 ∈ 핋2, 푢 ∈ ℝ2, there are at least 휏2 [ 휏2−1 2 ] points 푦 ∈ 푓 −1(푥) such that 푦 ∈ \ue233 ∗푦(푢) ℎ , where ∗푦 (푢) is as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 changing 푣푟 for 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' By the same argument used in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2, we can see that ∗푦 (푢) is constant for points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) lies in the same horizontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' There are exactly 휏2 pre-images 푦′ such that ℎ푡(푦) and ℎ푡(푦′) are in the same horizontal line, hence at least [ 휏2−1 2 ] of these lies in \ue233 ∗푦(푢) ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' As 푣−1◦퐸−1(푥) has 휏1 different vertical lines, we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1 Non-uniform hyperbolicity We end up having calculations completely mirrored in those made in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, and for that reason we will skip the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ, we define the sets 퐷푓 −푛(푥, 푢), \ue233푛, \ue22e푛, and the numbers 푔푛, 푏푛 = 푑푛 − 푔푛 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' From Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2 we deduce: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Let (푥, 푢) ∈ 푇핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훽, then at least 푑 − 1 of its pre-images under 퐷푓 are also in Δ푣 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훽, then at least 휏1 [ 휏2−1 2 ] of its pre-images under 퐷푓 are in Δℎ 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For that, we get for all 푛 ∈ ℕ: 푔푛+1 ≥ (푑 − 1 − 휏1 [ 휏2 − 1 2 ]) 푔푛 + 휏1 [ 휏2 − 1 2 ] 푑푛, hence, putting 푎푛 = 푔푛 푑푛 : 푎푛+1 ≥ ( 푑 − 1 푑 − 1 휏2 [ 휏2 − 1 2 ]) 푎푛 + 1 휏2 [ 휏2 − 1 2 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Thus, we get: 15 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0, it holds: lim inf 푎푛 ≥ 1 휏2 [ 휏2 − 1 2 ] 푑 1 + 휏1 [ 휏2−1 2 ] ∶= 퐿(휏1, 휏2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' This is where we are able to verify that this argument will work for the cases (휏1, 휏2) as (2, 4), (3, 3) and (4, 4), where we have 퐿(휏1, 휏2) as 2/3, 3/4 and 4/5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' And it won’t work for the other cases (1, 2), (1, 3), (1, 4) and (2, 2) where we will get 퐿(휏1, 휏2) as 0, 1/2, 1/2 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' As we will see, for the rest of the argument to work, we need this lower bound strictly greater than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' As another consequence of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='3 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='2, we get: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푡 > 2훽 푎 , then for all (푥, 푢) ∈ 푇핋2, it holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δ푣 훽, then: 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) ≥ 푑 − 1 푑 log 푡 + log ( 푒푣 훽 (푎 − 훽 푡 ) 푑−1 푑 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' If 푢 ∈ Δℎ 훽, then: 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) ≥ − (1 − 1 휏2 [ 휏2 − 1 2 ]) log 푡 + log (푒ℎ (푏 + 1 푡 ) −(1− 1 휏2[ 휏2−1 2 ]) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Again, by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='1, we have: 퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛) = 푛−1 ∑ 푖=0 ∑ 푦∈푓 −푖(푥) 퐼(푦, (퐷푦푓 푖)−1푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) 푘2푖 ∶= 푛−1 ∑ 푖=0 퐽푖, we compute, for 푡 > 2훽 푎 , for all 푖 ≥ 0: 퐽푖 = 1 푑 ∑ (푦,푤)∈\ue233푖 퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) + 1 푑 ∑ (푦,푤)∈\ue22e푖 퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓) ≥ 푎푖푉(푡, 휏1, 휏2) + (1 − 푎푖)퐻(푡, 휏1, 휏2), where 푎푖 is as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='5, 푉 and 퐻 are the right side of the inequalities obtained in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content='6 for 푢 ∈ Δ푣 훽 and 푢 ∈ Δℎ 훽 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' It follows: lim 푖→∞ 퐽푖 ≥ 퐿(휏1, 휏2)푉(푡, 휏1, 휏2) + (1 − 퐿(휏1, 휏2))퐻(푡, 휏1, 휏2) = (휏1 − 2 휏2) [ 휏2−1 2 ] − 1 1 + 휏1 [ 휏2−1 2 ] log 푡 + 퐶(푡, 휏1, 휏2), 16 where: 퐶(푡, 휏1, 휏2) =퐿(휏1, 휏2) log ( 푒푣 훽 (푎 − 훽 푡 ) 푑−1 푑 ) + (1 − 퐿(휏1, 휏2)) log (푒ℎ (푏 + 1 푡 ) −(1− 1 휏2[ 휏2−1 2 ]) ) > 퐶, for all 푡 > 2훽 푎 , that is, 퐶(푡, 휏1, 휏2) is uniformly bounded from below by some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Since 푑 = 휏1 ⋅ 휏2 > 4, the constant multiplying log 푡 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Therefore, since all the bounds above are uniform for all non-zero tangent vectors (푥, 푢), as in the homo- thety case we obtain that for 푡 sufficiently large, for all 푛 greater than some 푛0, and for all nonzero tangent vectors (푥, 푢): 1 푛퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 푓 푛) = 1 푛 푛−1 ∑ 푖=0 퐽푖(푥, 푢) > 0, hence, \ue22f\ue244(푓 ) > 0 which by Theorem 1 concludes the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Andersson, P.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' Zhu, Smooth Ergodic Theory for Endomorphisms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 1978 of Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 01 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'} diff --git a/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf b/AdFJT4oBgHgl3EQfrS3C/content/2301.11608v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f46bdcb9504e2f38b13a6750cbd60549c885df1f --- /dev/null +++ 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CdTe as a Tunnel Barrier at the α-Sn/InSb Interface +Malcolm J. A. Jardine,1, ∗ Derek Dardzinski,2, ∗ Maituo Yu,2 Amrita Purkayastha,1 +A.-H. Chen,3 Yu-Hao Chang,4 Aaron Engel,4 Vladimir N. Strocov,5 Mo¨ıra +Hocevar,3 Chris J. Palmstrøm,4, 6 Sergey M. Frolov,1 and Noa Marom2, 7, 8, † +1Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA +2Department of Materials Science and Engineering, +Carnegie Mellon University, Pittsburgh, PA 15213, USA +3Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France +4Materials Department, University of California-Santa Barbara, Santa Barbara, CA, USA +5Paul Scherrer Institut, Swiss Light Source, CH-5232 Villigen PSI, Switzerland +6Department of Electrical and Computer Engineering, +University of California-Santa Barbara, Santa Barbara, CA, USA +7Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA +8Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA 15213, USA +Majorana zero modes, with prospective applications in topological quantum computing, are ex- +pected to arise in superconductor/semiconductor interfaces, such as β-Sn and InSb. However, prox- +imity to the superconductor may also adversely affect the semiconductor’s local properties. A tunnel +barrier inserted at the interface could resolve this issue. We assess the wide band gap semiconduc- +tor, CdTe, as a candidate material to mediate the coupling at the lattice-matched interface between +α-Sn and InSb. To this end, we use density functional theory (DFT) with Hubbard U corrections, +whose values are machine-learned via Bayesian optimization (BO) [npj Computational Materials 6, +180 (2020)]. The results of DFT+U(BO) are validated against angle resolved photoemission spec- +troscopy (ARPES) experiments for α-Sn and CdTe. For CdTe, the z-unfolding method [Advanced +Quantum Technologies, 5, 2100033 (2022)] is used to resolve the contributions of different kz values +to the ARPES. We then study the band offsets and the penetration depth of metal-induced gap +states (MIGS) in bilayer interfaces of InSb/α-Sn, InSb/CdTe, and CdTe/α-Sn, as well as in tri-layer +interfaces of InSb/CdTe/α-Sn with increasing thickness of CdTe. We find that 16 atomic layers +(3.5 nm) of CdTe can serve as a tunnel barrier, effectively shielding the InSb from MIGS from the +α-Sn. This may guide the choice of dimensions of the CdTe barrier to mediate the coupling in +semiconductor-superconductor devices in future Majorana zero modes experiments. +I. +INTRODUCTION +A promising route toward the realization of fault- +tolerant quantum computing schemes is developing ma- +terials systems that can host topologically protected Ma- +jorana zero modes (MZMs) [1, 2]. +MZMs may ap- +pear in one-dimensional topological superconductors [3– +5], which can be effectively realized by proximity cou- +pling a conventional superconductor and a semiconduc- +tor nanowire that possesses strong spin-orbit coupling +(SOC). Adding in a magnetic field enables this system to +behave as an effective spinless p-wave topological super- +conductor, which allows for MZM states [6]. Recently, +there have been new developments in material choices +and experimental methods to identify MZMs in semicon- +ductor nanowire-superconductor systems [7], designed to +overcome challenges identified during the first wave of +experiments [8–10]. These include trying new combina- +tions of semiconductors and epitaxial superconductors, +e.g. Pb, Sn, Nb, to maximize the electron mobility and +utilize larger superconducting gaps and higher critical +magnetic fields [11–16]. Additionally, new proposed ar- +∗ These authors contributed equally to this work +† Corresponding author: nmarom@andrew.cmu.edu +chitectures include creating nanowire networks and in- +ducing the field via micromagnets [17, 18]. +One of the challenges presented by the superconduc- +tor/semiconductor nanowire construct, is that excessive +coupling between the superconducting metal and semi- +conductor may “metallize” the semiconductor, thus ren- +dering the topological phase out of reach. Theoretical +studies that treated the semiconducting and supercon- +ducting properties via the Poisson-Schr¨odinger equation, +have shown that excessive coupling between the mate- +rials may lead to the semiconductor’s requisite proper- +ties, such as the Lande´e g-factor and spin-orbit-coupling +(SOC), being renormalized to a value closer to the +metal’s. In addition, large unwanted band shifts may be +induced [12, 19–22]. Having a tunnel barrier could modu- +late the superconductor-semiconductor coupling strength +and thus the induced proximity effect, which is critical +for controlling experiments. It is currently unknown what +the required width range of a tunnel barrier is. Another +potential benefit of a CdTe layer is InSb surface passiva- +tion. +InSb and Sn are among the materials used to fabricate +devices for Majorana search [23]. InSb is the backbone of +such systems because it has the highest electron mobility, +strongest spin-orbit coupling (SOC) and a large Land´e +g-factor in the conduction band compared to other III- +V semiconductors. β−Sn has a bulk critical field of 30 +arXiv:2301.02879v1 [cond-mat.mtrl-sci] 7 Jan 2023 + +2 +mT and a superconducting critical temperature of 3.7 K, +higher than the 10 mT and 1 K, respectively, of Al. Re- +cently, β-Sn shells have been grown on InSb nanowires, +inducing a hard superconducting gap [12]. +The large +band gap semiconductor CdTe is a promising candidate +to serve as a tunnel barrier. Thanks to its relative in- +ertness, it may simultaneously act as a passivation layer +protecting the InSb from environmental effects and po- +tentially minimizing disorder [24, 25]. Advantageously, +CdTe is lattice matched to InSb [26]. +Sn has two al- +lotropes. The β form, with a BCT crystal structure, is +of direct relevance to MZM experiments thanks to its su- +perconducting properties. However, the semi-metallic α +form has a diamond structure, which is lattice matched +to InSb and CdTe, making it an ideal model system for +investigating, both theoretically and experimentally, the +electronic structure of Sn/InSb heterostructures. +Much experimental work, such as growth and ARPES +studies, has been undertaken on α-Sn. Previously, α-Sn +has been found to possess a topologically trivial band in- +version, with SOC inducing a second band inversion and +a topological surface state (TSS) [27, 28]. The effect of +strain on the topological properties of α-Sn has also been +studied [21, 29–38]. In-plane compressive strain has been +reported to make α-Sn a topological Dirac-semi-metal +and induce a second TSS to appear [27]. +Conversely, +tensile strain has been reported to induce a transition +to a topological insulator. CdTe [25] and α-Sn [12, 28] +have been epitaxially grown on InSb. Depositing Sn on +InSb often leads to growth of epitaxially matched α-Sn, +although β-Sn may appear under some conditions [39]. +In addition, α-Sn can transition to β-Sn if the Sn layer is +above a critical thickness or if heat is applied during fab- +rication processes [40, 41]. Studying the interface with +the lattice matched α-Sn may provide insight, which is +also pertinent to β-Sn as both could be present in hy- +brid systems. Therefore, these are promising materials +to investigate for future device construction. +MZM experiments rely on finely tuned proximity cou- +pling between a superconducting metal and a semicon- +ductor. By adding a tunnel barrier at the interface be- +tween the two materials and varying its width, one could +potentially mediate the proximity coupling strength to +achieve precise control over the interface transparency. +To the best of our knowledge, this idea has not yet been +tested in experiments and it is presently unknown which +material(s) would be the best choice for a barrier and +what would be the optimal thickness. +Simulations of +a tri-layer system with a tunnel barrier are therefore +needed to inform MZM experiments. Here, we use den- +sity functional theory (DFT) to study a tri-layer system, +in which InSb is separated from α-Sn by a CdTe tunnel +barrier. Despite recent progress towards treating super- +conductivity within the framework of DFT [42, 43] the +description of proximity-induced superconductivity at an +interface with a semiconductor is still outside the reach of +present-day methods. However, DFT can provide useful +information on properties, such as the band alignment at +the interface. Conduction band offsets are of particular +importance because the proximity effect in most experi- +ments on InSb primarily concerns the conduction band. +In addition, DFT can provide information on the pene- +tration depth of metal induced gap states (MIGS) into +the semiconductor [20, 25, 44, 45], which is important +for determining the appropriate thickness of the tunnel +barrier. +Within DFT, computationally efficient (semi-)local +exchange-correlation functionals severely underestimate +the band gap of semiconductors to the extent that +some narrow-gap semiconductors, such as InSb, are er- +roneously predicted to be metallic [46–49]. This is at- +tributed to the self-interaction error (SIE), a spurious +repulsion of an electron from its own charge density [50– +52]. Hybrid functionals, which include a fraction of exact +(Fock) exchange, mitigate the SIE and yield band gaps in +better agreement with experiment. However, their com- +putational cost is too high for simulations of large in- +terface systems, such as the α-Sn/CdTe/InSb tri-layer +system studied here. The DFT+U approach, whereby a +Hubbard U correction is added to certain atomic orbitals, +provides a good balance between accuracy and computa- +tional cost[46, 53, 54]. Recently, some of us have pro- +posed a method of machine learning the U parameter for +a given material by Bayesian optimization (BO) [55]. The +DFT+U(BO) method has been employed successfully for +InSb and CdTe [56]. +It has been shown that (semi-)local functionals fail +to describe the bulk band structure of α-Sn correctly, +specifically the band ordering and the orbital compo- +sition of the valence bands at the Γ point. +DFT+U, +hybrid functionals, or many-body perturbation theory +within the GW approximation are necessary to obtain a +correct description of the band structure [29, 35, 57–59]. +DFT+U simulations have required slab models of more +than 30 monolayers of Sn to converge towards a bulk +regime, where quantum confinement is no longer domi- +nant. With a small number of layers α-Sn may exhibit +topological properties [26, 60, 61]. +Some DFT studies +have considered slab models of bi-axially strained α-Sn. +DFT simulations of strained α-Sn on InSb have been con- +ducted with a small number of layers of both materials +[26, 62]. The DFT+U approach has reproduced the ef- +fects of strain and compared well with experimental data +[28, 60, 62]. +Here, +we perform first principles calculations us- +ing DFT+U(BO) for a (110) tri-layer semiconduc- +tor/tunnel barrier/metal interface composed of the ma- +terials InSb/CdTe/α-Sn, owing to their relevance to cur- +rent Majorana search experiments [12, 25]. To date, DFT +studies of large interface slab models with a vacuum re- +gion have not been conducted for these interfaces. Pre- +viously, the results of DFT+U(BO) for InSb(110) have +been shown to be in good agreement with angle-resolved +photoemission spectroscopy (ARPES) experiments [63]. +Here, we also compare the results of DFT+U(BO) to +ARPES for α-Sn (Section III A) and CdTe (Section + +3 +III B). Excellent agreement with experiment is obtained. +In particular, for CdTe the z-unfolding scheme (Section +II A) helps resolve the contributions of different kz values +and modelling the 2 × 2 surface reconstruction repro- +duces the spectral signatures of surface states. We then +proceed to study the bi-layer interfaces of InSb/CdTe, +CdTe/α-Sn, and InSb/α-Sn (Section III C). Finally, to +assess the effectiveness of the tunnel barrier, we study +tri-layer interfaces with 2 to 16 monolayers (0.5 nm to 3.5 +nm) of CdTe inserted between the InSb substrate and the +α-Sn (Section III D). This thickness is within the thick- +ness range of CdTe shells grown on InSb nanowires. For +all interfaces, our simulations provide information on the +band alignment and the presence of MIGS. We find that +16 layers of CdTe (about 3.5 nm) form an effective tunnel +barrier, insulating the InSb from the α-Sn. However, this +may be detrimental for transport at the interface. Based +on this, we estimate that the relevant thickness regime +for tuning the coupling between InSb and Sn may be in +the range of 6-10 layers of CdTe. +II. +METHODS +A. +Z-Unfolding +Simulations of large supercell models produce complex +band structures with a large number of bands, as shown +in Figure 1a,b for a CdTe(111) slab with 25 atomic layers, +whose band structure was calculated using PBE+U(BO), +as described in Section II B. Band structure unfolding +is a method of projecting the band structure of a su- +percell model onto the appropriate smaller cell ([63–68]. +This can help resolve the contributions of states emerg- +ing from of e.g., defects and surface reconstructions vs. +the bulk bands of the material. In addition, it can fa- +cilitate the comparison to angle-resolved photoemission +spectroscopy (ARPES) experiments. The “bulk band un- +folding” scheme [63] projects the supercell band struc- +ture onto the primitive unit cell, illustrated in Figure +1c. The resulting band structure, shown in Figure 1d, +appears bulk-like. Bulk-unfolded band structures have +been shown to compare well with ARPES experiments +using high photon energies, which are not surface sensi- +tive owing to the large penetration depth. +The “z-unfolding” scheme [63] projects the band struc- +ture of a slab model with a finite thickness onto the Bril- +louin zone (BZ) of a single layer of the slab supercell +with the same orientation, illustrated in Figure 1e. The +resulting band structure, shown in Figure 1f, contains ex- +tra bands that are not present in the bulk-unfolded band +structure. The extra bands originate from different kz +values in the 3D primitive Brillouin zone projecting onto +the surface Brillouin zone (SBZ), creating overlapping +paths. For example, panel Figure 1g shows cross sections +through the BZ at values of kz = 0 and kz = 0.5. The +bulk-paths of Γ − L, Γ − K and Γ − X all overlap with +the surface k-path Γ − M, possibly with contributions +from additional paths, such as X − U. The plane cuts at +different kz values are derived from the tessellated bulk +BZ structure, shown in Figure 1h. When z-unfolding is +performed, the value of kz may be treated as a free pa- +rameter. The dependence on kz manifests as a smooth +change in the spectral function over the possible range +of kz which varies the mixture of different constituent +bulk-paths that overlap the SBZ-path, as shown in Fig- +ure 1i for Γ − M. The BZ for z-unfolding is a surface BZ +with a finite thickness, shown in red in Figure 1j. The +simulation cell for the DFT calculations is set up to be +the corresponding real-space unit cell. The z-unfolded k- +paths are parallel to the (111) surface at a constant value +of kz. +In ARPES experiments, the relation of the experimen- +tal spectra to kz may be less straightforward. First, the +dependence of the inelastic mean free path of the elec- +trons on their kinetic energy is given by the universal +curve [69, 70]. +Using photon energies that correspond +to a small mean free path is advantageous for probing +surface states. +However, it can produce prominent kz +broadening due to the Heisenberg uncertainty principle +[71–75] that implies integration of the ARPES signal over +kz through the broadening interval. Second, deviations of +the photoemission final states from the free electron ap- +proximation can cause contributions from different values +of kz to appear in the ARPES spectra. The photoelec- +trons are often treated as free electrons, based on the as- +sumption that the photoelectron kinetic energy is much +larger than the modulations of the crystal potential. In +this case, kz for a given photoelectron kinetic energy, Ek, +and the in-plane momentum, K//, is one single value, +which is determined by: +kz = +√2m0 +ℏ +� +Ek − ℏ2 +2m0 +K2 +// − V0 +(1) +where m0 is the free-electron mass and V0 the inner po- +tential in the crystal. However, a considerable body of +evidence has accumulated that the final states even in +metals [76, 77] and to a greater extent in complex ma- +terials such as transition metal dichalcogenides [78, 79] +can significantly deviate from the free electron approx- +imation. +Such deviations can appear, first, as non- +parabolic dispersions of the final states and, second, as +their multiband composition. The latter means that for +given Ek and K// the final-state wavefunction Φf incor- +porates a few Bloch waves φkz with different kz values, +Φf = � +kz Akzφkz, which give comparable contributions +to the total photocurrent determined by the Akz ampli- +tudes [76]. A detailed theoretical description of the multi- +band final states, treated as the time-reversed low-energy +electron diffraction (LEED) states [73] within the wave- +function matching approach, as well as further examples +for various materials can be found in Refs. [78, 79] and +the references therein. An insightful analysis of the multi- +band final states extending into the soft-X-ray photon +energies can be found in Ref. [77]. A rigorous analysis of + +4 +FIG. 1. +(a) Side view of the CdTe(111) slab (b) Folded band structure of CdTe(111) 25 monolayer slab. (c) Primitive unit cell +of CdTe (d) bulk-unfolded band structure (e) unit cell of CdTe(111) slab used in z-unfolding. (f) Z-unfolded band structure +along k-path M − Γ − M for kz = 0.5, and (g) as a function of kz. (h) FCC bulk BZ (grey), (111) unit-cell BZ (red) and +(111) surface BZ (blue). (i) Intersecting planes slice through the bulk BZ for kz = 0 (green) and kz = 0.5 (red) with the SBZ +indicated. (j) tessellated bulk BZs showing (111) orientated intersecting planes for given kz values. +final state effects in ARPES is beyond the scope of this +work. Here, we will only mention that all these effects +trace back to hybridization of free-electron plane waves +through the higher Fourier components of the crystal po- +tential. In cases where significant kz broadening and/or +final states effects are present, z-unfolding, rather than +bulk unfolding, should be used in order to resolve the +contributions of different kz values to the measured spec- +trum. This is demonstrated for CdTe in Section III B, +where the final states appear to incorporate two Bloch +waves with kz = 0 and kz = 0.5. +B. +Computational Details +DFT calculations were conducted using the Vienna Ab +Initio Simulation Package (VASP) [80] with the projector +augmented wave method (PAW) [81, 82]. The general- +ized gradient approximation (GGA) of Perdew, Burke, +and Ernzerhof (PBE) [83] was employed to describe the +exchange-correlation interactions among electrons with a +Hubbard U correction [84]. The U values were machine +learned using Bayesian optimization (BO) [55]. Briefly, +the BO objective function is formulated to reproduce as +closely as possible the band structure obtained from the +Heyd-Scuseria-Ernzerhof (HSE) [85] hybrid functional. +The reference HSE calculations were conducted for bulk +CdTe with a lattice parameter of 6.482 ˚A and α-Sn with +a lattice parameter of 6.489 ˚A and compared to the re- +sults with the lattice constant of InSb, 6.479 ˚A, which +was used for interface models. It was verified that using +the lattice constant of InSb does not have an appreciable +effect on the electronic properties of CdTe and α-Sn, as +shown in the SI. +The hyperparameters of our BO implementation are +the coefficients α1 and α2, which assign different weights +to the band gap vs. the band structure in the objective +function, the number of valence and conduction bands +used for the calculation of the objective function, Nb, and +the parameter κ that controls the balance between explo- +ration and exploitation in the upper confidence bound ac- +quisition function. For InSb the values of U In,p +eff += −0.2 +and U Sb,p +eff += −6.1 were used, following Ref. [55, 63]. +It has been shown that PBE+U(BO) produces a band +structure in good agreement with ARPES for InSb [63]. +Because α-Sn is a semi-metal, only the band shape +was considered in the optimization, i.e. α1 was set to +0 and α2 = 1 [59].The other BO hyperparameters used +for Sn were κ = 7.5 and Nb = (5, 5). This resulted in a +value of U Sn,p +eff += −3.04 eV, slightly different than in Refs. +[29, 35, 61], which used empirical methods to choose a +U value that yields a correct band ordering. As shown +in Ref. [59], PBE+U(BO) reproduces the correct band +ordering of α-Sn with the band inversion at the Γ point, + +a) full slab side view +c) CdTe primitive cell +e) CdTe(111) unit cell +h) +↑[111] +g) +k²= 0.0 +k,= 0.5 +SBZ +k,= 0.5 +M- +0 +k_= 0.0 +CdO In + z unfolded - kz +0.00 +0.25 +0.50 +b) +d) +f) +folded +) +bulk unfolded +z unfolded - k_ = 0.5 +3 +3 +D +3 +[111] +2 +2 +2 +2 +1 +1 +M +「k +M +1 +02 +6 +0 +(eV) +0 +0 +0 +出 +-1 +-1 +.1 +1 +-2 +2 +-3 +-3 +-3 +-4 +-4 +-5 +5 +M +L +L +M +M +L +M5 +in agreement with other studies using DFT+U [27, 28]. +For CdTe, we applied a U correction to both the Cd-d +orbitals and Te-p orbitals, unlike earlier studies [56, 86]. +The hyperparameters used for CdTe were κ = 7.5, Nb = +(5, 5), α1 = 0.5 and α2 = 0.5. The latter two parameters +were chosen to assign equal weights to the band gap and +the band shape. This led to U values of U Cd,d +eff += 7.381 +and U T e,p +eff += −7.912. The Cd-d U value obtained here +is similar to the 7 eV used in Ref. [86] and somewhat +lower than U Cd,d +eff += 8.3 eV in Ref. [56]. The gap of 1.21 +eV, obtained here by applying the Hubbard U correction +to both the Te-p states and the Cd-d states is closer to +experimental values of around 1.5 eV [87, 88] and the +HSE value of 1.31 eV than previous calculations [56]. +Spin-orbit coupling (SOC) was used in all calculations +and dipole corrections were applied to slab models [89]. +The tags used for convergence of calculations were BMIX += 3, AMIN = 0.01, ALGO = Fast, and EDIFF = 1·10−5. +The kinetic energy cutoff was set to 400 eV for all bulk +calculations and 350 eV for surface and interface slab +models. +A 9 × 9 × 9 k-point mesh was used for bulk +calculations and a k-point mesh of 7 × 7 × 1 was used for +surface and interface calculations. All interface density +of states (DOS) calculations used a k-point mesh of 13 × +13 × 1. +All band structure and density of states plots were gen- +erated using the open-source Python package, VaspVis +[59], which is freely available from The Python Package +Index (PyPI) via the command: pip install vaspvis, or on +GitHub at: +https://github.com/DerekDardzinski/ +vaspvis +C. +Slab Construction +All slab models were constructed using the experimen- +tal InSb lattice constant value of 6.479 ˚A [90], assuming +that the epitaxial films of CdTe and α-Sn would conform +to the substrate. The length of two monolayers of a (110) +slab was 4.5815 ˚Ain the z-direction. A vacuum region of +around 40 ˚A was added to each slab model in the z- +direction to avoid spurious interactions between periodic +replicas. The surfaces of all slab models were passivated +by pseudo-hydrogen atoms such that there were no sur- +face states from dangling bonds [91]. Despite α-Sn being +a semi-metal passivation is required to remove spurious +surface states, as shown in the supplemental information +(SI). The pseudo-hydrogen fractional charges utilized to +passivate each atom were 1.25 for In and 0.75 for Sb +in InSb, 1.5 for Cd and 0.5 for Te in CdTe, and 1 for +Sn. Structural relaxation of the pseudo-hydrogen atoms +was performed until the maximal force was below 0.001 +eV/˚A. The InSb/CdTe interface structure has In-Te and +Sb-Cd bonds with each In interface atom connected to 3 +Sb and 1 Te. The configuration with In-Cd and Sb-Te +bonds was also considered but this was found to be less +stable by 1.33 eV. Ideal interfaces were considered with +no intermixing and no relaxation of the interface atoms +was performed. +When constructing such slab models, it is necessary +to converge the number of layers to avoid quantum size +effects and approach the bulk properties [92]. For InSb +it has previously been shown that 42 monolayers are suf- +ficiently converged [63]. Plots of the band gap vs. the +number of atomic layers for CdTe(110) and α-Sn (110) +slabs are provided in the SI. CdTe was deemed converged +with 42 monolayers with a gap value of 1.23 eV, which +is only slightly larger than the bulk PBE+U(BO) value. +The z-unfolded band structures of CdTe(111) were cal- +culated for a 40 monolayer slab. A 26 monolayer slab +model was used to simulate the 2 × 2 reconstruction, +due to the higher computational cost of the 2 × 2 su- +percell. Structural relaxation was performed for the top +two monolayers of the 2 × 2 reconstruction. For the slab +of unstrained (110) α-Sn, 70 monolayers were needed to +close the gap at the zero-gap point of the semi-metal, +which corresponds to around 16 nm. The tri-layer slab +models comprised 42 layers of InSb, 70 layers of α-Sn and +between 0 and 16 layers of CdTe in two-layer increments, +amounting to a total slab thickness of around 300 nm +(not including vacuum). The (110) bi-layer slab models +comprised 42 layers of CdTe and InSb, and 70 layers of +α-Sn as these were deemed converged. +D. +ARPES Experimental details +The α-Sn samples were grown by molecular beam epi- +taxy on an In-terminated c(8 × 2) InSb(001) surface pre- +pared by atomic hydrogen cleaning. 51 monolayers (16.5 +nm) of α-Sn were deposited as calibrated via Rutherford +backscattering spectrometry. Growth was performed at a +substrate temperature of -20 ◦C and a base pressure bet- +ter than 1·10−10 Torr. The ARPES measurements were +taken at Beamline 10.0.1.2 at the Advanced Light Source +in Berkeley. The base pressure was better than 5·10−11 +Torr while the sample temperature was held at 68 K. +The sample was illuminated with 63 eV p-polarized light +and spectra were collected using a Scienta R4000 detector +with energy resolution better than 40 meV and angular +resolution better than 0.1◦. The sample was transferred +via vacuum suitcase with a base pressure better than +·10−11 Torr between the growth chamber and beamline. +A photon energy of 63 eV corresponds to a kz approxi- +mately 0.15 ˚A−1 above the Γ002 point. +III. +RESULTS AND DISCUSSION +A. +α-Sn +Figure 2a shows the bulk unfolded PBE+U(BO) band +structure for a 51 monolayer thick α-Sn (001) slab, com- +pared to ARPES data for a sample of the same thickness +taken at a photon energy of 63 eV . The point M is at + +6 +FIG. 2. Electronic structure of α-Sn: (a) Bulk-unfolded band +structure of an α-Sn (001) slab with 51 atomic layers (light +blue) compared with ARPES data for a sample of the same +thickness. The point M is at 0.9298 ˚A−1. The ARPES data +is cutoff at 0.9 ˚A−1 due to experimental artifacts at the edges. +Spin-polarized band structures projected onto (b) the top sur- +face atoms and (c) the bottom surface atoms, indicated by the +green boxes on the slab structure illustrated in (d). +0.9298 ˚A−1. The ARPES data is cutoff at 0.9 ˚A−1 due to +experimental artifacts at the edges. The PBE+U (BO) +band structure is in excellent agreement with ARPES. +The top of the valence band in the ARPES and the sim- +ulated band structure lines up and the bulk bands are +reproduced well. The bandwidth of the heavy hole band, +Γ8, is slightly underestimated, consistent with Ref. [63]. +This is corrected by the HSE functional, as shown in the +SI for a bulk unit cell of α-Sn with a (001) orientation. +However, it is not feasible to use HSE for the large inter- +face models studied here, owing to its high computational +cost. +The previously reported topological properties of α- +Sn slabs are also observed here [27–31, 35, 36, 62]. The +spin-polarized topological surface state (TSS) is shown +in panels (b) and (c) of Fig. 2 for a (001) 51 monolayer +slab along the X − Γ − X k-path. As expected, the TSS +is characterized by a linear dispersion with the top and +bottom surfaces having opposite spin polarization. The +associated Rashba-like surface states are also observed +along the K − Γ − K k-path, as shown in the SI. This +linear surface state is also observed in the (110) slabs used +to construct the bilayer and tri-layer models. Notably +there is an energy gap between the top and bottom TSSs, +which closes at 70 layers, the same thickness at which +the band gap closes. +This gap is possibly induced by +the hybridization of the top and bottom surface states in +under-converged slabs. We note that the effect of strain +on the electronic structure of α-Sn is not studied here. +B. +CdTe +Fig. 3 shows a comparison of band structures obtained +using PBE+U(BO) to the ARPES experiments of Ren et +al. [93] for CdTe(111). Ren et al. collected ARPES data +at photon energies of 19, 25 and 30 eV . Here, we com- +pare our results with the second-derivative maps of the +ARPES data taken at 25 eV along the k-paths Γ − M +(panels (a) and (b)) and Γ − K − M (panels (c) and +(d)). The original data has been converted to gray scale +and reflected around kx = 0. +To facilitate the quali- +tative comparison of the DFT band structure features +with the ARPES experiment, we apply a Fermi energy +shift of 0.25 eV to line up the VBM and a stretch factor +of 1.22 to compensate for the bandwidth underestima- +tion of PBE+U(BO), particularly for bands deep below +the Fermi energy [94]. Bandwidth underestimation by +PBE+U(BO) compared with HSE and ARPES has also +been reported for InAs and InSb in [63, 95]. The original +computed band structure without the shift and stretch is +provided in the SI. +Owing to the low mean free path at this photon energy, +the spectrum appears integrated over a certain kz inter- +val and surface contributions are readily visible in the +ARPES [69, 70]. To account for the different kz contribu- +tions, the z-unfolding method was employed, as described +in Section II A. Panels (a) and (c) show the z-unfolded +band structures as a function of kz for slab models with- +out a surface reconstruction (figures with single values of +kz are provided in the SI). This is used determine which +kz values are likely present in the experiment. A mixture +of kz = 0 and k = 0.5 provides the best agreement with +the ARPES data. This combination of kz values is used +for the DFT data shown in cyan in panels (b) and (d). +This is consistent with the kz broadening with contribu- +tions centered around kz = 0 and k = 0.5 often present +in ARPES data taken at low mean field path energies in +gapped materials [71? , 72]. +To account for the presence of surface states, we mod- +eled the CdTe(111)A-(2 × 2) surface reconstruction [96], +illustrated in panel (e). The atom-projected band struc- +tures of the bottom layer (indicated by pink dashed box) +are plotted in pink in panels (b) and (d). +The addi- +tional bands arising from the surface reconstruction are +in close agreement with the bands in the ARPES labeled +as surface states by Ren et al., indicated by red arrows. +These surface states are unaffected by the choice of kz. +By accounting for the contributions of different kz values +and for the presence of surface states excellent agreement +with experiment is achieved, as the DFT band structures +reproduce all the features of the ARPES. +C. +Bilayer Interfaces +We begin by probing the local electronic structure at +the the InSb/α-Sn bi-layer interface. Fig. 4a shows the + +d) +0 +b) +UP +a) +↓ DOWN +(Λa) +0.0 +-1 +00 +1 +E-0.5 +00 +-2 +E(eV) +x +-3 +00 +UP +C) +(a) +DOWN +00 +0.0 +00 +-4 +1 +E -0.5 +0 +0.4 +-0.8-0.4 +0.0 +0.8 +T +↑M +M→ +X +k, (A-1)7 +FIG. 3. +Electronic structure of CdTe: Z-unfolded band structures of CdTe(111) compared with second-derivative map of +ARPES data (black and white), adapted with permission from “Spectroscopic studies of CdTe(111) bulk and surface electronic +structure” by J. Ren et al., Phys. Rev. B, 91, 235303 (2015); Copyright (2015) by the American Physical Society [93]. Z- +unfolded band structures compared to ARPES data along (a), (b) Γ − M and (c), (d) Γ − K − M. (a), (c) Dependence of the +band structure on kz. (b), (d) Mixture of kz = 0.0 and kz = 0.5 (cyan) for a model with a 2 × 2 surface reconstruction with +the contributions of the surface atoms shown in pink. DFT has shift of -0.25 eV and stretch factor of 1.22 for comparison. (e) +Illustration of the 2 × 2 surface reconstruction with the Cd atom removed indicated by a blue circle. The atoms used for the +surface projection are indicated by a pink dashed box +DOS as a function of position across the interface, in- +dicated by the atomic layer number. Fig. 4b shows the +local DOS at select positions. The Fermi level is posi- +tioned at the semi-metal point of the α-Sn and in the +gap of the InSb. We note that the α-Sn appears as if it +has a small gap due to an artifact of the 10−4 cutoff ap- +plied in the log plot in panels (a) and (d). The local DOS +plots shown in panels (b) and (e) and the band structure +plots shown in panels (c) and (f) clearly show the semi- +metal point. No significant band bending is found for +InSb, as expected from branching point theory [97, 98]. +Based on the element-projected band structure, shown +in panel (c), the InSb conduction band minimum (CBM) +lies 0.09 eV above the α-Sn semi-metal point and the +InSb valence band maximum (VBM) lies 0.16 eV below +it. A linear TSS is present in the α-Sn. Based on an +atom projected band structure, shown in the SI, the ori- +gin of this state is the top surface of α-Sn, adjacent to +the vacuum region. A TSS is no longer present in the +α-Sn layers at the interface with InSb, possibly owing to +hybridization between the α-Sn and InSb [62]. Metal- +induced gap states (MIGS) are an inherent property of +a metal/semiconductor interface, produced by the pen- +etration of exponentially decaying metallic Bloch states +into the gap of the semiconductor [99–102]. The pres- +ence of MIGS manifests in Figure Fig. 4a as a gradually +decaying non-zero DOS in the band gap of the InSb in +the vicinity of the interface. Figure 4b shows that the +MIGS are prominent in the first few atomic layers and +become negligible beyond 8 layers from the interface. +Fig. 4d shows the DOS as a function of position across +the CdTe/α-Sn interface, indicated by the atomic layer +number. Fig. 4e shows the local DOS at select positions. +The Fermi level is positioned at the semi-metal point of +the α-Sn and in the gap of the CdTe. +Based on the +projected band structure, shown in panel (f), the CdTe +CBM is positioned 0.18 eV above the Fermi level and +the CdTe VBM is located 1.03 eV below the Fermi level. +This agrees with previous reports that interfacing with +Sn brings the conduction band of the CdTe closer to the +Fermi energy, with downward band-bending of 0.25 eV +[103] and 0.1 eV [104]. +We find a valence band offset +of around 1 eV, similar to the (110) and (111) interface +reported in [33, 104–108]. Close to the interface there is +a significant density of MIGS, which decay within about +10 layers (3-4 nm) into the CdTe. This suggests that this +number of CdTe layers may be required for an effective +tunnel barrier. +Fig. 4g shows the DOS as a function of position across +the InSb/CdTe interface, indicated by the atomic layer +number. Fig. 4h shows the local DOS at select positions. +The band alignment is type-I with the CdTe band gap +straddling the InSb band-edges. The Fermi level is close +to the InSb VBM and around the middle of the gap of +the CdTe. No band bending is found in either material. +Based on the projected band structure, shown in panel +(i), the CdTe CBM lies 0.28 eV above the InSb CBM +and the CdTe VBM lies 0.75 eV below the InSb VBM. +These values are similar to the band offsets reported in +references [25, 88, 109]. Because the band gap of InSb +is significantly smaller than that of CdTe, states from +the InSb penetrate into the gap of the CdTe, similar to +MIGS. These states decay gradually and vanish at a dis- +tance greater than 12 layers from the interface. +D. +Tri-layer Interfaces +Fig. 5 shows the DOS as a function of position across +InSb/CdTe/α-Sn tri-layer interfaces with varying thick- + +Kz +0.2s +OCdOIn +0.25 +0.00 +0.50 +0.00 +0.50 +a) +b) +d) +c) +e) +. +0: +0 +0 +-1 +-1 +-2 +-2 +(na) +-3 +山 +4 +-4 +-4 +-5 +-5 +-5 +-5 +surface +states +surface +-6 +-6 +19- +-6. +states +12 +8 +0 +4 +12 +8 +4 +8 +4 +4 +12 +8 +12 +4 +0 +4 +8 +0 +8 +8 +8 +M +M +K +IF +K +M +M +M +M +T +M +K +K +M +k (A-1) +k (A-1) +k (A-1) +k (A-1)8 +FIG. 4. Electronic structure of bilayer interfaces: Density of states in the (a) InSb/α-Sn, (d) CdTe/α-Sn and (g) InSb/CdTe +interfaces as a function of position. The atomic layers are numbered based on distance from the interface, which is located at +zero. The structure of each interface is illustrated on top. (b Local density of states for selected layers in the (b) InSb/α-Sn, (e) +CdTe/α-Sn and (h) InSb/CdTe interfaces, indicated by dashed lines in the same colors in panels (a), (d), and (g), respectively. +Element projected band structures of the (c) InSb/α-Sn, (f) CdTe/α-Sn and (i) InSb/CdTe interfaces, with bands originating +from α-Sn colored in red, bands originating from InSb colored in light blue, and bands originating from CdTe colored in purple. +ness of the CdTe tunnel barrier. The position is indicated +by the atomic layer number, with the layer of InSb clos- +est to the CdTe considered as zero. Panels (a) and (b) +show that with 6 atomic layers of CdTe, the MIGS from +the α-Sn penetrate through the tunnel barrier into the +first 12 layers of the InSb. +For a thin layer of CdTe, +the band gap is expected to be significantly larger than +the bulk value because of the quantum size effect (see +the gap convergence plot in the SI). However, owing to +the presence of MIGS, the gap of the CdTe remains con- +siderably smaller than its bulk value. With 10 layers of +CdTe, shown in panels (c) and (d), there is still a sig- +nificant presence of MIGS throughout the CdTe, which +decay by 6 layers into the InSb. Panels (e) and (f) show +that with 16 layers of CdTe the InSb is completely insu- +lated from MIGS coming from the α-Sn. The gap of the +CdTe reaches a maximum of around 0.3 eV at a distance +of 5 layers from the InSb. This is because MIGS from +the α-Sn penetrate into the CdTe from one side, whereas +states from the InSb penetrate from the other side, such +that the band gap of the CdTe never reaches its expected +value. +Figure 6 summarizes the band alignment at the bilayer +and tri-layer interfaces studied here. For the tri-layer in- +terfaces, the band alignment between the InSb and the +α-Sn is not significantly affected by the presence of CdTe, +as shown in the element-projected band structures in the +SI. The α-Sn semi-metal point remains pinned at the +Fermi level, as in the bilayer InSb/α-Sn (see also Fig- +ure 4c). The InSb VBM remains at 0.17 eV below the +Fermi level, similar to its position in the bilayer interface, +regardless of the CdTe thickness. The InSb CBM posi- +tion decreases slightly with the thickness of the CdTe +from 0.09 eV above the Fermi level without CdTe, to + +8.8.8.8181818818.818 +.8.818.8181818.818 +.:1: +: +:1: +:1: +: +a)( +d) +g) +: +: +I +0.2 +CdTe/Sn +100 +InSb/Sn +InSb/CdTe +0.2 +0.4 +(arb. units) +0.0 +0.2 +EF (eV) +0.1 +-0.2 +0.0 +log(DOS)( +0.0 +-0.4 1 +-0.2 +E +-0.6- +10- +-3 +-0.1 +0.4 +-0.8 +-0.6 +-0.2 +8 -15 +3 +3 +18 +5 -12 +-9 +-6-3 +-18 +3 -15 -12 +9-　6 +m- +-3 +0 +3 +6 +12 +0 +0 +9 +一 +b) +Layers +e) +Layers +h) +Layers +5 J +5 +-17 +Sn +-12 +41 +41 +0 +4 +-8 +-17 +4 +(e-OL) +3 1 +-8 +3 1 +6 +-6 +Sb +DOS +-4 +8 +2 1 +2 1 +2 +-4 +0 +12 +Cd +一 +0 +4 +17 +一 +11 +1 +11 +4 +Te +01 +0, +0.2 +0.2 +-0.2 +-0.1 +0.0 +0.1 +-1.0 -0.8 -0.6 -0.4 -0.2 +0.2 +-0.2 +0.4 +-1.2 +0.0 +-0.6 +-0.4 +0.0 +0.6 +E-E (eV) +E-Ef (eV) +( +f) +E-Eε (eV) +i) +0.2 +0.18. +.0.38. +0.2 +0.4 +0.09 +Sn +0.1 +TSS +0.0 +0.2 +0.1. +0.0 +InSb +-0.2 +0.0 +(eV) +-0.1 +-0.1. +CdTe +-0.16 +-0.4 +-0.2 +-0.2 +TSS +-0.6 +-0.4 +E -0.3 +-0.8 +-0.6 +0.4 +-1.03 +-1.0 +-0.83 +-0.5 +-0.8 +-1.2 +-0.6 +1.0 +x +1X +1X +x9 +FIG. 5. +Electronic structure of InSb/CdTe/α-Sn tri-layer interfaces: Density of states as a function of distance from the +interface for (a) 6, (c) 10 and (e) 16 CdTe barrier layers. The atomic layers are numbered based on distance from the interface, +which is located at zero. Interface structures are illustrated on top. (b), (d), (f) Local density of states for selected layers, +indicated by dashed lines in the same colors in panels (a), (c), and (e), respectively. +FIG. 6. Valence and conduction band edge positions for InSb +and CdTe in the bilayer and tri-layer interfaces. The Fermi +level is at the semi-metal point of the α-Sn. +0.054 eV with 6 layers of CdTe, 0.04 eV with 10 layers, +and 0.037 eV with 16 layers. This may be attributed to +the quantum size effect, which causes a slight narrowing +of the InSb gap because of the increase in the overall +size of the system. Based on the element-projected band +structures provided in the SI, the band edge positions +of the CdTe are dominated by the interface with the α- +Sn, rather than the interface with the InSb. The CdTe +CBM remains at 0.18 eV above the Fermi level, as in +the bilayer CdTe/α-Sn interface (see also Figure 4f), re- +gardless of the number of layers. As the band gap of the +CdTe narrows with increasing thickness, the CdTe VBM +shifts from 1.24 eV below the Fermi level with 6 layers +to 1.105 eV with 10 layers, and 1.05 eV with 16 layers, +approaching the bilayer VBM position of 1.03 eV below +the Fermi level with 42 layers. Although the band gap of +the CdTe is significantly reduced due to MIGS, a type I +band alignment with the InSb is maintained, similar to +the bilayer InSb/CdTe interface (Figure 4g,i), as shown +in Fig. 5 panels (a), (c), and (e). +Figure 7 show the LDOS in the second layer of InSb +from the interface as a function of the number of CdTe +layers. Without CdTe and with two layers of CdTe, there +is no band gap in the InSb close to the interface, owing +to the significant density of MIGS. With 6 layers of CdTe +the gap of the InSb close to the interface is still consid- +erably narrower than its bulk value. The band gap in +the second layer of InSb from the interface approaches +its bulk value with 10 layers of CdTe and finally reaches +it with 16 layers of CdTe. This suggests that 16 CdTe +layers provide an effective barrier to electronically insu- +late the InSb from the α-Sn. It is reasonable to assume +that a barrier of this thickness or higher would all but +eliminate transport through the interface into the InSb. +Therefore, we estimate that the relevant barrier thickness +regime to modulate the coupling at an interface with a + +::*:18::+::+1:++:+::+::+:: (0 +a) +C) +8 +8 +： +:i: +::: +1100 +0.2 +0.2 +0.2 +(arb. units) +10-1 +(eV) +0.0 +0.0 +0.0 +10-2 +log(DOS) ( +-3 +-0.2 +-0.2 +-0.2 +-0.4 +-0.4 +-0.4 +10-4 +-12-9 +-12 -8 -4０　4　8　121620 +-12 +-9 +-6-30 +3 +6 +9 +-6-3０36 +９1215 +Layers +Layers +Layers +b) +d) +f) +5 +5 +5 +-12 +-12 +-12 +Sn +-6 +-6 +-6 +4 1 +4 +4 +0 +0 +0 +In +(t-OL) +2 +31 +2 +2 +3 +3 +3 +5 +Sb +DOS +6 +9 +8 +2 1 +21 +2 +15 +Cd +11 +1 +Te +0 - +0 +0 +-0.15-0.10-0.05 0.00 +0.05 +0.100.15 +-0.15-0.10-0.05 0.00 0.05 +0.10 +0.15 +-0.15-0.10-0.05 0.00 0.05 +0.100.15 +E- EF (eV) +E- EF (eV) +E- EF (eV)0.5 +0.0 +(eV) +InSb +CdTe +-0.5 +-1.0 +CdTe/α-Sn - +InSb/α-Sn +InSb/(CdTe)6/α-Sn +InSb/(CdTe)10/α-Sn +InSb/(CdTe)16/α-Sn +InSb/CdTe +Interface10 +FIG. 7. Density of states in the second InSb layer from the +interface (layer -2 in Figure 5) as a function of the number of +CdTe barrier layers. +superconductor and tune the proximity effect would be +in the range of 6-10 layers, where MIGS still exist. We +note, however, that the interface with β-Sn may have +somewhat different characteristics in terms of the band +alignment and the penetration depth of MIGS. +IV. +CONCLUSION +In summary, we have used DFT with a Hubbard +U correction machine-learned by Bayesian optimization +to study CdTe as a prospective tunnel barrier at the +InSb/α-Sn interface. The results of PBE+U(BO) were +validated by comparing the band structures of slab mod- +els of α-Sn(001) and CdTe(111) with ARPES experi- +ments (the PBE+U(BO) band structure of InSb(110) +had been compared to ARPES experiments previously +[63]). Excellent agreement with experiment is obtained +for both materials. In particular, for the low-mean-free- +path ARPES of CdTe, the z-unfolding scheme success- +fully reproduces the contributions of different kz values +and modelling the 2 × 2 surface reconstruction success- +fully reproduces the contributions of surface states. +We then proceeded to use PBE+U(BO) to calculate +the electronic structure of bilayer InSb/α-Sn, CdTe/α- +Sn, and InSb/CdTe, as well as tri-layer InSb/CdTe/α-Sn +interfaces with varying thickness of CdTe. Simulations +of these very large interface models were possible thanks +to the balance between accuracy and computational cost +provided by PBE+U(BO). We find that the most stable +configuration of the InSb/CdTe interface is with In-Te +and Sb-Cd bonding. MIGS penetrate from the α-Sn into +the InSn and CdTe. Similarly, states from the band edges +of InSb penetrate into the larger gap of the CdTe. No +interface states are found in any of the interfaces studied +here, in contrast to the EuS/InAs interface, for example, +in which a quantum well interface state emerges [110]. +For all interfaces comprising α-Sn, the semi-metal +point is pinned at the Fermi level. For the tri-layer inter- +face, the band alignment between the InSb and the α-Sn +remains the same as in the bilayer interface regardless of +the thickness of the CdTe barrier, with the Fermi level +closer to the conduction band edge of the InSb. The band +edge positions of the CdTe are dominated by the inter- +face with the α-Sn rather than the interface with InSb, +with the conduction band edge being closer to the Fermi +level. A type-I band alignment is maintained between +CdTe and InSb with the gap of the former straddling +the latter. The CBM of the CdTe is pinned whereas the +VBM shifts upwards towards the Fermi level as the gap +narrows with the increase in thickness. +We find that 16 layers of CdTe (about 3.5 nm) serve as +an effective barrier, preventing the penetration of MIGS +from the α-Sn into the InSb. However, in the context of +Majorana experiments, it is possible that a barrier thick +enough to completely insulate the semiconductor from +the superconductor would also all but eliminate trans- +port. +Therefore, we estimate that the relevant regime +for tuning the coupling at the interface would be in the +thickness range where some MIGS are still present, while +thicker CdTe layers could be used to passivate exposed +InSb surfaces. We note, however, that the interface with +the superconducting β-Sn, which is not lattice matched +to InSb and CdTe, may have different characteristics than +the interface with α-Sn. In practice, careful experimen- +tation with varying barrier thickness would be needed to +determine the optimal configuration for MZM devices. +We have thus demonstrated that DFT simulations +can provide useful insight into the electronic properties +of semiconductor/tunnel barrier/metal interfaces. This +includes the interface bonding configuration, the band +alignment, and the presence of MIGS (and, possibly, of +interface states). Such simulations may be conducted for +additional interfaces to explore other prospective mate- +rial combinations. This may inform the choice of inter- +face systems and the design of future Majorana experi- +ments. More broadly, similar DFT simulations of inter- +faces may be performed to evaluate prospective tunnel +barriers e.g., for semiconductor devices. +V. +ACKNOWLEDGEMENTS +We thank Guang Bian from the University of Mis- +souri, Li Fu from Northwestern Polytechnical University, +China, and Tai C. Chiang from the University of Illinois +at Urbana-Champaign for sharing their ARPES data for +CdTe. Work at the University of Pittsburgh was sup- +ported by the Department of Energy through grant DE- +SC-0019274. Work at CMU and UCSB was funded by +the National Science Foundation (NSF) through grant +OISE-1743717. Work in Grenoble is supported by the +ANR-NSF PIRE:HYBRID, Transatlantic Research Part- +nership and IRP-CNRS HYNATOQ. This research used +computing resources of the University of Pittsburgh Cen- +ter for Research Computing, which is supported by NIH +award number S10OD028483 and of the National Energy + +10 +0 + CdTe +Density of States (10-4) +2 +CdTe +8 +6 CdTe +10 CdTe +6 +16 CdTe +4 +2 +0 +-0.1 +0.0 +0.1 +0.2 +E- Er (eV)11 +Research Scientific Computing Center (NERSC), a U.S. +Department of Energy Office of Science User Facility op- +erated under Contract No. DE-AC02-05CH11231. +[1] D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, +J. Danon, M. Leijnse, T. S. 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Mater. 5, 064606 (2021). + diff --git a/BtE1T4oBgHgl3EQfDgPC/content/tmp_files/load_file.txt b/BtE1T4oBgHgl3EQfDgPC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bd7d65e9cc79cea31457a2042eaeba1d89b232e --- /dev/null +++ b/BtE1T4oBgHgl3EQfDgPC/content/tmp_files/load_file.txt @@ -0,0 +1,1786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf,len=1785 +page_content='First Principles Assessment of CdTe as a Tunnel Barrier at the α-Sn/InSb Interface Malcolm J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Jardine,1, ∗ Derek Dardzinski,2, ∗ Maituo Yu,2 Amrita Purkayastha,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Chen,3 Yu-Hao Chang,4 Aaron Engel,4 Vladimir N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Strocov,5 Mo¨ıra Hocevar,3 Chris J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Palmstrøm,4, 6 Sergey M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Frolov,1 and Noa Marom2, 7, 8, † 1Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA 2Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA 3Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Grenoble Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Grenoble INP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Institut N´eel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 38000 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' France 4Materials Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' University of California-Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' USA 5Paul Scherrer Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Swiss Light Source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CH-5232 Villigen PSI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Switzerland 6Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' University of California-Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' USA 7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' PA 15213,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' USA 8Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' PA 15213,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' USA Majorana zero modes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' with prospective applications in topological quantum computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' are ex- pected to arise in superconductor/semiconductor interfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' such as β-Sn and InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, prox- imity to the superconductor may also adversely affect the semiconductor’s local properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A tunnel barrier inserted at the interface could resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We assess the wide band gap semiconduc- tor, CdTe, as a candidate material to mediate the coupling at the lattice-matched interface between α-Sn and InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' To this end, we use density functional theory (DFT) with Hubbard U corrections, whose values are machine-learned via Bayesian optimization (BO) [npj Computational Materials 6, 180 (2020)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The results of DFT+U(BO) are validated against angle resolved photoemission spec- troscopy (ARPES) experiments for α-Sn and CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For CdTe, the z-unfolding method [Advanced Quantum Technologies, 5, 2100033 (2022)] is used to resolve the contributions of different kz values to the ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We then study the band offsets and the penetration depth of metal-induced gap states (MIGS) in bilayer interfaces of InSb/α-Sn, InSb/CdTe, and CdTe/α-Sn, as well as in tri-layer interfaces of InSb/CdTe/α-Sn with increasing thickness of CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We find that 16 atomic layers (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 nm) of CdTe can serve as a tunnel barrier, effectively shielding the InSb from MIGS from the α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This may guide the choice of dimensions of the CdTe barrier to mediate the coupling in semiconductor-superconductor devices in future Majorana zero modes experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' INTRODUCTION A promising route toward the realization of fault- tolerant quantum computing schemes is developing ma- terials systems that can host topologically protected Ma- jorana zero modes (MZMs) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' MZMs may ap- pear in one-dimensional topological superconductors [3– 5], which can be effectively realized by proximity cou- pling a conventional superconductor and a semiconduc- tor nanowire that possesses strong spin-orbit coupling (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Adding in a magnetic field enables this system to behave as an effective spinless p-wave topological super- conductor, which allows for MZM states [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Recently, there have been new developments in material choices and experimental methods to identify MZMs in semicon- ductor nanowire-superconductor systems [7], designed to overcome challenges identified during the first wave of experiments [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' These include trying new combina- tions of semiconductors and epitaxial superconductors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Pb, Sn, Nb, to maximize the electron mobility and utilize larger superconducting gaps and higher critical magnetic fields [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Additionally, new proposed ar- ∗ These authors contributed equally to this work † Corresponding author: nmarom@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='edu chitectures include creating nanowire networks and in- ducing the field via micromagnets [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' One of the challenges presented by the superconduc- tor/semiconductor nanowire construct, is that excessive coupling between the superconducting metal and semi- conductor may “metallize” the semiconductor, thus ren- dering the topological phase out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Theoretical studies that treated the semiconducting and supercon- ducting properties via the Poisson-Schr¨odinger equation, have shown that excessive coupling between the mate- rials may lead to the semiconductor’s requisite proper- ties, such as the Lande´e g-factor and spin-orbit-coupling (SOC), being renormalized to a value closer to the metal’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In addition, large unwanted band shifts may be induced [12, 19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Having a tunnel barrier could modu- late the superconductor-semiconductor coupling strength and thus the induced proximity effect, which is critical for controlling experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' It is currently unknown what the required width range of a tunnel barrier is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Another potential benefit of a CdTe layer is InSb surface passiva- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' InSb and Sn are among the materials used to fabricate devices for Majorana search [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' InSb is the backbone of such systems because it has the highest electron mobility, strongest spin-orbit coupling (SOC) and a large Land´e g-factor in the conduction band compared to other III- V semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' β−Sn has a bulk critical field of 30 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='02879v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='mtrl-sci] 7 Jan 2023 2 mT and a superconducting critical temperature of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='7 K, higher than the 10 mT and 1 K, respectively, of Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Re- cently, β-Sn shells have been grown on InSb nanowires, inducing a hard superconducting gap [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The large band gap semiconductor CdTe is a promising candidate to serve as a tunnel barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Thanks to its relative in- ertness, it may simultaneously act as a passivation layer protecting the InSb from environmental effects and po- tentially minimizing disorder [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Advantageously, CdTe is lattice matched to InSb [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Sn has two al- lotropes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The β form, with a BCT crystal structure, is of direct relevance to MZM experiments thanks to its su- perconducting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, the semi-metallic α form has a diamond structure, which is lattice matched to InSb and CdTe, making it an ideal model system for investigating, both theoretically and experimentally, the electronic structure of Sn/InSb heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Much experimental work, such as growth and ARPES studies, has been undertaken on α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Previously, α-Sn has been found to possess a topologically trivial band in- version, with SOC inducing a second band inversion and a topological surface state (TSS) [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The effect of strain on the topological properties of α-Sn has also been studied [21, 29–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In-plane compressive strain has been reported to make α-Sn a topological Dirac-semi-metal and induce a second TSS to appear [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Conversely, tensile strain has been reported to induce a transition to a topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CdTe [25] and α-Sn [12, 28] have been epitaxially grown on InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Depositing Sn on InSb often leads to growth of epitaxially matched α-Sn, although β-Sn may appear under some conditions [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In addition, α-Sn can transition to β-Sn if the Sn layer is above a critical thickness or if heat is applied during fab- rication processes [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Studying the interface with the lattice matched α-Sn may provide insight, which is also pertinent to β-Sn as both could be present in hy- brid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Therefore, these are promising materials to investigate for future device construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' MZM experiments rely on finely tuned proximity cou- pling between a superconducting metal and a semicon- ductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' By adding a tunnel barrier at the interface be- tween the two materials and varying its width, one could potentially mediate the proximity coupling strength to achieve precise control over the interface transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' To the best of our knowledge, this idea has not yet been tested in experiments and it is presently unknown which material(s) would be the best choice for a barrier and what would be the optimal thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Simulations of a tri-layer system with a tunnel barrier are therefore needed to inform MZM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Here, we use den- sity functional theory (DFT) to study a tri-layer system, in which InSb is separated from α-Sn by a CdTe tunnel barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Despite recent progress towards treating super- conductivity within the framework of DFT [42, 43] the description of proximity-induced superconductivity at an interface with a semiconductor is still outside the reach of present-day methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, DFT can provide useful information on properties, such as the band alignment at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Conduction band offsets are of particular importance because the proximity effect in most experi- ments on InSb primarily concerns the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In addition, DFT can provide information on the pene- tration depth of metal induced gap states (MIGS) into the semiconductor [20, 25, 44, 45], which is important for determining the appropriate thickness of the tunnel barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Within DFT, computationally efficient (semi-)local exchange-correlation functionals severely underestimate the band gap of semiconductors to the extent that some narrow-gap semiconductors, such as InSb, are er- roneously predicted to be metallic [46–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This is at- tributed to the self-interaction error (SIE), a spurious repulsion of an electron from its own charge density [50– 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Hybrid functionals, which include a fraction of exact (Fock) exchange, mitigate the SIE and yield band gaps in better agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, their com- putational cost is too high for simulations of large in- terface systems, such as the α-Sn/CdTe/InSb tri-layer system studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The DFT+U approach, whereby a Hubbard U correction is added to certain atomic orbitals, provides a good balance between accuracy and computa- tional cost[46, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Recently, some of us have pro- posed a method of machine learning the U parameter for a given material by Bayesian optimization (BO) [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The DFT+U(BO) method has been employed successfully for InSb and CdTe [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' It has been shown that (semi-)local functionals fail to describe the bulk band structure of α-Sn correctly, specifically the band ordering and the orbital compo- sition of the valence bands at the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' DFT+U, hybrid functionals, or many-body perturbation theory within the GW approximation are necessary to obtain a correct description of the band structure [29, 35, 57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' DFT+U simulations have required slab models of more than 30 monolayers of Sn to converge towards a bulk regime, where quantum confinement is no longer domi- nant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' With a small number of layers α-Sn may exhibit topological properties [26, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Some DFT studies have considered slab models of bi-axially strained α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' DFT simulations of strained α-Sn on InSb have been con- ducted with a small number of layers of both materials [26, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The DFT+U approach has reproduced the ef- fects of strain and compared well with experimental data [28, 60, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Here, we perform first principles calculations us- ing DFT+U(BO) for a (110) tri-layer semiconduc- tor/tunnel barrier/metal interface composed of the ma- terials InSb/CdTe/α-Sn, owing to their relevance to cur- rent Majorana search experiments [12, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' To date, DFT studies of large interface slab models with a vacuum re- gion have not been conducted for these interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Pre- viously, the results of DFT+U(BO) for InSb(110) have been shown to be in good agreement with angle-resolved photoemission spectroscopy (ARPES) experiments [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Here, we also compare the results of DFT+U(BO) to ARPES for α-Sn (Section III A) and CdTe (Section 3 III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Excellent agreement with experiment is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In particular, for CdTe the z-unfolding scheme (Section II A) helps resolve the contributions of different kz values and modelling the 2 × 2 surface reconstruction repro- duces the spectral signatures of surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We then proceed to study the bi-layer interfaces of InSb/CdTe, CdTe/α-Sn, and InSb/α-Sn (Section III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Finally, to assess the effectiveness of the tunnel barrier, we study tri-layer interfaces with 2 to 16 monolayers (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 nm to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 nm) of CdTe inserted between the InSb substrate and the α-Sn (Section III D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This thickness is within the thick- ness range of CdTe shells grown on InSb nanowires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For all interfaces, our simulations provide information on the band alignment and the presence of MIGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We find that 16 layers of CdTe (about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 nm) form an effective tunnel barrier, insulating the InSb from the α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, this may be detrimental for transport at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Based on this, we estimate that the relevant thickness regime for tuning the coupling between InSb and Sn may be in the range of 6-10 layers of CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Z-Unfolding Simulations of large supercell models produce complex band structures with a large number of bands, as shown in Figure 1a,b for a CdTe(111) slab with 25 atomic layers, whose band structure was calculated using PBE+U(BO), as described in Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Band structure unfolding is a method of projecting the band structure of a su- percell model onto the appropriate smaller cell ([63–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This can help resolve the contributions of states emerg- ing from of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=', defects and surface reconstructions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' the bulk bands of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In addition, it can fa- cilitate the comparison to angle-resolved photoemission spectroscopy (ARPES) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The “bulk band un- folding” scheme [63] projects the supercell band struc- ture onto the primitive unit cell, illustrated in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The resulting band structure, shown in Figure 1d, appears bulk-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Bulk-unfolded band structures have been shown to compare well with ARPES experiments using high photon energies, which are not surface sensi- tive owing to the large penetration depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The “z-unfolding” scheme [63] projects the band struc- ture of a slab model with a finite thickness onto the Bril- louin zone (BZ) of a single layer of the slab supercell with the same orientation, illustrated in Figure 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The resulting band structure, shown in Figure 1f, contains ex- tra bands that are not present in the bulk-unfolded band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The extra bands originate from different kz values in the 3D primitive Brillouin zone projecting onto the surface Brillouin zone (SBZ), creating overlapping paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For example, panel Figure 1g shows cross sections through the BZ at values of kz = 0 and kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The bulk-paths of Γ − L, Γ − K and Γ − X all overlap with the surface k-path Γ − M, possibly with contributions from additional paths, such as X − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The plane cuts at different kz values are derived from the tessellated bulk BZ structure, shown in Figure 1h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' When z-unfolding is performed, the value of kz may be treated as a free pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The dependence on kz manifests as a smooth change in the spectral function over the possible range of kz which varies the mixture of different constituent bulk-paths that overlap the SBZ-path, as shown in Fig- ure 1i for Γ − M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The BZ for z-unfolding is a surface BZ with a finite thickness, shown in red in Figure 1j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The simulation cell for the DFT calculations is set up to be the corresponding real-space unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The z-unfolded k- paths are parallel to the (111) surface at a constant value of kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In ARPES experiments, the relation of the experimen- tal spectra to kz may be less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' First, the dependence of the inelastic mean free path of the elec- trons on their kinetic energy is given by the universal curve [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Using photon energies that correspond to a small mean free path is advantageous for probing surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, it can produce prominent kz broadening due to the Heisenberg uncertainty principle [71–75] that implies integration of the ARPES signal over kz through the broadening interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Second, deviations of the photoemission final states from the free electron ap- proximation can cause contributions from different values of kz to appear in the ARPES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The photoelec- trons are often treated as free electrons, based on the as- sumption that the photoelectron kinetic energy is much larger than the modulations of the crystal potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In this case, kz for a given photoelectron kinetic energy, Ek, and the in-plane momentum, K//, is one single value, which is determined by: kz = √2m0 ℏ � Ek − ℏ2 2m0 K2 // − V0 (1) where m0 is the free-electron mass and V0 the inner po- tential in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, a considerable body of evidence has accumulated that the final states even in metals [76, 77] and to a greater extent in complex ma- terials such as transition metal dichalcogenides [78, 79] can significantly deviate from the free electron approx- imation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Such deviations can appear, first, as non- parabolic dispersions of the final states and, second, as their multiband composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The latter means that for given Ek and K// the final-state wavefunction Φf incor- porates a few Bloch waves φkz with different kz values, Φf = � kz Akzφkz, which give comparable contributions to the total photocurrent determined by the Akz ampli- tudes [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A detailed theoretical description of the multi- band final states, treated as the time-reversed low-energy electron diffraction (LEED) states [73] within the wave- function matching approach, as well as further examples for various materials can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [78, 79] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' An insightful analysis of the multi- band final states extending into the soft-X-ray photon energies can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A rigorous analysis of 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (a) Side view of the CdTe(111) slab (b) Folded band structure of CdTe(111) 25 monolayer slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (c) Primitive unit cell of CdTe (d) bulk-unfolded band structure (e) unit cell of CdTe(111) slab used in z-unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (f) Z-unfolded band structure along k-path M − Γ − M for kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5, and (g) as a function of kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (h) FCC bulk BZ (grey), (111) unit-cell BZ (red) and (111) surface BZ (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (i) Intersecting planes slice through the bulk BZ for kz = 0 (green) and kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 (red) with the SBZ indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (j) tessellated bulk BZs showing (111) orientated intersecting planes for given kz values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' final state effects in ARPES is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Here, we will only mention that all these effects trace back to hybridization of free-electron plane waves through the higher Fourier components of the crystal po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In cases where significant kz broadening and/or final states effects are present, z-unfolding, rather than bulk unfolding, should be used in order to resolve the contributions of different kz values to the measured spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This is demonstrated for CdTe in Section III B, where the final states appear to incorporate two Bloch waves with kz = 0 and kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Computational Details DFT calculations were conducted using the Vienna Ab Initio Simulation Package (VASP) [80] with the projector augmented wave method (PAW) [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The general- ized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [83] was employed to describe the exchange-correlation interactions among electrons with a Hubbard U correction [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The U values were machine learned using Bayesian optimization (BO) [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Briefly, the BO objective function is formulated to reproduce as closely as possible the band structure obtained from the Heyd-Scuseria-Ernzerhof (HSE) [85] hybrid functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The reference HSE calculations were conducted for bulk CdTe with a lattice parameter of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='482 ˚A and α-Sn with a lattice parameter of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='489 ˚A and compared to the re- sults with the lattice constant of InSb, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='479 ˚A, which was used for interface models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' It was verified that using the lattice constant of InSb does not have an appreciable effect on the electronic properties of CdTe and α-Sn, as shown in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The hyperparameters of our BO implementation are the coefficients α1 and α2, which assign different weights to the band gap vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' the band structure in the objective function, the number of valence and conduction bands used for the calculation of the objective function, Nb, and the parameter κ that controls the balance between explo- ration and exploitation in the upper confidence bound ac- quisition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For InSb the values of U In,p eff = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 and U Sb,p eff = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 were used, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [55, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' It has been shown that PBE+U(BO) produces a band structure in good agreement with ARPES for InSb [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Because α-Sn is a semi-metal, only the band shape was considered in the optimization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' α1 was set to 0 and α2 = 1 [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='The other BO hyperparameters used for Sn were κ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 and Nb = (5, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This resulted in a value of U Sn,p eff = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='04 eV, slightly different than in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [29, 35, 61], which used empirical methods to choose a U value that yields a correct band ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [59], PBE+U(BO) reproduces the correct band ordering of α-Sn with the band inversion at the Γ point, a) full slab side view c) CdTe primitive cell e) CdTe(111) unit cell h) ↑[111] g) k²= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 k,= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 SBZ k,= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 M- 0 k_= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 CdO In z unfolded - kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='50 b) d) f) folded ) bulk unfolded z unfolded - k_ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 3 3 D 3 [111] 2 2 2 2 1 1 M 「k M 1 02 6 0 (eV) 0 0 0 出 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 1 2 2 3 3 3 4 4 5 5 M L L M M L M5 in agreement with other studies using DFT+U [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For CdTe, we applied a U correction to both the Cd-d orbitals and Te-p orbitals, unlike earlier studies [56, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The hyperparameters used for CdTe were κ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5, Nb = (5, 5), α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 and α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The latter two parameters were chosen to assign equal weights to the band gap and the band shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This led to U values of U Cd,d eff = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='381 and U T e,p eff = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The Cd-d U value obtained here is similar to the 7 eV used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [86] and somewhat lower than U Cd,d eff = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='3 eV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='21 eV, obtained here by applying the Hubbard U correction to both the Te-p states and the Cd-d states is closer to experimental values of around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 eV [87, 88] and the HSE value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='31 eV than previous calculations [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Spin-orbit coupling (SOC) was used in all calculations and dipole corrections were applied to slab models [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The tags used for convergence of calculations were BMIX = 3, AMIN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='01, ALGO = Fast, and EDIFF = 1·10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The kinetic energy cutoff was set to 400 eV for all bulk calculations and 350 eV for surface and interface slab models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A 9 × 9 × 9 k-point mesh was used for bulk calculations and a k-point mesh of 7 × 7 × 1 was used for surface and interface calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' All interface density of states (DOS) calculations used a k-point mesh of 13 × 13 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' All band structure and density of states plots were gen- erated using the open-source Python package, VaspVis [59], which is freely available from The Python Package Index (PyPI) via the command: pip install vaspvis, or on GitHub at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='com/DerekDardzinski/ vaspvis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Slab Construction All slab models were constructed using the experimen- tal InSb lattice constant value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='479 ˚A [90], assuming that the epitaxial films of CdTe and α-Sn would conform to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The length of two monolayers of a (110) slab was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5815 ˚Ain the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A vacuum region of around 40 ˚A was added to each slab model in the z- direction to avoid spurious interactions between periodic replicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The surfaces of all slab models were passivated by pseudo-hydrogen atoms such that there were no sur- face states from dangling bonds [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Despite α-Sn being a semi-metal passivation is required to remove spurious surface states, as shown in the supplemental information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The pseudo-hydrogen fractional charges utilized to passivate each atom were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='25 for In and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='75 for Sb in InSb, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 for Cd and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 for Te in CdTe, and 1 for Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Structural relaxation of the pseudo-hydrogen atoms was performed until the maximal force was below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='001 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The InSb/CdTe interface structure has In-Te and Sb-Cd bonds with each In interface atom connected to 3 Sb and 1 Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The configuration with In-Cd and Sb-Te bonds was also considered but this was found to be less stable by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='33 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Ideal interfaces were considered with no intermixing and no relaxation of the interface atoms was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' When constructing such slab models, it is necessary to converge the number of layers to avoid quantum size effects and approach the bulk properties [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For InSb it has previously been shown that 42 monolayers are suf- ficiently converged [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Plots of the band gap vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' the number of atomic layers for CdTe(110) and α-Sn (110) slabs are provided in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CdTe was deemed converged with 42 monolayers with a gap value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='23 eV, which is only slightly larger than the bulk PBE+U(BO) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The z-unfolded band structures of CdTe(111) were cal- culated for a 40 monolayer slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A 26 monolayer slab model was used to simulate the 2 × 2 reconstruction, due to the higher computational cost of the 2 × 2 su- percell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Structural relaxation was performed for the top two monolayers of the 2 × 2 reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For the slab of unstrained (110) α-Sn, 70 monolayers were needed to close the gap at the zero-gap point of the semi-metal, which corresponds to around 16 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The tri-layer slab models comprised 42 layers of InSb, 70 layers of α-Sn and between 0 and 16 layers of CdTe in two-layer increments, amounting to a total slab thickness of around 300 nm (not including vacuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The (110) bi-layer slab models comprised 42 layers of CdTe and InSb, and 70 layers of α-Sn as these were deemed converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' ARPES Experimental details The α-Sn samples were grown by molecular beam epi- taxy on an In-terminated c(8 × 2) InSb(001) surface pre- pared by atomic hydrogen cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 51 monolayers (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 nm) of α-Sn were deposited as calibrated via Rutherford backscattering spectrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Growth was performed at a substrate temperature of -20 ◦C and a base pressure bet- ter than 1·10−10 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The ARPES measurements were taken at Beamline 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 at the Advanced Light Source in Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The base pressure was better than 5·10−11 Torr while the sample temperature was held at 68 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The sample was illuminated with 63 eV p-polarized light and spectra were collected using a Scienta R4000 detector with energy resolution better than 40 meV and angular resolution better than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The sample was transferred via vacuum suitcase with a base pressure better than 10−11 Torr between the growth chamber and beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A photon energy of 63 eV corresponds to a kz approxi- mately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15 ˚A−1 above the Γ002 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' α-Sn Figure 2a shows the bulk unfolded PBE+U(BO) band structure for a 51 monolayer thick α-Sn (001) slab, com- pared to ARPES data for a sample of the same thickness taken at a photon energy of 63 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The point M is at 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Electronic structure of α-Sn: (a) Bulk-unfolded band structure of an α-Sn (001) slab with 51 atomic layers (light blue) compared with ARPES data for a sample of the same thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The point M is at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='9298 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The ARPES data is cutoff at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='9 ˚A−1 due to experimental artifacts at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Spin-polarized band structures projected onto (b) the top sur- face atoms and (c) the bottom surface atoms, indicated by the green boxes on the slab structure illustrated in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='9298 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The ARPES data is cutoff at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='9 ˚A−1 due to experimental artifacts at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The PBE+U (BO) band structure is in excellent agreement with ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The top of the valence band in the ARPES and the sim- ulated band structure lines up and the bulk bands are reproduced well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The bandwidth of the heavy hole band, Γ8, is slightly underestimated, consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This is corrected by the HSE functional, as shown in the SI for a bulk unit cell of α-Sn with a (001) orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, it is not feasible to use HSE for the large inter- face models studied here, owing to its high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The previously reported topological properties of α- Sn slabs are also observed here [27–31, 35, 36, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The spin-polarized topological surface state (TSS) is shown in panels (b) and (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 2 for a (001) 51 monolayer slab along the X − Γ − X k-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' As expected, the TSS is characterized by a linear dispersion with the top and bottom surfaces having opposite spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The associated Rashba-like surface states are also observed along the K − Γ − K k-path, as shown in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This linear surface state is also observed in the (110) slabs used to construct the bilayer and tri-layer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Notably there is an energy gap between the top and bottom TSSs, which closes at 70 layers, the same thickness at which the band gap closes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This gap is possibly induced by the hybridization of the top and bottom surface states in under-converged slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We note that the effect of strain on the electronic structure of α-Sn is not studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CdTe Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 3 shows a comparison of band structures obtained using PBE+U(BO) to the ARPES experiments of Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' [93] for CdTe(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' collected ARPES data at photon energies of 19, 25 and 30 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Here, we com- pare our results with the second-derivative maps of the ARPES data taken at 25 eV along the k-paths Γ − M (panels (a) and (b)) and Γ − K − M (panels (c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The original data has been converted to gray scale and reflected around kx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' To facilitate the quali- tative comparison of the DFT band structure features with the ARPES experiment, we apply a Fermi energy shift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='25 eV to line up the VBM and a stretch factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='22 to compensate for the bandwidth underestima- tion of PBE+U(BO), particularly for bands deep below the Fermi energy [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Bandwidth underestimation by PBE+U(BO) compared with HSE and ARPES has also been reported for InAs and InSb in [63, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The original computed band structure without the shift and stretch is provided in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Owing to the low mean free path at this photon energy, the spectrum appears integrated over a certain kz inter- val and surface contributions are readily visible in the ARPES [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' To account for the different kz contribu- tions, the z-unfolding method was employed, as described in Section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Panels (a) and (c) show the z-unfolded band structures as a function of kz for slab models with- out a surface reconstruction (figures with single values of kz are provided in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This is used determine which kz values are likely present in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A mixture of kz = 0 and k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 provides the best agreement with the ARPES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This combination of kz values is used for the DFT data shown in cyan in panels (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This is consistent with the kz broadening with contribu- tions centered around kz = 0 and k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 often present in ARPES data taken at low mean field path energies in gapped materials [71?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' , 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' To account for the presence of surface states, we mod- eled the CdTe(111)A-(2 × 2) surface reconstruction [96], illustrated in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The atom-projected band struc- tures of the bottom layer (indicated by pink dashed box) are plotted in pink in panels (b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The addi- tional bands arising from the surface reconstruction are in close agreement with the bands in the ARPES labeled as surface states by Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=', indicated by red arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' These surface states are unaffected by the choice of kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' By accounting for the contributions of different kz values and for the presence of surface states excellent agreement with experiment is achieved, as the DFT band structures reproduce all the features of the ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Bilayer Interfaces We begin by probing the local electronic structure at the the InSb/α-Sn bi-layer interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4a shows the d) 0 b) UP a) ↓ DOWN (Λa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 1 00 1 E-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 00 2 E(eV) x 3 00 UP C) (a) DOWN 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 00 4 1 E -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8 T ↑M M→ X k, (A-1)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Electronic structure of CdTe: Z-unfolded band structures of CdTe(111) compared with second-derivative map of ARPES data (black and white), adapted with permission from “Spectroscopic studies of CdTe(111) bulk and surface electronic structure” by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' B, 91, 235303 (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Copyright (2015) by the American Physical Society [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Z- unfolded band structures compared to ARPES data along (a), (b) Γ − M and (c), (d) Γ − K − M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (a), (c) Dependence of the band structure on kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (b), (d) Mixture of kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 and kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 (cyan) for a model with a 2 × 2 surface reconstruction with the contributions of the surface atoms shown in pink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' DFT has shift of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='25 eV and stretch factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='22 for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (e) Illustration of the 2 × 2 surface reconstruction with the Cd atom removed indicated by a blue circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The atoms used for the surface projection are indicated by a pink dashed box DOS as a function of position across the interface, in- dicated by the atomic layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4b shows the local DOS at select positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The Fermi level is posi- tioned at the semi-metal point of the α-Sn and in the gap of the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We note that the α-Sn appears as if it has a small gap due to an artifact of the 10−4 cutoff ap- plied in the log plot in panels (a) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The local DOS plots shown in panels (b) and (e) and the band structure plots shown in panels (c) and (f) clearly show the semi- metal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' No significant band bending is found for InSb, as expected from branching point theory [97, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Based on the element-projected band structure, shown in panel (c), the InSb conduction band minimum (CBM) lies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='09 eV above the α-Sn semi-metal point and the InSb valence band maximum (VBM) lies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='16 eV below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A linear TSS is present in the α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Based on an atom projected band structure, shown in the SI, the ori- gin of this state is the top surface of α-Sn, adjacent to the vacuum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A TSS is no longer present in the α-Sn layers at the interface with InSb, possibly owing to hybridization between the α-Sn and InSb [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Metal- induced gap states (MIGS) are an inherent property of a metal/semiconductor interface, produced by the pen- etration of exponentially decaying metallic Bloch states into the gap of the semiconductor [99–102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The pres- ence of MIGS manifests in Figure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4a as a gradually decaying non-zero DOS in the band gap of the InSb in the vicinity of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Figure 4b shows that the MIGS are prominent in the first few atomic layers and become negligible beyond 8 layers from the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4d shows the DOS as a function of position across the CdTe/α-Sn interface, indicated by the atomic layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4e shows the local DOS at select positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The Fermi level is positioned at the semi-metal point of the α-Sn and in the gap of the CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Based on the projected band structure, shown in panel (f), the CdTe CBM is positioned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='18 eV above the Fermi level and the CdTe VBM is located 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='03 eV below the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This agrees with previous reports that interfacing with Sn brings the conduction band of the CdTe closer to the Fermi energy, with downward band-bending of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='25 eV [103] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 eV [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We find a valence band offset of around 1 eV, similar to the (110) and (111) interface reported in [33, 104–108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Close to the interface there is a significant density of MIGS, which decay within about 10 layers (3-4 nm) into the CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This suggests that this number of CdTe layers may be required for an effective tunnel barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4g shows the DOS as a function of position across the InSb/CdTe interface, indicated by the atomic layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4h shows the local DOS at select positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The band alignment is type-I with the CdTe band gap straddling the InSb band-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The Fermi level is close to the InSb VBM and around the middle of the gap of the CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' No band bending is found in either material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Based on the projected band structure, shown in panel (i), the CdTe CBM lies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='28 eV above the InSb CBM and the CdTe VBM lies 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='75 eV below the InSb VBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' These values are similar to the band offsets reported in references [25, 88, 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Because the band gap of InSb is significantly smaller than that of CdTe, states from the InSb penetrate into the gap of the CdTe, similar to MIGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' These states decay gradually and vanish at a dis- tance greater than 12 layers from the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Tri-layer Interfaces Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 5 shows the DOS as a function of position across InSb/CdTe/α-Sn tri-layer interfaces with varying thick- Kz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2s OCdOIn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='50 a) b) d) c) e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 0: 0 0 1 1 2 2 (na) 3 山 4 4 4 5 5 5 5 surface states surface 6 6 19- 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' states 12 8 0 4 12 8 4 8 4 4 12 8 12 4 0 4 8 0 8 8 8 M M K IF K M M M M T M K K M k (A-1) k (A-1) k (A-1) k (A-1)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Electronic structure of bilayer interfaces: Density of states in the (a) InSb/α-Sn, (d) CdTe/α-Sn and (g) InSb/CdTe interfaces as a function of position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The atomic layers are numbered based on distance from the interface, which is located at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The structure of each interface is illustrated on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (b Local density of states for selected layers in the (b) InSb/α-Sn, (e) CdTe/α-Sn and (h) InSb/CdTe interfaces, indicated by dashed lines in the same colors in panels (a), (d), and (g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Element projected band structures of the (c) InSb/α-Sn, (f) CdTe/α-Sn and (i) InSb/CdTe interfaces, with bands originating from α-Sn colored in red, bands originating from InSb colored in light blue, and bands originating from CdTe colored in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' ness of the CdTe tunnel barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The position is indicated by the atomic layer number, with the layer of InSb clos- est to the CdTe considered as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Panels (a) and (b) show that with 6 atomic layers of CdTe, the MIGS from the α-Sn penetrate through the tunnel barrier into the first 12 layers of the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For a thin layer of CdTe, the band gap is expected to be significantly larger than the bulk value because of the quantum size effect (see the gap convergence plot in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, owing to the presence of MIGS, the gap of the CdTe remains con- siderably smaller than its bulk value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' With 10 layers of CdTe, shown in panels (c) and (d), there is still a sig- nificant presence of MIGS throughout the CdTe, which decay by 6 layers into the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Panels (e) and (f) show that with 16 layers of CdTe the InSb is completely insu- lated from MIGS coming from the α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The gap of the CdTe reaches a maximum of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='3 eV at a distance of 5 layers from the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This is because MIGS from the α-Sn penetrate into the CdTe from one side, whereas states from the InSb penetrate from the other side, such that the band gap of the CdTe never reaches its expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Figure 6 summarizes the band alignment at the bilayer and tri-layer interfaces studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For the tri-layer in- terfaces, the band alignment between the InSb and the α-Sn is not significantly affected by the presence of CdTe, as shown in the element-projected band structures in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The α-Sn semi-metal point remains pinned at the Fermi level, as in the bilayer InSb/α-Sn (see also Fig- ure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The InSb VBM remains at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='17 eV below the Fermi level, similar to its position in the bilayer interface, regardless of the CdTe thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The InSb CBM posi- tion decreases slightly with the thickness of the CdTe from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='09 eV above the Fermi level without CdTe, to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8181818818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='818 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8181818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='818 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' :1: : :1: :1: : a)( d) g) : : I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 CdTe/Sn 100 InSb/Sn InSb/CdTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 EF (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 log(DOS)( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6- 10- 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 8 -15 3 3 18 5 -12 9 6-3 18 3 -15 -12 9- 6 m- 3 0 3 6 12 0 0 9 一 b) Layers e) Layers h) Layers 5 J 5 17 Sn 12 41 41 0 4 8 17 4 (e-OL) 3 1 8 3 1 6 6 Sb DOS 4 8 2 1 2 1 2 4 0 12 Cd 一 0 4 17 一 11 1 11 4 Te 01 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6 E-E (eV) E-Ef (eV) ( f) E-Eε (eV) i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='09 Sn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 TSS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 InSb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CdTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 TSS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 E -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 x 1X 1X x9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Electronic structure of InSb/CdTe/α-Sn tri-layer interfaces: Density of states as a function of distance from the interface for (a) 6, (c) 10 and (e) 16 CdTe barrier layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The atomic layers are numbered based on distance from the interface, which is located at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Interface structures are illustrated on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' (b), (d), (f) Local density of states for selected layers, indicated by dashed lines in the same colors in panels (a), (c), and (e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Valence and conduction band edge positions for InSb and CdTe in the bilayer and tri-layer interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The Fermi level is at the semi-metal point of the α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='054 eV with 6 layers of CdTe, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='04 eV with 10 layers, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='037 eV with 16 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This may be attributed to the quantum size effect, which causes a slight narrowing of the InSb gap because of the increase in the overall size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Based on the element-projected band structures provided in the SI, the band edge positions of the CdTe are dominated by the interface with the α- Sn, rather than the interface with the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The CdTe CBM remains at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='18 eV above the Fermi level, as in the bilayer CdTe/α-Sn interface (see also Figure 4f), re- gardless of the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' As the band gap of the CdTe narrows with increasing thickness, the CdTe VBM shifts from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='24 eV below the Fermi level with 6 layers to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='105 eV with 10 layers, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 eV with 16 layers, approaching the bilayer VBM position of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='03 eV below the Fermi level with 42 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Although the band gap of the CdTe is significantly reduced due to MIGS, a type I band alignment with the InSb is maintained, similar to the bilayer InSb/CdTe interface (Figure 4g,i), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 5 panels (a), (c), and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Figure 7 show the LDOS in the second layer of InSb from the interface as a function of the number of CdTe layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Without CdTe and with two layers of CdTe, there is no band gap in the InSb close to the interface, owing to the significant density of MIGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' With 6 layers of CdTe the gap of the InSb close to the interface is still consid- erably narrower than its bulk value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The band gap in the second layer of InSb from the interface approaches its bulk value with 10 layers of CdTe and finally reaches it with 16 layers of CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This suggests that 16 CdTe layers provide an effective barrier to electronically insu- late the InSb from the α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' It is reasonable to assume that a barrier of this thickness or higher would all but eliminate transport through the interface into the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Therefore, we estimate that the relevant barrier thickness regime to modulate the coupling at an interface with a ::*:18::+::+1:++:+::+::+:: (0 a) C) 8 8 ： :i: ::: 1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' units) 10-1 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 10-2 log(DOS) ( 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='4 10-4 12-9 12 -8 -4０ 4 8 121620 12 9 6-30 3 6 9 6-3０36 ９1215 Layers Layers Layers b) d) f) 5 5 5 12 12 12 Sn 6 6 6 4 1 4 4 0 0 0 In (t-OL) 2 31 2 2 3 3 3 5 Sb DOS 6 9 8 2 1 21 2 15 Cd 11 1 Te 0 - 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='15 E- EF (eV) E- EF (eV) E- EF (eV)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 (eV) InSb CdTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 CdTe/α-Sn - InSb/α-Sn InSb/(CdTe)6/α-Sn InSb/(CdTe)10/α-Sn InSb/(CdTe)16/α-Sn InSb/CdTe Interface10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Density of states in the second InSb layer from the interface (layer -2 in Figure 5) as a function of the number of CdTe barrier layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' superconductor and tune the proximity effect would be in the range of 6-10 layers, where MIGS still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We note, however, that the interface with β-Sn may have somewhat different characteristics in terms of the band alignment and the penetration depth of MIGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' CONCLUSION In summary, we have used DFT with a Hubbard U correction machine-learned by Bayesian optimization to study CdTe as a prospective tunnel barrier at the InSb/α-Sn interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The results of PBE+U(BO) were validated by comparing the band structures of slab mod- els of α-Sn(001) and CdTe(111) with ARPES experi- ments (the PBE+U(BO) band structure of InSb(110) had been compared to ARPES experiments previously [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Excellent agreement with experiment is obtained for both materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In particular, for the low-mean-free- path ARPES of CdTe, the z-unfolding scheme success- fully reproduces the contributions of different kz values and modelling the 2 × 2 surface reconstruction success- fully reproduces the contributions of surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We then proceeded to use PBE+U(BO) to calculate the electronic structure of bilayer InSb/α-Sn, CdTe/α- Sn, and InSb/CdTe, as well as tri-layer InSb/CdTe/α-Sn interfaces with varying thickness of CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Simulations of these very large interface models were possible thanks to the balance between accuracy and computational cost provided by PBE+U(BO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We find that the most stable configuration of the InSb/CdTe interface is with In-Te and Sb-Cd bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' MIGS penetrate from the α-Sn into the InSn and CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Similarly, states from the band edges of InSb penetrate into the larger gap of the CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' No interface states are found in any of the interfaces studied here, in contrast to the EuS/InAs interface, for example, in which a quantum well interface state emerges [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For all interfaces comprising α-Sn, the semi-metal point is pinned at the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' For the tri-layer inter- face, the band alignment between the InSb and the α-Sn remains the same as in the bilayer interface regardless of the thickness of the CdTe barrier, with the Fermi level closer to the conduction band edge of the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The band edge positions of the CdTe are dominated by the inter- face with the α-Sn rather than the interface with InSb, with the conduction band edge being closer to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' A type-I band alignment is maintained between CdTe and InSb with the gap of the former straddling the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' The CBM of the CdTe is pinned whereas the VBM shifts upwards towards the Fermi level as the gap narrows with the increase in thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We find that 16 layers of CdTe (about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='5 nm) serve as an effective barrier, preventing the penetration of MIGS from the α-Sn into the InSb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' However, in the context of Majorana experiments, it is possible that a barrier thick enough to completely insulate the semiconductor from the superconductor would also all but eliminate trans- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Therefore, we estimate that the relevant regime for tuning the coupling at the interface would be in the thickness range where some MIGS are still present, while thicker CdTe layers could be used to passivate exposed InSb surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We note, however, that the interface with the superconducting β-Sn, which is not lattice matched to InSb and CdTe, may have different characteristics than the interface with α-Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' In practice, careful experimen- tation with varying barrier thickness would be needed to determine the optimal configuration for MZM devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' We have thus demonstrated that DFT simulations can provide useful insight into the electronic properties of semiconductor/tunnel barrier/metal interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This includes the interface bonding configuration, the band alignment, and the presence of MIGS (and, possibly, of interface states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Such simulations may be conducted for additional interfaces to explore other prospective mate- rial combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This may inform the choice of inter- face systems and the design of future Majorana experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' More broadly, similar DFT simulations of inter- faces may be performed to evaluate prospective tunnel barriers e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=', for semiconductor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Guang Bian from the University of Mis- souri, Li Fu from Northwestern Polytechnical University, China, and Tai C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Chiang from the University of Illinois at Urbana-Champaign for sharing their ARPES data for CdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Work at the University of Pittsburgh was sup- ported by the Department of Energy through grant DE- SC-0019274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Work at CMU and UCSB was funded by the National Science Foundation (NSF) through grant OISE-1743717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Work in Grenoble is supported by the ANR-NSF PIRE:HYBRID, Transatlantic Research Part- nership and IRP-CNRS HYNATOQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' This research used computing resources of the University of Pittsburgh Cen- ter for Research Computing, which is supported by NIH award number S10OD028483 and of the National Energy 10 0 CdTe Density of States (10-4) 2 CdTe 8 6 CdTe 10 CdTe 6 16 CdTe 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='2 E- Er (eV)11 Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfDgPC/content/2301.02879v1.pdf'} +page_content=' Department of Energy Office of Science User Facility op- erated under Contract No.' metadata={'source': 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0000000000000000000000000000000000000000..0ea9ec30ee0f9422152e249ebaffde409861f29a --- /dev/null +++ b/CtE1T4oBgHgl3EQfpwVv/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e72ed39744a5ff25861eee0773f1283471b04a97acfcfb746343bc7c9be77836 +size 7340077 diff --git a/CtFAT4oBgHgl3EQftB7D/content/tmp_files/2301.08662v1.pdf.txt b/CtFAT4oBgHgl3EQftB7D/content/tmp_files/2301.08662v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8367073582ed23701132703dc499fbda6f7de95f --- /dev/null +++ b/CtFAT4oBgHgl3EQftB7D/content/tmp_files/2301.08662v1.pdf.txt @@ -0,0 +1,1047 @@ +arXiv:2301.08662v1 [math.PR] 20 Jan 2023 +On the construction and identification of Boltzmann +processes +S. Albeverio∗, B. R¨udiger†and P. Sundar ‡ +January 23, 2023 +Abstract +Given the existence of a solution {f(t, x, v)}t≥0 of the Boltzmann equation +for hard spheres, we introduce a stochastic differential equation driven by +a Poisson random measure that depends on f(t, x, v). The marginal distri- +butions of its solution solves a linearized Boltzmann equation in the weak +form. Further, if the distributions admit a probability density, we establish, +under suitable conditions, that the density at each t coincides with f(t, x, v). +The stochastic process is therefore called the Boltzmann process. +AMS Subject Classification: 35Q20; 60H20; 60H30 +Keywords: +Boltzmann equation; Poisson random measures; stochastic +differential equations; relative entropy +1 +Introduction +The Boltzmann equation describes the time evolution of the density of molecules +in dilute (or rarified) gas for a given initial distribution. +Each molecule (or +particle) moves in a straight line without any external forces acting on it until it +collides with another particle and gets deflected. The Boltzmann equation forms +the basis for the kinetic theory of gases [6]. +∗Institute of Applied Mathematics and HCM, BiBoS, IZKS, University of Bonn, Germany. +Email: albeverio@iam.uni-bonn.de +†Bergische Universit¨at Wuppertal Fakult¨at 4 -Fachgruppe Mathematik-Informatik, Gauss +str. 20, 42097 Wuppertal, Germany. Email: ruediger@uni-wuppertal.de +‡Department of Mathematics, Louisiana State University, Baton Rouge, La 70803, USA. +Email: psundar@lsu.edu +1 + +The Boltzmann equation has the general form +∂f +∂t (t, x, z) + z · ∇xf(t, x, z) = Q(f, f)(t, x, z), +(1.1) +where f is a probability density function that depends on time t ≥ 0, the space +(location) variable x ∈ R3, and velocity, z ∈ R3. The function Q is a certain +quadratic form in f, called collision operator (or integral). +Set Ξ := (0, π] × [0, 2π). Then Q can be written in the general form +Q(f, f)(t, x, z) = +� +R3×Ξ +{f(t, x, z⋆)f(t, x, v⋆) − f(t, x, z)f(t, x, v)}B(z, dv, dθ)dφ. +(1.2) +The dynamics of collisions are encoded in B(z, dv, dθ). +Each v ∈ R3 in (1.2) +denotes the velocity of an incoming particle which may hit, at the fixed loca- +tion x ∈ R3, particles whose velocity is z. Let z⋆ ∈ R3 and v⋆ ∈ R3 denote +the resulting outgoing (post-collision) velocities corresponding to the incoming +(pre-collision) velocities z and v respectively. The angle θ ∈ (0, π] denotes the az- +imuthal or colatitude angle of the deflected velocity, v⋆, and φ ∈ [0, 2π) measures +the longitude of v∗. +In the Boltzmann equation, the collisions are assumed to be elastic and hence, +conservation of momentum and kinetic energy hold, i.e. considering particles of +mass m = 1, the following equalities hold: +� +z⋆ + v⋆ = z + v +|z⋆|2 + |v⋆|2 = |z|2 + |v|2 +(1.3) +In fact, +� +z⋆ = z + (n, v − z)n +v⋆ = v − (n, v − z)n +(1.4) +where +n = z⋆ − z +|z⋆ − z| +(1.5) +where (·, ·) denotes the scalar product, and | · |, the Euclidean norm in R3. +Remark 1.1. The Jacobian of the transformation (1.4) is 1 in magnitude, and +(z⋆)⋆ = z since the collision dynamics are reversible. +The outgoing velocity z∗ is then uniquely determined in terms of the colatitude +angle θ ∈ (0, π] measured from the center, and longitude angle φ ∈ [0, 2π) of +the deflection vector n in a sphere with north-pole z and south-pole v centered +2 + +at z+v +2 +(and with radius determined by the conserved kinetic energy) which are +used in equation (1.1) and (1.2) (see e.g. the article by H. Tanaka [19] for futher +details). It follows +(v − z, n) = |v − z| cos(π +2 − θ +2) = |v − z| sin(θ +2), +(1.6) +where θ is the angle between v − z and v⋆ − z⋆, and π +2 − θ +2 is the angle between +n and v − z. In polar coordinates we obtain +(v − z, n)dn = |v − z| sin(θ +2) cos(θ +2)dθdφ +(1.7) +The collision measure B(z, dv, dθ) is a σ-additive positive measure defined on the +Borel σ-field B(R3) × B((0, π]), for each z, and measurable in z for each fixed set +in the above Borel σ-field. The form of B depends on the version of Boltzmann +equation one has in mind. +There are several models of the Boltzmann equation (see e.g. [20]) In general, +one sets +B(z, dv, dθ) = σ(|v − z|)dvQ(dθ) +(1.8) +where σ, known as velocity cross-section, is a positive function on R+, and Q is +a σ-finite measure on B((0, π]). If Q is a finite measure, it is called the cut-off case. +To write (1.1), (1.2) in its weak form (in the functional analytic sense), we need +a result of Tanaka [18]: +Proposition 1.1. Let ψ(u, v, y, z) ∈ C0(R12), as a function of u, v, y, z ∈ R3. For +each θ ∈ (0, π] fixed +� +R6×[0,2π) +ψ(u, v, u⋆, v⋆)dφdudv = +� +R6×[0,2π) +ψ(u⋆, v⋆, u, v)dφdudv +(1.9) +Consider the Boltzmann equation (1.1) with collision operator (1.2). Multiply +(1.1) by a function ψ (of (x, z) ∈ R6) belonging to C2 +0(R6), and integrate with +respect to x and z, using integration by parts, we arrive at the weak formulation +of the Boltzmann equation: +� +R6 ψ(x, z)∂f +∂t (t, x, z)dxdz − +� +R6 f(t, x, z)(z, ∇xψ(x, z))dxdz += +� +R6 f(t, x, z)Lfψ(x, z)dxdz +(1.10) +3 + +for all t ∈ R+ with +Lfψ(x, z) = +� +R3×(0,π]×[0,2π) +{ψ(x, z⋆) − ψ(x, z)}f(t, x, v)B(z, dv, dθ)dφ, +where B is as in (1.8). +In its weak form, the Boltzmann equation cannot be treated as the Kolmogorov +equation that corresponds to a Markov process with jumps. In fact, to over- +come this obstacle we proposed and studied the Boltzmann-Enskog equation [2] +which corresponds to the dynamics of moderately dense gases. For hard and soft +potentials, solvability and uniqueness of the Boltzmann-Enskog equation were +carried out in subsequent works [11] and [12]. Indeed, there is a vast literature +on various aspects of the Boltzmann equation and we refer the reader to [7], [9], +[10], [15] [18, 19], [20], and the references therein. Standing on the shoulders of +these giants, our aim is to build a stochastic analytic treatment of the (spatially +non-homogeneous) Boltzmann equation in the non-cutoff case for hard spheres +with σ(|z−v|)= |z−v|. An ingenious suggestion given to us by Professor Presutti +provided the impetus to this work, and made the problem tractable. +Definition 1.1. A collection of densities {f(t, x, z}t∈[0,T], with x, z ∈ R3, is +a strong (resp. +weak) solution of the Boltzmann equation in [0, T] if for any +t ∈ [0, T] it solves (1.1) (resp. (1.10)). +We denote by D := D([0, T], R3) the space of all right continuous functions with +left limits on [0, T] taking values in R3, and equipped with the topology induced +by the Skorohod metric. +Definition 1.2. A stochastic process (Xs, Zs)s∈[0,T] with values on D × D, and +having time t marginals with density denoted by f(t, x, z) which solve (1.10) for +all t ∈ [0, T], is called a ”Boltzmann process”. +We remark that the infinitesimal generator of a Boltzmann process is given by +(z, ∇x) + Lf. Costantini and Marra [8] analyzed hydrodynamical limits of a pro- +cess given by a the drift term involving (z, ∇x) and Lf in addition to a martingale. +We use the following notation R0 ++ := {t ∈ R : t ≥ 0}. In this article we assume +the following hypothesis. +Hypotheses A: +A1. The measure Q on [0, π) is finite outside any neighbourhood of 0, and for +all ǫ > 0, it satisfies +� ǫ +0 +θQ(dθ) < ∞. +4 + +A2. The function σ : R0 ++ → R0 ++ (entering (1.8)) is given by σ(z) := czγ, with +c > 0, γ ∈ (−1, 1] fixed. +There are many useful consequences of A1. Let us set +α(z, v, θ, φ) := (n, (v − z))n +(1.11) +Define ˆα(z, v, θ, φ) := α(z, v, θ, φ)σ(|z − v|). +Condition A1 implies that there +exists a constant C such that the following estimates hold. +� +Ξ +|ˆα(z, v, θ, φ)|Q(dθ)dφ ≤ C|z − v|1+γ, +(1.12) +and hence +� +Ξ +|ˆα(z, v, θ, φ)|Q(dθ)dφ ≤ C(|z|1+γ + |v|1+γ). +(1.13) +Moreover, the following parameter transformation was introduced for each z ̸= v +in [18] (see also [9], Section 3, or [10]). +α(z, v, θ, φ) = 1 − cos(θ) +2 +(v − z) + sin(θ) +2 +Γ(v − z, φ) += sin2(θ +2)(v − z) + sin(θ) +2 +Γ(v − z, φ) +(1.14) +for all φ ∈ [0, 2π), where +Γ(v − z, φ) = I(v − z) cos(φ) + J(v − z) sin(φ) +and +v−z +|v−z|, I(v−z) +|v−z| , J(v−z) +|v−z| form an orthogonal basis. It follows in particular that +� 2π +0 +Γ(v − z, φ)dφ = 0. +(1.15) +In order to study solutions to the Boltzmann equation it is feasible to study +continuity properties of (u − v, n)n in u, v for fixed θ, φ. However, it was already +pointed out by Tanaka that (u, v) �−→ (u−v, n)n cannot be smooth. To overcome +this problem Tanaka introduced in Lemma 3.1 of [18] another transformation of +parameters, which describes a rotation around the longitude angle, is bijective and +has Jacobian 1. As a consequence of this transformation φ0 he proved following +Lemma 1.1 (see also Lemma 2.6 in [16]). +Lemma 1.1. [18][10] There exists a measurable function φ0 : R12 → (0, 2π] such +that +|Γ(v − z, φ) − Γ(v′ − z′, φ + φ0(z, v, z′, v′))| ≤ 3|z − v − (z′ − v′)| +(1.16) +5 + +and hence +|α(z, v, θ, φ) − α(z′, v′, θ, φ + φ0(z, v, z′, v′))| ≤ 2θ(|z − z′| + |v − v′|) +(1.17) +and +|α(z, v, θ, φ)| ≤ 2θ(|z| + |v|) +(1.18) +Moreover by his transformation [18] (see also [10]), Section 3) and using (1.14) +Tanaka proved the following inequality: +� π +0 +��� +� 2π +0 +α(z, v, θ, φ) − α(z′, v′, θ, φ)dφ +��� Q(dθ) +≤ C(|z − z′| + |v − v′|). +(1.19) +where by an abuse of notation we use the same symbol C > 0 in (1.13) and (1.19), +even though the constants are different. +Let {f(t, x, z)}t∈R0 ++ be a collection of densities on (R3 × R3, B(R3 × R3)). Let +us introduce the operator Qt(f, f)(·) defined through the right side of equation +(1.10) +Qt(f, f)(ψ) := +� +R6 f(t, x, z)Lfψ(x, z)dxdz +(1.20) +It is easy to verify that +Qt(f, f)(ψ) = 0 +for +ψ(x, z) = a + (b, z) + c|z|2 +(1.21) +∀ a, c ∈ R , b ∈ R3. (For a rigorous proof see Chapter II.7 [5] or [7], [4].) +The integral form of equation (1.10) corresponds to +� +R6 ψ(x, z)f(t, x, z)dxdz = +� +R6 ψ(x, z)f(0, x, z)dxdz ++ +� t +0 +� +R6 f(t, x, z){(z, ∇xψ(x, z)) + Lfψ(x, z)}dxdzds, +(1.22) +It is worthwhile to note that if a second collection of densities {g(t, x, z)}t∈R0 ++ on +(R3 × R3, B(R3 × R3)) is given, then +QS +t (f, g)(ψ) = 0 +for +ψ(x, z) = a + (b, z) + c|z|2 +(1.23) +∀ a, c ∈ R , b ∈ R3 with the operator QS +t defined through +QS +t (f, g)(·) := Qt(f, g)(·) + Qt(g, f)(·) +(1.24) +6 + +with +Qt(f, g)(ψ) := +� +R6 g(t, x, z)Lf ψ(x, z)dxdz. +(1.25) +The following Povzner type inequality is essentially contained in [15, Lemma 3.6]. +(See also [7], Theorem 6.2.1 and Appendix B of Chapter 6 for p ≥ 2 and references +there.) +Lemma 1.2. For all θ ∈ (0, π], p ≥ 2 and γ ∈ (0, 1], +� 2π +0 +� +⟨z + α(z, v, θ, φ)⟩2p + ⟨v − α(z, v, θ, φ)⟩2p − ⟨z⟩2p − ⟨v⟩2p� +dφ +≤ −sin2(θ) +2 +� +⟨v⟩2p + ⟨z⟩2p� ++ Cp sin2(θ) +⌊ p+1 +2 ⌋ +� +k=1 +� +⟨v⟩2k⟨z⟩2p−2k + ⟨v⟩2p−2k⟨z⟩2k� +, +where ⟨v⟩ := (1 + |v|2) +1 +2 , ⌊x⌋ ∈ Z is defined by ⌊x⌋ ≤ x < ⌊x⌋ + 1 and Cp > 0 is +some constant. +Using conservation laws and Lemma 1.2, it can be proven that if {f(t, x, u)}t∈[0,T] +is a weak solution of the Boltzmann equation in [0, T], with initial finite second +moment, i.e. +� +R6 |z|2f(0, x, z)dxdz < ∞, then for all p ≥ 1 +� +R6 |z|pf(t, x, z)dxdz < ∞ +∀t ∈ [0, T]. +(1.26) +For a proof we refer the reader to [15, Theorem 3.6]. +Let {µt(dx, dz)}t∈R0 ++ be a collection of probabilities on (R3 × R3, B(R3 × R3)). +Let us define the operator +Qt(f, µ)(ψ) := +� +R6 Lfψ(x, z)µt(dx, dz). +(1.27) +acting on all ψ for which the integral on the right side is finite. +Lemma 1.3. +Qt(f, µ)(|z|2) += +� +R9×Ξ +(|v|2 − |z|2)σ(|z − v|) sin2(θ +2)Q(dθ)dφf(t, x, v)dvµt(dx, dz) +(1.28) +Proof. +Qt(f, µ)(|z|2) = +� +R6 Lf|z|2µt(dx, dz) +(1.29) += +� +R9×Ξ +(|z⋆|2 − |z|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz). +7 + +Moreover, +Qt(f, µ)(|z|2) = +� +R9×Ξ +(|z⋆|2 + |v⋆|2 − |z|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) +− +� +R9×Ξ +(|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) += − +� +R9×Ξ +(|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz). +(1.30) +where in the last equality we used that the kinetic energy is conserved during the +elastic collision, see (1.3). +Combining equation (1.29) and (1.30), we obtain +Qt(f, µ)(|z|2) = +1 +2 +� +R9×Ξ +(|z⋆|2 − |z|2 − (|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) += 1 +2 +� +R9×Ξ +(|z|2 + 2(z, α) + |α|2 − |z|2) − (|v|2 − 2(v, α) + |α|2 − |v|2) +× σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) += +� +R9×Ξ +(z + v, α)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz), +(1.31) +Using the parametrization (1.14) for α = α(z, v, θ, φ) and (1.15) we obtain +Qt(f, µ)(|z|2) = ++ +� +R9×Ξ +(z + v, v − z) sin2(θ +2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) += − +� +R9×Ξ +(|z|2 − |v|2) sin2(θ +2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) +2 +The Boltzmann process +We use the following notation throughout the paper. U0 = D × [0, π) × (0, 2π]. +Let {f(t, x, v)}t∈R0 ++ be a collection of densities on (R3 × R3, B(R3 × R3)). Then +m(t, v) denotes the marginal density of velocity v at time t, i.e. +m(t, v) := +� +R3 f(t, x, v)dx so that f(t, x|v)m(t, v) := f(t, x, v), upon disintagration of mea- +sures. +8 + +Hypotheses B: We assume that t → f(t, x, v) is differentiable for each x, v +∈ R3 fixed, and satisfies +B0. |∂f +∂t | is bounded on any compact subset of R0 ++ × R6. +B1. +∂f +∂t (t, ·) ∈ L1(R6), +∀t ∈ R0 ++, +B2. supx∈R3 +� +R3 |u|1+γf(s, x, u)du ∈ C([0, T]) +∀ T > 0. +B3. supx∈R3 +� +R3 |u|1+γ ∂ +∂tf(t, x, u)du ∈ L1([0, T]) +∀ T > 0. +Theorem 2.1. Let {f(t, x, v)}t∈R0 ++ be a collection of densities which satisfies +hypothesis B. Suppose hypothesis A hold. +Let X0 and Z0 be R3- valued ran- +dom variables. Suppose that for any fixed T > 0 there exists a stochastic basis +(Ω, F, (Ft)t∈[0,T], P), an adapted process (Xt, Zt)t∈[0,T] with values on D × D, +which has time marginals with density f(t, x, u), and such that it satisfies a.s. +the following stochastic equation for t ∈ [0, T]: + + + + + + + + + +Xt = X0 + +� t +0 +Zsds +Zt = Z0 + +� t +0 +� +U0×R0 ++ +α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)dN, +(2.1) +where in the above equation, dN := N(dv, dθ, dφ, dr, ds) is a Poisson random +measure with compensator m(s, v)dvQ(dθ)dφdsdr. Then (Xt, Zt)t∈[0,T] is a Boltz- +mann process. +Proof. From (1.13) it follows for each T > 0 +� T +0 +E[ +� +U0×R0 ++ +|α(Zs, vs, θ, φ)|1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)m(s, v)dvQ(dθ)dφdr]ds += +� T +0 +E[ +� +U0 +|ˆα(Zs, vs, θ, φ)|f(s, Xs, v)dvQ(dθ)dφ]ds +≤ C +� T +0 +� +R9(|z|1+γ + |v|1+γ)f(s, x, z)f(s, x, v)dxdzdvds, +≤ 2C +� T +0 +sup +x∈R3 +� +R6 +� +|z|1+γf(s, x, z)dz +� +f(s, x, v)dvdxds < ∞. +for some constant C > 0. In the above estimates we have used that the function +f(t) is the probability density of the process (Xt, Zt), as well the assumption A2 +and B2. It follows that we can apply the Itˆo formula to (Xs, Zs)s∈R+ [17]. In +9 + +fact let t, ∆t > 0, ψ ∈ C2 +0(R3 × R3), then +ψ(Xt+∆t, Zt+∆t) += ψ(Xt, Zt) + +� t+∆t +t +(Zs, ∇xψ(Xs, Zs))ds ++ +� t+∆t +t +� +U0×R+ +0 +{ψ(Xs, Zs + α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)) − ψ(Xs, Zs)}dN +It follows +E[ψ(Xt+∆t, Zt+∆t) − ψ(Xt, Zt)] = +E +�� t+∆t +t +(Zs, ∇xψ(Xs, Zs))ds +� ++ +E + + +� t+∆t +t +� +U0 +{ψ(Xs, Zs+ α(Zs, vs, θ, φ))−ψ(Xs, Zs)}σ(|Zs−vs|)f(s, Xs,vs)dvQ(dθ)dφds + + +Upon dividing by ∆t on both sides, we obtain +lim +∆t↓0 +1 +∆t +� +R6 ψ(x, u){f(t + ∆t, x, u) − f(t, x, u)}dxdu += lim +∆t↓0 +1 +∆t +� t+∆t +t +� +R6(u, ∇xψ(x, u))f(s, x, u)dxduds + +lim +∆t↓0 +1 +∆t +� t+∆t +t +� +R6×R3×[0,π)×(0,2π] +{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)} +× σ(|u − v|)f(s, x, v)f(s, x, u)dvQ(dθ)dφdxduds +(2.2) +Letting ∆t → 0 in every term of (2.2) we obtain (1.10). Indeed, for e.g., the +second term on the right side of (2.2) and prove the continuity of the function +g(s) := +� +R6×R3×[0,π)×(0,2π]{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)} +×σ(|u − v|)f(s, x, v)f(s, x, u)dvQ(dθ)dφdxdu +Since +{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)} ≃ ∇uψ(x, z)α(u, v, θ, φ) +with (x.z) ∈ K compact set, and +|α(u, v, θ, φ)|σ(|u − v|) ≤ |u − v|1+γ| sin(θ +2)|, +by denoting with F a compact set in R3 which includes all projections x of +(x.z) ∈ K, it follows that +|g(s) − g(s0)| ≤ C +� +F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)| +×(|u|γ+1 + |v|γ+1)dxdudv, +(2.3) +10 + +with +C := ∥∇uψ∥∞2π +� π +0 +θQ(dθ) +We split the integral on the right side of (2.3) into two terms, one with |u|γ+1 +(resp. |u|γ+1), and get +� +F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)||u|γ+1dxdudv += +� +F ×R6 |f(s, x, u)f(s, x, v) − f(s, x, u)f(s0, x, v)||u|γ+1dxdudv ++ +� +F ×R6 |f(s, x, u)f(s0, x, v) − f(s0, x, u)f(s0, x, v)||u|γ+1dxdudv += J1(s) + J2(s) +(2.4) +where J1(s) (resp. J2(s)) is the first (resp. second) term on the right side of +(2.4). +J1(s) = +� +F ×R6 |u|γ+1f(s, x, u)| +� s +s0 +∂f +∂r (r, x, v)dr|dxdudv +≤ +� +supx∈R3 +� +R3 |u|γ+1f(s, x, u)du +� � s +s0 +� +R6 |∂f +∂r (r, x, v)|dxdvdr +By B1 and B2 lims→s0 J1(s) = 0. +Let us consider J2(s). +J2(s) = +� +F ×R6 |u|γ+1f(s0, x, v)| +� s +s0 +∂f +∂r (r, x, u)dr|dxdudv +≤ +� s +s0 +� +F ×R3 +�� +R3 |u|γ+1|∂f +∂r (r, x, u)|du +� +f(s0, x, v)dxdvdr +Since supx∈R3 +� +R3 |u|γ+1|∂f +∂r (r, x, u)|du is integrable in [s0, s] by B3, we obtain +lims→s0 J2(s) = 0. +Likewise, and without any changes in the arguments it follows +lim +s→s0 C +� +F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)||v|γ+1dxdudv = 0 +Hence lims→s0 g(s) = g(s0), so that g is a continuous function and +lim +∆t↓0 +1 +∆t +� t+∆t +t +g(s)ds = g(t). +Note that in the above arguments we have taken s > s0 for simplicity. One may +also take s0 > s. +Theorem 2.1 motivates the following Definition. +Definition 2.1. Let {f(t, x, v)}t∈R0 ++ be a collection of densities satisfying Hy- +pothesis B. Suppose that for any fixed T > 0, there exists a stochastic basis +(Ω, F, (Ft)t∈[0,T], P) and an adapted process (Xt, Zt)t∈[0,T] with values on D × D +such that +11 + +i) (Xt, Zt)t∈[0,T] has time marginals with density f(t, x, u), for t ∈ [0, T], +ii) (Xt, Zt)t∈[0,T] is a solution of the McKean -Vlasov SDE (2.1). +Then we say that “the McKean -Vlasov equation (2.1) with density functions +{f(t, x, v)}t∈R0 ++ is associated to the the Boltzmann equation (1.1)”. +If the above property holds for T ∈ [0, S] with S > 0, then the McKean -Vlasov +SDE (2.1) with density functions {f(t, x, v)}t∈[0,S] is associated to the Boltzmann +equation (1.1) up to time S. +Remark 2.1. Let us assume hypothesis A. From Theorem 2.1 it follows that +any stochastic process (Xt, Zt)t∈[0,T] solving a McKean -Vlasov equation (2.1) +associated to the Boltzmann equation (1.10) is (according to Definition 1.2) a +Boltzmann process. +The Boltzmann equation (1.10) is hence the Kolmogorov +equation associated to the McKean -Vlasov equation (2.1). +3 +Existence of the Boltzmann process +In Theorem 2.1 we proved that any process (Xt, Zt)t∈[0,T] solving the McKean +-Vlasov equation (2.1) associated to (1.10) in [0, T] is a Boltzmann process. In +this section we analyze the following: given a strong solution {f(t, x, z)}t∈[0,T] of +the Boltzmann equation (1.1), we find sufficient conditions for the existence of +a solution of the McKean -Vlasov equation (2.1) with density {f(t, x, z)}t∈[0,T]. +The solution process (Xt, Zt)t∈[0,T] is then a Boltzmann process. +We present an overview on the construction of Boltzmann processes. We briefly +outline the construction of the process (Xt, Zt)t∈[0,T] under suitable conditions +before stating the main result on Boltzmann processes. The proofs of the ensuing +results on the existence of a solution to a certain linearized stochastic system will +appear in a separate paper [1]. +3.1 +Construction of a solution of a SDE defined through a col- +lection of densities solving (1.1) +In this paragraph, we assume that {f(t, x, z)}t∈[0,T] is a collection of densities +which solves the Boltzmann equation (1.1) and satisfies the following conditions: +B4. sups∈[0,T],x∈R3 +� +R3 f(s, x, v)dv ≤ CT < ∞. +12 + +B5. There exists for every K > 0 a constant CK +T > 0 such that +sup +s∈[0,T],|x|≤K +� +R3 max(1, |v|1+γ)|∇xf(s, x, v)|dv ≤ CK +T < ∞. +B6. sups∈[0,T],x∈R3 +� +R3 |v|γ+2f(s, x, v)dv ≤ cT < ∞. +On any fixed filtered probability space (Ω, F, (Ft)t∈[0,T], P) satisfying the usual +conditions, let ST := S1 +T (Rd) denote the linear space of all adapted c`adl`ag pro- +cesses (Xt)t∈[0,T] with values on Rd equipped with norm +∥X∥S1 +T := E[ sup +s∈[0,T] +|Xs|]. +(3.1) +Under hypotheses B4 - B6, and adopting the notation f(s, x, v) = f(s, x | v)m(s, v) +upon disintegration of measures, we first prove the existence of a weak solution +to the stochastic system +Xt = X0 + +� t +0 +Zsds, +∀t ∈ [0, T] +(3.2) +Zt = Z0 + +� t +0 +� +U0×R+ +0 +α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)dN +∀t ∈ [0, T] +(3.3) +for t ∈ [0, T] where dN := N(dv, dθ, dφ, dr, ds) with its compensator given by +m(s, v)dvQ(dθ)dφdsdr with values in S1 +T := S1 +T (R3 × R3). +Here we do not assume that (3.3) is of McKean -Vlasov type. +First, we recall the definition of weak solutions in the context of stochastic analysis +[14]. +Definition 3.1. A ”weak solution” of equation ((3.3), (3.2)) in the time inter- +val [0, T] is a triplet ((Ω, F, (Ft)t∈[0,T], P), N(dv, dθ, dφ, dr, ds), (Xt, Zt)t∈[0,T]) +for which the following properties hold: +• (Ω, F, (Ft)t∈[0,T], P) is a stochastic basis; +• N(dv, dθ, dφ, dr, ds) is an adapted Poisson random measure with compen- +sator m(s, v)dvQ(dθ)dφdsdr; +• (X·, Z·) := (Xt, Zt)t∈[0,T]) is an adapted c`adl`ag stochastic process with val- +ued in Rd × Rd which satisfies ((3.3), (3.2)) P -a.s. +The existence of solutions to the stochastic system (3.3),(3.2) is stated in the +following theorem, proven in [1]. +13 + +Theorem 3.1. Let γ = 1 and Hypothesis A be satisfied. +Let T > 0 and +{f(t, x, v)}t∈[0,T] be a collection of densities which satisfy f(t, x, u) ∈ C([0, T] × +R6) and Hypotheses B. Let the initial distribution of (X0, Z0) admit finite second +moment. There exists a weak solution +((Ω, F, (Ft)t∈[0,T], P), N(dv, dθ, dφ, dr, ds), (Xt, Zt)t∈[0,T]) +of (3.2), (3.3) such that (X·, Z·) ∈ S1 +T . Moreover, +sup +t∈[0,T] +E[|Xt|2] + sup +t∈[0,T] +E[|Zt|2] < ∞ +(3.4) +We remark that the estimate (3.4) is proven by symmetry arguments similar to +those appearing in the proof of Lemma 1.2. The form of the stochastic system +(3.2), (3.3) with the process taking values in R6 at each t ∈ [0, T], one obtains +that the solution lies in D × D. +3.2 +Construction of Boltzmann processes with densities satisfy- +ing (1.1) +We recall the concept of relative entropy which plays a key role in the proof of +the following theorem. Recall that for any two probability measures µ, ν on a +common measurable space (X, X), the relative entropy of ν with respect to µ, +denoted R(ν || µ), is defined by +R(ν || µ) = +� +X +� +log dν +dµ +� +dν +if ν is absolutely continuous with respect to µ. Otherwise, we set R(ν || µ) = ∞. +The following Lemma is well known. +Lemma 3.2. Let µ, ν be two probability measures on a measurable space (X, X). +Then R(ν || µ) ≥ 0 and R(ν || µ) = 0 if and only if µ = ν. +We assume that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the +Boltzmann equation (1.1) and satisfies hypotheses B as well as the following +condition: +C1. The densities f(t, x, z) and g(t, x, z) are in C1,2([0, T] × R6) and are strictly +positive-valued functions with g log g, g log f ∈ L1(R6) for each t ∈ [0, T] and +lim|x|→∞ g(t, x, z) = 0 and g(0, x, z) = f(0, x, z) a.s. +Theorem 3.3. Let (Xt, Zt)t∈[0,T] be a stochastic process that solves the stochastic +system (3.2), (3.3). Suppose that (Xt, Zt)t∈[0,T] has time marginals with density +g(t, x, z), for each t ∈ [0, T]. Suppose that {f(t, x, z)}t∈[0,T] and {g(t, x, z)}t∈[0,T] +satisfy hypotheses B0 − B6 and C1. Then g(t, x, z) = f(t, x, z) +a.e. for all +t ∈ [0, T]. +14 + +Proof. We will write R(g || f) for the relative entropy of the measure with prob- +ability density g with respect to the measure with probability density f. The +theorem is proved by establishing the following equality. +Rt(g|f) := +� +R6 log +� g(t.x.z) +f(t, x, z) +� +g(t, x, z)dxdz = 0 +∀t ∈ [0, T] +(3.5) +We first apply the Itˆo formula [13] to log(g(t, Xt, Zt)), where (Xt, Zt)t∈[0,T] solves +(3.3), (3.2) and take expectation to obtain +� +R6 log (g(t, x, z))g(t, x, z)dxdz − +� +R6 log (g(0, x, z))g(0, x, z)dxdz += +� t +0 +� +R6×R3×Ξ +{log (g(s, x, z⋆)) − log (g(s, x, z))} +× σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds. +(3.6) +Indeed, in arriving at (3.6), we have used the following two calculations: +(i) +� t +0 +� +R6 +∂ +∂s log (g(s, x, z))g(s, x, z)dxdzds = +� t +0 +� +R6 +∂ +∂sg(s, x, z)dxdzds += +� +R6(g(t, x, z) − g(0, x, z))dxdz = 0 +since g is a probability density. +(ii) +� +R6 z · ∇x log (g(s, x, z))g(s, x, z)dxdzds = +� +R6 z · ∇xg(s, x, z)dxdzds = 0 +where the last equality is obtained by integrating and using the condition that +lim +|x|→∞g(t, x, z) = 0. Likewise, one obtains upon taking expectation and recall- +ing that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the Boltzmann +equation (1.1), +� +R6 log (f(t, x, z))g(t, x, z)dxdz − +� +R6 log (f(0, x, z))g(0, x, z)dxdz += +� t +0 +� +R6 +Q(f, f)(s, x, z) +f(s, x, z) +g(s, x, z)dxdzds ++ +� t +0 +� +R6×R3×Ξ +{log (f(s, x, z⋆)) − log (f(s, x, z))} +× σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds += +� t +0 +� +R9×Ξ +{g(s, x, z⋆) +f(s, x, z⋆) − g(s, x, z) +f(s, x, z)} +× σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds , ++ +� t +0 +� +R6×R3×Ξ +{log (f(s, x, z⋆)) − log (f(s, x, z))} +× σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds. +(3.7) +15 + +It is worthwhile to note that the last equality in the above display results upon +rewriting +� t +0 +� +R6 +Q(f, f)(s, x, z) +f(s, x, z) +g(s, x, z)dxdzds += +� t +0 +� +R9×Ξ +{f(s, x, z⋆)f(s, x, v⋆) − f(s, x, z)f(s, x, v)} +× σ(|z − v|) g(s, x, z) +f(s, x, z)Q(dθ)dφdvdxdzds += +� t +0 +� +R9×Ξ +{g(s, x, z⋆) +f(s, x, z⋆) − g(s, x, z) +f(s, x, z)} +× σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds , +by using Proposition 1.1. +Combining equations (3.6) with (3.7) we obtain that +Rt(g|f) = +� +R6 log (g(t, x, z))g(t, x, z)dxdz − +� +R6 log (f(t, x, z))g(t, x, z)dxdz += +� t +0 +� +R9×Ξ +� g(s, x, z) +f(s, x, z){1 + log +�g(s, x, z⋆)/f(s, x, z⋆) +g(s, x, z)/f(s, x, z) +� +} − g(s, x, z⋆) +f(s, x, z⋆) +� +× σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds +(3.8) +Transforming (3.8) in the equivalent equation below, and recalling that for x ≥ 0 +we have 1 + log (x) − x ≤ 0, we easily see that +Rt(g|f) = +� t +0 +� +R9×Ξ +� +1 + log +�g(s, x, z⋆)/f(s, x, z⋆) +g(s, x, z)/f(s, x, z) +� +− g(s, x, z⋆)/f(s, x, z⋆) +g(s, x, z)/f(s, x, z) +� +× g(s, x, z) +f(s, x, z)σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds +≤ 0. +However, by Lemma 3.2, Rt(g|f) ≥ 0, and hence, Rt(g|f) = 0. +From Theorem 3.3 it follows that (Xt, Zt)t∈[0,T] in Theorem 3.3 solves the McKean- +Vlasov equation associated to the Boltzmann equation (1.1) and is a Boltzmann +process with densities {f(t, x, z)}t∈[0,T] up to time T. +Based on Theorem 3.1, the main result of this work is given below: +Theorem 3.4. Let Hypotheses A be satisfied and γ = 1. Assume that {f(t, x, u)}t∈[0,T] +is a collection of densities which solves the Boltzmann equation (1.1), and satis- +fies the hypotheses B0 − B6. Let the random vector (X0, Z0) have finite second +16 + +moment. Suppose that the weak solution of the stochastic system (3.3), (3.2) has +its distribution that admits a probability density at each time t ∈ [0, T] given by +g(t, x, u). If condition C1 is satisfied by {f(t, x, u)}t∈[0,T] and {g(t, x, u)}t∈[0,T], +then the McKean-Vlasov equation (2.1) (that involves {f(t, x, u)}t∈[0,T]) has a +weak solution in [0, T] with values in D × D, and its Kolmogorov equation solves +equation (1.1). +Proof. The result follows from Theorem 3.1 and Theorem 3.3. +Acknowledgments: The authors are very thankful to Professor Errico Presutti +for suggesting that a given solution of the Boltzmann equation be used in order to +construct a Boltzmann process. The second author considers herself blessed for +having had the opportunity to write her Doctoral Thesis under the supervision +of Errico Presutti. +References +[1] S. Albeverio, B. R¨udiger, P. Sundar, Boltzmann processes and their construc- +tion. In preparation (2023). +[2] S. Albeverio, B. R¨udiger, P. Sundar, The Enskog Process, J. Stat. Phys. +167(1), 90-122 (2017). +[3] Boltzmann, L.: Vorlesungen ¨uber Gastheorie. (1896) J. A. Barth, Leipzig, +Part I; Part II. (1898) transl. by S. B. Brush, Lectures on Gas Theory. Univ. +Calif. Press, Berkeley (1964). +[4] Bressan, A. : Notes on the Boltzmann equation. Lecture Notes for a Summer +Course given at S.I.S.S. A. 2005. +[5] Cercignani, C.: Theory and application of the Boltzmann Equation and its +Applications. Scottish Academic Press Edinburgh and london (1975). +[6] Cercignani C.: The Boltzmann Equation and its Applications. Springer Ver- +lag, New York (1988). +[7] Cercignani, C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute +Gases. Applied Mathematical Sciences Vol. 106, Springer Verlag (1994). +[8] Costantini, C., Marra, R.: Hydrodynamic limits for the Boltzmann process, +J. Stat. Phys. (1-2), 67, 229–249 (1992). +[9] Fournier N., Finiteness of entropy for the homogenous Boltzmann equation +with measure initial condition, The Annals of Applied Probabilty Vol. 25. No +2. 860 -897 (2015). +17 + +[10] Fournier N., Mouhot C., On the Well -Posedness of the Spatially Homoge- +nous Boltzmann Equation with a moderate Angular Singularity. Commun. +Math. Phys. 289, 803 -824 (2009). +[11] Friesen, M., R¨udiger, B., Sundar, P., The Enskog process for hard and soft +potentials, Nonlinear Differential Equations and Applications, 26, Art. no. 20 +(42 pages) (2019). +[12] Friesen, M., R¨udiger, B., Sundar, P., On uniqueness and stability for the +Boltzmann–Enskog equation, Nonlinear Differential Equations and Applica- +tions, 29, Art. no. 25 (25 pages) (2022). +[13] Ikeda, N., Watanabe, S., Stochastic Differential Equations and Diffusion Pro- +cesses (second edition), North-Holland Publishing Co., Amsterdam, Oxford, +New York (1989). +[14] Karatzas I., Shreve S.E.: Brownian motion and stochastic calculus (second +edition). Graduate Texts in Mathematics 113. Springer Verlag, Berlin, New +York (1991) +[15] Lu, X., , Mouhot C.: On measure solutions of the Boltzmann equation, part +I: moment production and stability estimates. J. Diff. Equ. 252 (4), 3305 -3363 +(2012) +[16] Mandrekar V. , R¨udiger B.: Stochastic Integration in Banach spaces, Theory +and Applications. Probability Theory and Stochastic Modelling, Springer, +Berlin (2015). +[17] R¨udiger, B., Ziglio, G.: Itˆo formula for stochastic integrals w.r.t. compen- +sated Poisson random measures on separable Banach spaces. Stochastics 78 +(6), 377–410 (2006). +[18] Tanaka, +H.: +Probabilistic treatment +of the Boltzmann +equation of +Maxwellian molecules. Z. Wahr. verw. Gebiete 46, 67-105 (1978). +[19] Tanaka, H.: Stochastic differential equations corresponding to the spatially +homogeneous Boltzmann equation of Maxwellian and non cut-off type. J. Fac. +Sci. Univ Tokyo, Sect. A, Math. 34, 351-369 (1987). +[20] Villani, C.: A review of mathematical topics in collision kinetic theory. Hand- +book of mathematical fluid dynamics, Vol. I. pages 71 -305. North -Holland, +Amsterdam 2002. +18 + diff --git a/CtFAT4oBgHgl3EQftB7D/content/tmp_files/load_file.txt b/CtFAT4oBgHgl3EQftB7D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2cd6d28c92dcc3f502edc0bfe1ff8102442e82a --- /dev/null +++ b/CtFAT4oBgHgl3EQftB7D/content/tmp_files/load_file.txt @@ -0,0 +1,552 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf,len=551 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='08662v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='PR] 20 Jan 2023 On the construction and identification of Boltzmann processes S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Albeverio∗, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' R¨udiger†and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Sundar ‡ January 23, 2023 Abstract Given the existence of a solution {f(t, x, v)}t≥0 of the Boltzmann equation for hard spheres, we introduce a stochastic differential equation driven by a Poisson random measure that depends on f(t, x, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The marginal distri- butions of its solution solves a linearized Boltzmann equation in the weak form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Further, if the distributions admit a probability density, we establish, under suitable conditions, that the density at each t coincides with f(t, x, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The stochastic process is therefore called the Boltzmann process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' AMS Subject Classification: 35Q20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 60H20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 60H30 Keywords: Boltzmann equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Poisson random measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' stochastic differential equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' relative entropy 1 Introduction The Boltzmann equation describes the time evolution of the density of molecules in dilute (or rarified) gas for a given initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Each molecule (or particle) moves in a straight line without any external forces acting on it until it collides with another particle and gets deflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The Boltzmann equation forms the basis for the kinetic theory of gases [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ∗Institute of Applied Mathematics and HCM, BiBoS, IZKS, University of Bonn, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Email: albeverio@iam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='uni-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='de †Bergische Universit¨at Wuppertal Fakult¨at 4 -Fachgruppe Mathematik-Informatik, Gauss str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 20, 42097 Wuppertal, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Email: ruediger@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='de ‡Department of Mathematics, Louisiana State University, Baton Rouge, La 70803, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Email: psundar@lsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='edu 1 The Boltzmann equation has the general form ∂f ∂t (t, x, z) + z · ∇xf(t, x, z) = Q(f, f)(t, x, z), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) where f is a probability density function that depends on time t ≥ 0, the space (location) variable x ∈ R3, and velocity, z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The function Q is a certain quadratic form in f, called collision operator (or integral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Set Ξ := (0, π] × [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Then Q can be written in the general form Q(f, f)(t, x, z) = � R3×Ξ {f(t, x, z⋆)f(t, x, v⋆) − f(t, x, z)f(t, x, v)}B(z, dv, dθ)dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) The dynamics of collisions are encoded in B(z, dv, dθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Each v ∈ R3 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) denotes the velocity of an incoming particle which may hit, at the fixed loca- tion x ∈ R3, particles whose velocity is z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let z⋆ ∈ R3 and v⋆ ∈ R3 denote the resulting outgoing (post-collision) velocities corresponding to the incoming (pre-collision) velocities z and v respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The angle θ ∈ (0, π] denotes the az- imuthal or colatitude angle of the deflected velocity, v⋆, and φ ∈ [0, 2π) measures the longitude of v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In the Boltzmann equation, the collisions are assumed to be elastic and hence, conservation of momentum and kinetic energy hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' considering particles of mass m = 1, the following equalities hold: � z⋆ + v⋆ = z + v |z⋆|2 + |v⋆|2 = |z|2 + |v|2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) In fact, � z⋆ = z + (n, v − z)n v⋆ = v − (n, v − z)n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4) where n = z⋆ − z |z⋆ − z| (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='5) where (·, ·) denotes the scalar product, and | · |, the Euclidean norm in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The Jacobian of the transformation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4) is 1 in magnitude, and (z⋆)⋆ = z since the collision dynamics are reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The outgoing velocity z∗ is then uniquely determined in terms of the colatitude angle θ ∈ (0, π] measured from the center, and longitude angle φ ∈ [0, 2π) of the deflection vector n in a sphere with north-pole z and south-pole v centered 2 at z+v 2 (and with radius determined by the conserved kinetic energy) which are used in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' the article by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Tanaka [19] for futher details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' It follows (v − z, n) = |v − z| cos(π 2 − θ 2) = |v − z| sin(θ 2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6) where θ is the angle between v − z and v⋆ − z⋆, and π 2 − θ 2 is the angle between n and v − z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In polar coordinates we obtain (v − z, n)dn = |v − z| sin(θ 2) cos(θ 2)dθdφ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='7) The collision measure B(z, dv, dθ) is a σ-additive positive measure defined on the Borel σ-field B(R3) × B((0, π]), for each z, and measurable in z for each fixed set in the above Borel σ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The form of B depends on the version of Boltzmann equation one has in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' There are several models of the Boltzmann equation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' [20]) In general, one sets B(z, dv, dθ) = σ(|v − z|)dvQ(dθ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='8) where σ, known as velocity cross-section, is a positive function on R+, and Q is a σ-finite measure on B((0, π]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' If Q is a finite measure, it is called the cut-off case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' To write (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) in its weak form (in the functional analytic sense), we need a result of Tanaka [18]: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let ψ(u, v, y, z) ∈ C0(R12), as a function of u, v, y, z ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' For each θ ∈ (0, π] fixed � R6×[0,2π) ψ(u, v, u⋆, v⋆)dφdudv = � R6×[0,2π) ψ(u⋆, v⋆, u, v)dφdudv (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='9) Consider the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) with collision operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Multiply (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) by a function ψ (of (x, z) ∈ R6) belonging to C2 0(R6), and integrate with respect to x and z, using integration by parts, we arrive at the weak formulation of the Boltzmann equation: � R6 ψ(x, z)∂f ∂t (t, x, z)dxdz − � R6 f(t, x, z)(z, ∇xψ(x, z))dxdz = � R6 f(t, x, z)Lfψ(x, z)dxdz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) 3 for all t ∈ R+ with Lfψ(x, z) = � R3×(0,π]×[0,2π) {ψ(x, z⋆) − ψ(x, z)}f(t, x, v)B(z, dv, dθ)dφ, where B is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In its weak form, the Boltzmann equation cannot be treated as the Kolmogorov equation that corresponds to a Markov process with jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In fact, to over- come this obstacle we proposed and studied the Boltzmann-Enskog equation [2] which corresponds to the dynamics of moderately dense gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' For hard and soft potentials, solvability and uniqueness of the Boltzmann-Enskog equation were carried out in subsequent works [11] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Indeed, there is a vast literature on various aspects of the Boltzmann equation and we refer the reader to [7], [9], [10], [15] [18, 19], [20], and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Standing on the shoulders of these giants, our aim is to build a stochastic analytic treatment of the (spatially non-homogeneous) Boltzmann equation in the non-cutoff case for hard spheres with σ(|z−v|)= |z−v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' An ingenious suggestion given to us by Professor Presutti provided the impetus to this work, and made the problem tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' A collection of densities {f(t, x, z}t∈[0,T], with x, z ∈ R3, is a strong (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' weak) solution of the Boltzmann equation in [0, T] if for any t ∈ [0, T] it solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We denote by D := D([0, T], R3) the space of all right continuous functions with left limits on [0, T] taking values in R3, and equipped with the topology induced by the Skorohod metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' A stochastic process (Xs, Zs)s∈[0,T] with values on D × D, and having time t marginals with density denoted by f(t, x, z) which solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) for all t ∈ [0, T], is called a ”Boltzmann process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We remark that the infinitesimal generator of a Boltzmann process is given by (z, ∇x) + Lf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Costantini and Marra [8] analyzed hydrodynamical limits of a pro- cess given by a the drift term involving (z, ∇x) and Lf in addition to a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We use the following notation R0 + := {t ∈ R : t ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In this article we assume the following hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Hypotheses A: A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The measure Q on [0, π) is finite outside any neighbourhood of 0, and for all ǫ > 0, it satisfies � ǫ 0 θQ(dθ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 4 A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The function σ : R0 + → R0 + (entering (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='8)) is given by σ(z) := czγ, with c > 0, γ ∈ (−1, 1] fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' There are many useful consequences of A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let us set α(z, v, θ, φ) := (n, (v − z))n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='11) Define ˆα(z, v, θ, φ) := α(z, v, θ, φ)σ(|z − v|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Condition A1 implies that there exists a constant C such that the following estimates hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' � Ξ |ˆα(z, v, θ, φ)|Q(dθ)dφ ≤ C|z − v|1+γ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='12) and hence � Ξ |ˆα(z, v, θ, φ)|Q(dθ)dφ ≤ C(|z|1+γ + |v|1+γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='13) Moreover, the following parameter transformation was introduced for each z ̸= v in [18] (see also [9], Section 3, or [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' α(z, v, θ, φ) = 1 − cos(θ) 2 (v − z) + sin(θ) 2 Γ(v − z, φ) = sin2(θ 2)(v − z) + sin(θ) 2 Γ(v − z, φ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='14) for all φ ∈ [0, 2π), where Γ(v − z, φ) = I(v − z) cos(φ) + J(v − z) sin(φ) and v−z |v−z|, I(v−z) |v−z| , J(v−z) |v−z| form an orthogonal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' It follows in particular that � 2π 0 Γ(v − z, φ)dφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='15) In order to study solutions to the Boltzmann equation it is feasible to study continuity properties of (u − v, n)n in u, v for fixed θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' However, it was already pointed out by Tanaka that (u, v) �−→ (u−v, n)n cannot be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' To overcome this problem Tanaka introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 of [18] another transformation of parameters, which describes a rotation around the longitude angle, is bijective and has Jacobian 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' As a consequence of this transformation φ0 he proved following Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 (see also Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' [18][10] There exists a measurable function φ0 : R12 → (0, 2π] such that |Γ(v − z, φ) − Γ(v′ − z′, φ + φ0(z, v, z′, v′))| ≤ 3|z − v − (z′ − v′)| (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='16) 5 and hence |α(z, v, θ, φ) − α(z′, v′, θ, φ + φ0(z, v, z′, v′))| ≤ 2θ(|z − z′| + |v − v′|) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='17) and |α(z, v, θ, φ)| ≤ 2θ(|z| + |v|) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='18) Moreover by his transformation [18] (see also [10]), Section 3) and using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='14) Tanaka proved the following inequality: � π 0 ��� � 2π 0 α(z, v, θ, φ) − α(z′, v′, θ, φ)dφ ��� Q(dθ) ≤ C(|z − z′| + |v − v′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='19) where by an abuse of notation we use the same symbol C > 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='19), even though the constants are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let {f(t, x, z)}t∈R0 + be a collection of densities on (R3 × R3, B(R3 × R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let us introduce the operator Qt(f, f)(·) defined through the right side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) Qt(f, f)(ψ) := � R6 f(t, x, z)Lfψ(x, z)dxdz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='20) It is easy to verify that Qt(f, f)(ψ) = 0 for ψ(x, z) = a + (b, z) + c|z|2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='21) ∀ a, c ∈ R , b ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (For a rigorous proof see Chapter II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='7 [5] or [7], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=') The integral form of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) corresponds to � R6 ψ(x, z)f(t, x, z)dxdz = � R6 ψ(x, z)f(0, x, z)dxdz + � t 0 � R6 f(t, x, z){(z, ∇xψ(x, z)) + Lfψ(x, z)}dxdzds, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='22) It is worthwhile to note that if a second collection of densities {g(t, x, z)}t∈R0 + on (R3 × R3, B(R3 × R3)) is given, then QS t (f, g)(ψ) = 0 for ψ(x, z) = a + (b, z) + c|z|2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='23) ∀ a, c ∈ R , b ∈ R3 with the operator QS t defined through QS t (f, g)(·) := Qt(f, g)(·) + Qt(g, f)(·) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='24) 6 with Qt(f, g)(ψ) := � R6 g(t, x, z)Lf ψ(x, z)dxdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='25) The following Povzner type inequality is essentially contained in [15, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (See also [7], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 and Appendix B of Chapter 6 for p ≥ 2 and references there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=') Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' For all θ ∈ (0, π], p ≥ 2 and γ ∈ (0, 1], � 2π 0 � ⟨z + α(z, v, θ, φ)⟩2p + ⟨v − α(z, v, θ, φ)⟩2p − ⟨z⟩2p − ⟨v⟩2p� dφ ≤ −sin2(θ) 2 � ⟨v⟩2p + ⟨z⟩2p� + Cp sin2(θ) ⌊ p+1 2 ⌋ � k=1 � ⟨v⟩2k⟨z⟩2p−2k + ⟨v⟩2p−2k⟨z⟩2k� , where ⟨v⟩ := (1 + |v|2) 1 2 , ⌊x⌋ ∈ Z is defined by ⌊x⌋ ≤ x < ⌊x⌋ + 1 and Cp > 0 is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Using conservation laws and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2, it can be proven that if {f(t, x, u)}t∈[0,T] is a weak solution of the Boltzmann equation in [0, T], with initial finite second moment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' � R6 |z|2f(0, x, z)dxdz < ∞, then for all p ≥ 1 � R6 |z|pf(t, x, z)dxdz < ∞ ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='26) For a proof we refer the reader to [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let {µt(dx, dz)}t∈R0 + be a collection of probabilities on (R3 × R3, B(R3 × R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let us define the operator Qt(f, µ)(ψ) := � R6 Lfψ(x, z)µt(dx, dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='27) acting on all ψ for which the integral on the right side is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Qt(f, µ)(|z|2) = � R9×Ξ (|v|2 − |z|2)σ(|z − v|) sin2(θ 2)Q(dθ)dφf(t, x, v)dvµt(dx, dz) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Qt(f, µ)(|z|2) = � R6 Lf|z|2µt(dx, dz) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='29) = � R9×Ξ (|z⋆|2 − |z|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 7 Moreover, Qt(f, µ)(|z|2) = � R9×Ξ (|z⋆|2 + |v⋆|2 − |z|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) − � R9×Ξ (|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) = − � R9×Ξ (|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='30) where in the last equality we used that the kinetic energy is conserved during the elastic collision, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Combining equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='29) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='30), we obtain Qt(f, µ)(|z|2) = 1 2 � R9×Ξ (|z⋆|2 − |z|2 − (|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) = 1 2 � R9×Ξ (|z|2 + 2(z, α) + |α|2 − |z|2) − (|v|2 − 2(v, α) + |α|2 − |v|2) × σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) = � R9×Ξ (z + v, α)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='31) Using the parametrization (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='14) for α = α(z, v, θ, φ) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='15) we obtain Qt(f, µ)(|z|2) = + � R9×Ξ (z + v, v − z) sin2(θ 2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) = − � R9×Ξ (|z|2 − |v|2) sin2(θ 2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz) 2 The Boltzmann process We use the following notation throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' U0 = D × [0, π) × (0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let {f(t, x, v)}t∈R0 + be a collection of densities on (R3 × R3, B(R3 × R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Then m(t, v) denotes the marginal density of velocity v at time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' m(t, v) := � R3 f(t, x, v)dx so that f(t, x|v)m(t, v) := f(t, x, v), upon disintagration of mea- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 8 Hypotheses B: We assume that t → f(t, x, v) is differentiable for each x, v ∈ R3 fixed, and satisfies B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' |∂f ∂t | is bounded on any compact subset of R0 + × R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ∂f ∂t (t, ·) ∈ L1(R6), ∀t ∈ R0 +, B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' supx∈R3 � R3 |u|1+γf(s, x, u)du ∈ C([0, T]) ∀ T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' supx∈R3 � R3 |u|1+γ ∂ ∂tf(t, x, u)du ∈ L1([0, T]) ∀ T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let {f(t, x, v)}t∈R0 + be a collection of densities which satisfies hypothesis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Suppose hypothesis A hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let X0 and Z0 be R3- valued ran- dom variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Suppose that for any fixed T > 0 there exists a stochastic basis (Ω, F, (Ft)t∈[0,T], P), an adapted process (Xt, Zt)t∈[0,T] with values on D × D, which has time marginals with density f(t, x, u), and such that it satisfies a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' the following stochastic equation for t ∈ [0, T]: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Xt = X0 + � t 0 Zsds Zt = Z0 + � t 0 � U0×R0 + α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)dN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) where in the above equation, dN := N(dv, dθ, dφ, dr, ds) is a Poisson random measure with compensator m(s, v)dvQ(dθ)dφdsdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Then (Xt, Zt)t∈[0,T] is a Boltz- mann process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='13) it follows for each T > 0 � T 0 E[ � U0×R0 + |α(Zs, vs, θ, φ)|1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)m(s, v)dvQ(dθ)dφdr]ds = � T 0 E[ � U0 |ˆα(Zs, vs, θ, φ)|f(s, Xs, v)dvQ(dθ)dφ]ds ≤ C � T 0 � R9(|z|1+γ + |v|1+γ)f(s, x, z)f(s, x, v)dxdzdvds, ≤ 2C � T 0 sup x∈R3 � R6 � |z|1+γf(s, x, z)dz � f(s, x, v)dvdxds < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In the above estimates we have used that the function f(t) is the probability density of the process (Xt, Zt), as well the assumption A2 and B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' It follows that we can apply the Itˆo formula to (Xs, Zs)s∈R+ [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In 9 fact let t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ∆t > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ψ ∈ C2 0(R3 × R3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' then ψ(Xt+∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zt+∆t) = ψ(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zt) + � t+∆t t (Zs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ∇xψ(Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zs))ds + � t+∆t t � U0×R+ 0 {ψ(Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zs + α(Zs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' vs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' φ)1[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' σ(|Zs−vs|)f(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='Xs|vs)](r)) − ψ(Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zs)}dN It follows E[ψ(Xt+∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zt+∆t) − ψ(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zt)] = E �� t+∆t t (Zs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ∇xψ(Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zs))ds � + E \uf8ee \uf8f0 � t+∆t t � U0 {ψ(Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zs+ α(Zs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' vs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' φ))−ψ(Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Zs)}σ(|Zs−vs|)f(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='vs)dvQ(dθ)dφds \uf8f9 \uf8fb Upon dividing by ∆t on both sides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' we obtain lim ∆t↓0 1 ∆t � R6 ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u){f(t + ∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u) − f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u)}dxdu = lim ∆t↓0 1 ∆t � t+∆t t � R6(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' ∇xψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u))f(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u)dxduds + lim ∆t↓0 1 ∆t � t+∆t t � R6×R3×[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='π)×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2π] {ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u + α(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' φ)) − ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u)} × σ(|u − v|)f(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' v)f(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' u)dvQ(dθ)dφdxduds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) Letting ∆t → 0 in every term of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Indeed, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=', the second term on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) and prove the continuity of the function g(s) := � R6×R3×[0,π)×(0,2π]{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)} ×σ(|u − v|)f(s, x, v)f(s, x, u)dvQ(dθ)dφdxdu Since {ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)} ≃ ∇uψ(x, z)α(u, v, θ, φ) with (x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='z) ∈ K compact set, and |α(u, v, θ, φ)|σ(|u − v|) ≤ |u − v|1+γ| sin(θ 2)|, by denoting with F a compact set in R3 which includes all projections x of (x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='z) ∈ K, it follows that |g(s) − g(s0)| ≤ C � F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)| ×(|u|γ+1 + |v|γ+1)dxdudv, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) 10 with C := ∥∇uψ∥∞2π � π 0 θQ(dθ) We split the integral on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) into two terms, one with |u|γ+1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' |u|γ+1), and get � F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)||u|γ+1dxdudv = � F ×R6 |f(s, x, u)f(s, x, v) − f(s, x, u)f(s0, x, v)||u|γ+1dxdudv + � F ×R6 |f(s, x, u)f(s0, x, v) − f(s0, x, u)f(s0, x, v)||u|γ+1dxdudv = J1(s) + J2(s) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4) where J1(s) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' J2(s)) is the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' second) term on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' J1(s) = � F ×R6 |u|γ+1f(s, x, u)| � s s0 ∂f ∂r (r, x, v)dr|dxdudv ≤ � supx∈R3 � R3 |u|γ+1f(s, x, u)du � � s s0 � R6 |∂f ∂r (r, x, v)|dxdvdr By B1 and B2 lims→s0 J1(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let us consider J2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' J2(s) = � F ×R6 |u|γ+1f(s0, x, v)| � s s0 ∂f ∂r (r, x, u)dr|dxdudv ≤ � s s0 � F ×R3 �� R3 |u|γ+1|∂f ∂r (r, x, u)|du � f(s0, x, v)dxdvdr Since supx∈R3 � R3 |u|γ+1|∂f ∂r (r, x, u)|du is integrable in [s0, s] by B3, we obtain lims→s0 J2(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Likewise, and without any changes in the arguments it follows lim s→s0 C � F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)||v|γ+1dxdudv = 0 Hence lims→s0 g(s) = g(s0), so that g is a continuous function and lim ∆t↓0 1 ∆t � t+∆t t g(s)ds = g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Note that in the above arguments we have taken s > s0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' One may also take s0 > s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 motivates the following Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let {f(t, x, v)}t∈R0 + be a collection of densities satisfying Hy- pothesis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Suppose that for any fixed T > 0, there exists a stochastic basis (Ω, F, (Ft)t∈[0,T], P) and an adapted process (Xt, Zt)t∈[0,T] with values on D × D such that 11 i) (Xt, Zt)t∈[0,T] has time marginals with density f(t, x, u), for t ∈ [0, T], ii) (Xt, Zt)t∈[0,T] is a solution of the McKean -Vlasov SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Then we say that “the McKean -Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) with density functions {f(t, x, v)}t∈R0 + is associated to the the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' If the above property holds for T ∈ [0, S] with S > 0, then the McKean -Vlasov SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) with density functions {f(t, x, v)}t∈[0,S] is associated to the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) up to time S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let us assume hypothesis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 it follows that any stochastic process (Xt, Zt)t∈[0,T] solving a McKean -Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) associated to the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) is (according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) a Boltzmann process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) is hence the Kolmogorov equation associated to the McKean -Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 3 Existence of the Boltzmann process In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 we proved that any process (Xt, Zt)t∈[0,T] solving the McKean Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) associated to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='10) in [0, T] is a Boltzmann process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In this section we analyze the following: given a strong solution {f(t, x, z)}t∈[0,T] of the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1), we find sufficient conditions for the existence of a solution of the McKean -Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) with density {f(t, x, z)}t∈[0,T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The solution process (Xt, Zt)t∈[0,T] is then a Boltzmann process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We present an overview on the construction of Boltzmann processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We briefly outline the construction of the process (Xt, Zt)t∈[0,T] under suitable conditions before stating the main result on Boltzmann processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The proofs of the ensuing results on the existence of a solution to a certain linearized stochastic system will appear in a separate paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 Construction of a solution of a SDE defined through a col- lection of densities solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) In this paragraph, we assume that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) and satisfies the following conditions: B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' sups∈[0,T],x∈R3 � R3 f(s, x, v)dv ≤ CT < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 12 B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' There exists for every K > 0 a constant CK T > 0 such that sup s∈[0,T],|x|≤K � R3 max(1, |v|1+γ)|∇xf(s, x, v)|dv ≤ CK T < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' B6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' sups∈[0,T],x∈R3 � R3 |v|γ+2f(s, x, v)dv ≤ cT < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' On any fixed filtered probability space (Ω, F, (Ft)t∈[0,T], P) satisfying the usual conditions, let ST := S1 T (Rd) denote the linear space of all adapted c`adl`ag pro- cesses (Xt)t∈[0,T] with values on Rd equipped with norm ∥X∥S1 T := E[ sup s∈[0,T] |Xs|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) Under hypotheses B4 - B6, and adopting the notation f(s, x, v) = f(s, x | v)m(s, v) upon disintegration of measures, we first prove the existence of a weak solution to the stochastic system Xt = X0 + � t 0 Zsds, ∀t ∈ [0, T] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) Zt = Z0 + � t 0 � U0×R+ 0 α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)dN ∀t ∈ [0, T] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) for t ∈ [0, T] where dN := N(dv, dθ, dφ, dr, ds) with its compensator given by m(s, v)dvQ(dθ)dφdsdr with values in S1 T := S1 T (R3 × R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Here we do not assume that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) is of McKean -Vlasov type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' First, we recall the definition of weak solutions in the context of stochastic analysis [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' A ”weak solution” of equation ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2)) in the time inter- val [0, T] is a triplet ((Ω, F, (Ft)t∈[0,T], P), N(dv, dθ, dφ, dr, ds), (Xt, Zt)t∈[0,T]) for which the following properties hold: (Ω, F, (Ft)t∈[0,T], P) is a stochastic basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' N(dv, dθ, dφ, dr, ds) is an adapted Poisson random measure with compen- sator m(s, v)dvQ(dθ)dφdsdr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (X·, Z·) := (Xt, Zt)t∈[0,T]) is an adapted c`adl`ag stochastic process with val- ued in Rd × Rd which satisfies ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2)) P -a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The existence of solutions to the stochastic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) is stated in the following theorem, proven in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let γ = 1 and Hypothesis A be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let T > 0 and {f(t, x, v)}t∈[0,T] be a collection of densities which satisfy f(t, x, u) ∈ C([0, T] × R6) and Hypotheses B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let the initial distribution of (X0, Z0) admit finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' There exists a weak solution ((Ω, F, (Ft)t∈[0,T], P), N(dv, dθ, dφ, dr, ds), (Xt, Zt)t∈[0,T]) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) such that (X·, Z·) ∈ S1 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Moreover, sup t∈[0,T] E[|Xt|2] + sup t∈[0,T] E[|Zt|2] < ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4) We remark that the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4) is proven by symmetry arguments similar to those appearing in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The form of the stochastic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3) with the process taking values in R6 at each t ∈ [0, T], one obtains that the solution lies in D × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2 Construction of Boltzmann processes with densities satisfy- ing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) We recall the concept of relative entropy which plays a key role in the proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Recall that for any two probability measures µ, ν on a common measurable space (X, X), the relative entropy of ν with respect to µ, denoted R(ν || µ), is defined by R(ν || µ) = � X � log dν dµ � dν if ν is absolutely continuous with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Otherwise, we set R(ν || µ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The following Lemma is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let µ, ν be two probability measures on a measurable space (X, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Then R(ν || µ) ≥ 0 and R(ν || µ) = 0 if and only if µ = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We assume that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) and satisfies hypotheses B as well as the following condition: C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The densities f(t, x, z) and g(t, x, z) are in C1,2([0, T] × R6) and are strictly positive-valued functions with g log g, g log f ∈ L1(R6) for each t ∈ [0, T] and lim|x|→∞ g(t, x, z) = 0 and g(0, x, z) = f(0, x, z) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let (Xt, Zt)t∈[0,T] be a stochastic process that solves the stochastic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Suppose that (Xt, Zt)t∈[0,T] has time marginals with density g(t, x, z), for each t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Suppose that {f(t, x, z)}t∈[0,T] and {g(t, x, z)}t∈[0,T] satisfy hypotheses B0 − B6 and C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Then g(t, x, z) = f(t, x, z) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' We will write R(g || f) for the relative entropy of the measure with prob- ability density g with respect to the measure with probability density f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The theorem is proved by establishing the following equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Rt(g|f) := � R6 log � g(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='z) f(t, x, z) � g(t, x, z)dxdz = 0 ∀t ∈ [0, T] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='5) We first apply the Itˆo formula [13] to log(g(t, Xt, Zt)), where (Xt, Zt)t∈[0,T] solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) and take expectation to obtain � R6 log (g(t, x, z))g(t, x, z)dxdz − � R6 log (g(0, x, z))g(0, x, z)dxdz = � t 0 � R6×R3×Ξ {log (g(s, x, z⋆)) − log (g(s, x, z))} × σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6) Indeed, in arriving at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6), we have used the following two calculations: (i) � t 0 � R6 ∂ ∂s log (g(s, x, z))g(s, x, z)dxdzds = � t 0 � R6 ∂ ∂sg(s, x, z)dxdzds = � R6(g(t, x, z) − g(0, x, z))dxdz = 0 since g is a probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (ii) � R6 z · ∇x log (g(s, x, z))g(s, x, z)dxdzds = � R6 z · ∇xg(s, x, z)dxdzds = 0 where the last equality is obtained by integrating and using the condition that lim |x|→∞g(t, x, z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Likewise, one obtains upon taking expectation and recall- ing that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1), � R6 log (f(t, x, z))g(t, x, z)dxdz − � R6 log (f(0, x, z))g(0, x, z)dxdz = � t 0 � R6 Q(f, f)(s, x, z) f(s, x, z) g(s, x, z)dxdzds + � t 0 � R6×R3×Ξ {log (f(s, x, z⋆)) − log (f(s, x, z))} × σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds = � t 0 � R9×Ξ {g(s, x, z⋆) f(s, x, z⋆) − g(s, x, z) f(s, x, z)} × σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds , + � t 0 � R6×R3×Ξ {log (f(s, x, z⋆)) − log (f(s, x, z))} × σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='7) 15 It is worthwhile to note that the last equality in the above display results upon rewriting � t 0 � R6 Q(f, f)(s, x, z) f(s, x, z) g(s, x, z)dxdzds = � t 0 � R9×Ξ {f(s, x, z⋆)f(s, x, v⋆) − f(s, x, z)f(s, x, v)} × σ(|z − v|) g(s, x, z) f(s, x, z)Q(dθ)dφdvdxdzds = � t 0 � R9×Ξ {g(s, x, z⋆) f(s, x, z⋆) − g(s, x, z) f(s, x, z)} × σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds , by using Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Combining equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='6) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='7) we obtain that Rt(g|f) = � R6 log (g(t, x, z))g(t, x, z)dxdz − � R6 log (f(t, x, z))g(t, x, z)dxdz = � t 0 � R9×Ξ � g(s, x, z) f(s, x, z){1 + log �g(s, x, z⋆)/f(s, x, z⋆) g(s, x, z)/f(s, x, z) � } − g(s, x, z⋆) f(s, x, z⋆) � × σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='8) Transforming (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='8) in the equivalent equation below, and recalling that for x ≥ 0 we have 1 + log (x) − x ≤ 0, we easily see that Rt(g|f) = � t 0 � R9×Ξ � 1 + log �g(s, x, z⋆)/f(s, x, z⋆) g(s, x, z)/f(s, x, z) � − g(s, x, z⋆)/f(s, x, z⋆) g(s, x, z)/f(s, x, z) � × g(s, x, z) f(s, x, z)σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' However, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2, Rt(g|f) ≥ 0, and hence, Rt(g|f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3 it follows that (Xt, Zt)t∈[0,T] in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3 solves the McKean- Vlasov equation associated to the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) and is a Boltzmann process with densities {f(t, x, z)}t∈[0,T] up to time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Based on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1, the main result of this work is given below: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let Hypotheses A be satisfied and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Assume that {f(t, x, u)}t∈[0,T] is a collection of densities which solves the Boltzmann equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1), and satis- fies the hypotheses B0 − B6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Let the random vector (X0, Z0) have finite second 16 moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Suppose that the weak solution of the stochastic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='2) has its distribution that admits a probability density at each time t ∈ [0, T] given by g(t, x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' If condition C1 is satisfied by {f(t, x, u)}t∈[0,T] and {g(t, x, u)}t∈[0,T], then the McKean-Vlasov equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1) (that involves {f(t, x, u)}t∈[0,T]) has a weak solution in [0, T] with values in D × D, and its Kolmogorov equation solves equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Acknowledgments: The authors are very thankful to Professor Errico Presutti for suggesting that a given solution of the Boltzmann equation be used in order to construct a Boltzmann process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' The second author considers herself blessed for having had the opportunity to write her Doctoral Thesis under the supervision of Errico Presutti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Albeverio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' R¨udiger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Sundar, Boltzmann processes and their construc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' In preparation (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Albeverio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' R¨udiger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Sundar, The Enskog Process, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFAT4oBgHgl3EQftB7D/content/2301.08662v1.pdf'} +page_content=' Stat.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..8c09dc00b0097f285d5f21a06cc41b6555c12425 --- /dev/null +++ b/FdAyT4oBgHgl3EQfSvci/content/tmp_files/2301.00090v1.pdf.txt @@ -0,0 +1,2719 @@ +Four-body Semileptonic Charm Decays D → P1P2ℓ+νℓ Based on +SU(3) Flavor Analysis +Ru-Min Wang1,†, +Yi Qiao1, +Yi-Jie Zhang1, +Xiao-Dong Cheng2,§, +Yuan-Guo Xu1,♯ +1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China +2College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang, Henan 464000, China +†ruminwang@sina.com +§chengxd@mails.ccnu.edu.cn +♯yuanguoxu@jxnu.edu.cn +Motivated by the significant experimental progress in probing semileptonic decays D +→ +P1P2ℓ+νℓ (ℓ = µ, e), we analyze the branching ratios of the D → P1P2ℓ+νℓ decays with the non- +resonant, the light scalar meson resonant and the vector meson resonant contributions in this work. +We obtain the hadronic amplitude relations between different decay modes by the SU(3) flavor +analysis, and then predict relevant branching ratios of the D → P1P2ℓ+νℓ decays by the present ex- +perimental data with 2σ errors. Most of our predicted branching ratios are consistent with present +experimental data within 2σ error bars, and others are consistent with the data within 3σ error +bars. We find that the branching ratios of the non-resonant decays D0 → π−K +0ℓ+νℓ, π0K−ℓ+νℓ, +D+ → π+K−ℓ+νℓ, π0K +0ℓ+νℓ, π+π−ℓ+νℓ, π0π0ℓ+νℓ, and D+ +s +→ K+K−ℓ+νℓ, K0K +0ℓ+νℓ are on +the order of O(10−3 − 10−4). +The vector meson resonant contributions are dominant in the +D0 → π−K +0ℓ+νℓ, π0K−ℓ+νℓ, π0π−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K +0ℓ+νℓ, π+π−ℓ+νℓ, and D+ +s +→ +K+K−ℓ+νℓ, K0K +0ℓ+νℓ, K+π−ℓ+νℓ, K0π0ℓ+νℓ decays. The non-resonant, the vector meson reso- +nant and the scalar resonant contributions are all important in the D0 → ηπ−ℓ+νℓ decays. The +D0 → K−K0ℓ+νℓ, η′π−ℓ+νℓ and D+ → K +0K0ℓ+νℓ, π0π0ℓ+νℓ, ηπ0ℓ+νℓ, η′π0ℓ+νℓ decays only receive +both the non-resonant and the scalar resonant contributions, and both contributions are important +in their branching ratios. According to our predictions, many decay modes could be observed in +the experiments like BESIII, LHCb and BelleII, and some decay modes might be measured in these +experiments in near future. +arXiv:2301.00090v1 [hep-ph] 31 Dec 2022 + +2 +I. +INTRODUCTION +Semileptonic heavy meson decays dominated by tree-level exchange of W-bosons in the SM are very important +processes in testing the stand model and in searching for the new physics beyond the stand model, for example, the +extraction of the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements. Four-body semileptonic exclusive decays +D → P1P2ℓ+νℓ are generated by the c → s/dℓ+νℓ transitions, and they can receive contributions from the non- +resonant, the light scalar meson resonant and the vector meson resonant contributions, etc. Therefore, these decays +are also a good laboratory for probing the internal structure of light hadrons [1–3]. Some non-resonant D → P1P2ℓ+νℓ +decays, the light scalar meson resonant decays D → S(S → P1P2)ℓ+νℓ and the vector meson resonant decays D → +S(S → P1P2)ℓ+νℓ have been observed by BESIII, BABAR, CLEO and MARKIII, etc [4–11]. Present experimental +measurements give us an opportunity to additionally test theoretical approaches. +Experimental backgrounds of the semileptonic decays are cleaner than ones of the hadronic decays, and theoretical +description of the semileptonic exclusive decays are relatively simple. Since leptons do not participate in the strong +interaction, the weak and strong dynamics can be separated in these processes. +All the strong dynamics in the +initial and final hadrons is included in the hadronic transition form factors, which are important for testing the +theoretical calculations of the involved strong interaction. The form factors can be calculated, for examples, by the +chiral perturbation theory [12], the unitarized chiral perturbation theory [13, 14], the light-cone sum rules [15–17] and +the QCD factorization [18]. Nevertheless, due to our poor understanding of hadronic interactions, the evaluations +of the form factors are difficult and often plugged with large uncertainties. One needs to find ways to minimize the +uncertainties to extract useful information. +In the lack of reliable calculations, symmetries provide very important information for particle physics. SU(3) +flavor symmetry is a symmetry in QCD for strong interaction. From the perspective of the SU(3) flavor symmetry, +the leptonic part of the D → P1P2ℓ+νℓ decay is SU(3) flavor singlet, which makes no difference between different +decay modes with certain lepton (e or µ). The different hadronic parts (the hadronic amplitudes or the hadronic form +factors) of the D → P1P2ℓ+νℓ decays could be related by the SU(3) flavor symmetry without the detailed dynamics. +Nevertheless, the size of the hadronic amplitudes or the form factors can not be determined by itself in the SU(3) +flavor symmetry approach. However, if experimental data are enough, one may use the data to extract the hadronic +amplitudes or the form factors, which can be viewed as predictions based on symmetry, has a smaller dependency +on estimated form factors. Although the SU(3) flavor symmetry is only an approximate symmetry because up, down +and strange quarks have different masses, it still provides some very useful information about the decays. The SU(3) +flavor symmetry has been widely used to study hadron decays, for instance, b-hadron decays [19–32], c-hadron decays +[31–46] and light hadron decays [31, 47–52]. +Although the SU(3) flavor symmetry works well in heavy hadron decays, the calculations of SU(3) flavor breaking +effects would play a key role in the precise theoretical predictions of the observables and a precise test of the the +unitarity of the CKM matrix. If up and down quark masses are neglected, a non-zero strange quark mass breaks +the SU(3) flavor symmetry down to the isospin symmetry. When up and down quark mass difference is kept, isospin +symmetry is also broken. Applications of the SU(3) flavor breaking approach on hadron decays can be found in Refs. + +3 +[53–60]. The SU(3) flavor breaking effects due to the fact of ms ≫ mu,d will be considered in our analysis of the +non-resonant D → P1P2ℓ+νℓ decays. +Four body semileptonic decay D → P1P2ℓ+νℓ have been studied, for instance, in Refs. [13, 61–66]. In this work, we +will study the D → P1P2ℓ+νℓ decays with the SU(3) flavor symmetry/breaking. In three cases of the non-resonant +decays, the light scalar meson resonant decays and the vector meson resonant decays, we will firstly construct the +hadronic amplitude relations between different decay modes, use the available data to extract the hadronic amplitudes, +then predict the not-yet-measured modes for further tests in experiments, and finally analyze the contributions with +the non-resonance, the light scalar meson resonances and the vector meson resonances in the branching ratios. +This paper is organized as follows. In Sec. II, the expressions of the branching ratios are given. In Sec. III, we will +give our numerical results of the D → P1P2ℓ+ν decays with the non-resonant, the light scalar meson resonant and +the vector meson resonant contributions. Our conclusions are given in Sec. IV. +II. +Theoretical frame +A. +Decay branching ratios +The effective Hamiltonian for c → qiℓ+νℓ transition can be written as +Heff(c → qiℓ+νℓ) = GF +√ +2 Vcqi ¯qiγµ(1 − γ5)c ¯νℓγµ(1 − γ5)ℓ, +(1) +where GF is the Fermi constant, Vcqi is the CKM matrix element, and qi = d, s for i = 2, 3. The decay amplitude of +the D(p) → P1(k1)P2(k2)ℓ+(q1)νℓ(q2) decay can be divided into leptonic and hadronic parts +A(D → P1P2ℓ+νℓ) = ⟨P1(k1)P2(k2)ℓ+(q1)νℓ(q2)|Heff(c → qiℓ+νℓ)|D(p)⟩ +(2) += GF +√ +2 VcqiLµHµ, +(3) +where Lµ = ¯νℓγµ(1 − γ5)ℓ is leptonic charged current, and Hµ = ⟨P1(k1)P2(k2)|¯s/ ¯dγµ(1 − γ5)c|D(p)⟩ is hadronic +matrix element. The leptonic part Lµ is calculable using the perturbation theory, while the hadronic part Hµ are +encoded into the transition form factors. Following Refs. [18, 67], the D → P1P2 form factors are given as +⟨P1(k1)P2(k2)|¯s/ ¯dγµc|D(p)⟩ = iF⊥ +1 +√ +k2 qµ +⊥, +(4) +−⟨P1(k1)P2(k2)|¯s/ ¯dγµγ5c|D(p)⟩ = Ft +qµ +� +q2 + F0 +2 +� +q2 +√ +λ +kµ +0 + F∥ +1 +√ +k2 +¯kµ +∥ , +(5) +with +kµ +0 = kµ − k · q +q2 qµ, +(6) +¯kµ +∥ = ¯kµ − 4(k · q)(q · ¯k) +λ +kµ + 4k2(q · ¯k) +λ +qµ, +(7) +qµ +⊥ = 2ϵµαβγ qαkβ¯kγ +√ +λ +, +(8) + +4 +where k ≡ k1+k2, q ≡ q1+q2, ¯k ≡ k1−k2, ¯q ≡ q2−q1, and λ = λ(m2 +D, q2, k2) with λ(a, b, c) = a2+b2+c2−2ab−2bc−2ac. +In terms of the form factors, the differential branching ratio of the non-resonant D → P1P2ℓ+νℓ decays can be +written as [18] +dB(D → P1P2ℓ+ν)N +dq2 dk2 += 1 +2τD|N|2βℓ(3 − βℓ)|FA|2, +(9) +with +|N|2 = G2 +F |Vcq|2 βℓq2� +λ(m2 +D, q2, k2) +3 · 210π5m3 +D +with +βℓ = 1 − m2 +ℓ +q2 , +|FA|2 = |F0|2 + 2 +3(|F∥|2 + |F⊥|2) + +3m2 +ℓ +q2(3 − βℓ)|Ft|2, +(10) +where τM(mM) is lifetime(mass) of M particle. In this work, we ignore the small contributions of |Ft|2 term, which +is proportional to m2 +ℓ. The corresponding limits of integration are given by (mP1 + mP2)2 ≤ k2 ≤ (mDq − mℓ)2 +and m2 +ℓ ≤ q2 ≤ (mDq − +√ +k2)2. The calculations of the form factors F0, F∥, F⊥ and Ft are quite complicated, and +their specific expressions in the QCD factorization limit can be found in Ref. [18]. Nevertheless, we will not use +the specific expressions in this work, and we will relate the different hadronic decay amplitudes or the different form +factors between different decay modes by the SU(3) flavor symmetry/breaking, which are discussed in later Sec. II C. +Except for the non-resonant D → P1P2ℓ+νℓ decays, the resonant D → R(R → P1P2)ℓ+νℓ decays with the +scalar(R = S) resonance and the vector(R = V ) resonance are also studied in this work. In the case of the de- +cay widths of the resonances are very narrow, the resonant decay branching ratios respect a simple factorization +relation +B(D → Rℓ+νℓ, R → P1P2) = B(D → Rℓ+νℓ) × B(R → P1P2), +(11) +and this result is also a good approximation for wider resonances. Above Eq. (11) will be used in our analysis for the +scalar resonant D → S(S → P1P2)ℓ+νℓ decays and the vector resonant D → V (V → P1P2)ℓ+νℓ decays in Sec. III B +and III C, respectively. Relevant B(D → Rℓ+νℓ) and B(R → P1P2) are also obtained by the SU(3) flavor symmetry +in our later analysis. +B. +Meson multiplets +Before giving the hadronic amplitudes based on the SU(3) flavor analysis, we will collect the representations for the +multiplets of the SU(3) flavor group first in this subsection. +Charmed mesons containing one heavy c quark are flavor SU(3) anti-triplets +Di = +� +D0(c¯u), D+(c ¯d), D+ +s (c¯s) +� +. +(12) +Light pseudoscalar meson (P) and vector meson (V ) octets and singlets under the SU(3) flavor symmetry of light +u, d, s quarks are [68] +P = +� +� +� +� +� +π0 +√ +2 + η8 +√ +6 + η1 +√ +3 +π+ +K+ +π− +− π0 +√ +2 + η8 +√ +6 + η1 +√ +3 +K0 +K− +K +0 +− 2η8 +√ +6 + η1 +√ +3 +� +� +� +� +� , +(13) + +5 +V += +� +� +� +� +� +ρ0 +√ +2 + +ω +√ +2 +ρ+ +K∗+ +ρ− +− ρ0 +√ +2 + +ω +√ +2 +K∗0 +K∗− +K +∗0 +φ +� +� +� +� +� , +(14) +where the η and η′ are mixtures of η1 = u¯u+d ¯d+s¯s +√ +3 +and η8 = u¯u+d ¯d−2s¯s +√ +6 +with the mixing angle θP +� +� η +η′ +� +� = +� +� cosθP −sinθP +sinθP +cosθP +� +� +� +� η8 +η1 +� +� . +(15) +And θP = [−20◦, −10◦] from Particle Data Group (PDG) [11] will be used in our numerical analysis. +The structures of the light scalar mesons are not fully understood yet. Many suggestions are discussed, such as +ordinary two quark state, four quark state, meson-meson bound state, molecular state, glueball state or hybrid state, +for examples, in Refs. [69–77]. In this work, we will consider the two quark and the four quark scenarios for the scalar +mesons below or near 1 GeV . In the two quark picture, the light scalar mesons can be written as [78] +S = +� +� +� +� +� +a0 +0 +√ +2 + +σ +√ +2 +a+ +0 +K+ +0 +a− +0 +− a0 +0 +√ +2 + +σ +√ +2 +K0 +0 +K− +0 +K +0 +0 +f0 +� +� +� +� +� . +(16) +The two isoscalars f0(980) and f0(500) are obtained by the mixing of σ = u¯u+d ¯d +√ +2 +and f0 = s¯s, +� +� f0(980) +f0(500) +� +� = +� +� cosθS +sinθS +−sinθS cosθS +� +� +� +� f0 +σ +� +� , +(17) +where the three possible ranges of the mixing angle θS [69, 79], 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < +30◦ will be analyzed in our numerical results. In the four quark picture, the light scalar mesons are given as [11, 80] +σ = u¯ud ¯d, +f0 = (u¯u + d ¯d)s¯s/ +√ +2, +a0 +0 = (u¯u − d ¯d)s¯s/ +√ +2, +a+ +0 = u ¯ds¯s, +a− +0 = d¯us¯s, +K+ +0 = u¯sd ¯d, +K0 +0 = d¯su¯u, +K +0 +0 = s ¯du¯u, +K+ +0 = s¯ud ¯d, +(18) +and the two isoscalars are expressed as +� +� f0(980) +f0(500) +� +� = +� +� cosφS +sinφS +−sinφS cosφS +� +� +� +� f0 +σ +� +� , +(19) +where the constrained mixing angle φS = (174.6+3.4 +−3.2)◦ [70]. +C. +Non-resonant hadronic amplitudes +Since the hadronic amplitudes of the semileptonic D → V/Sℓ+νℓ decays based on the SU(3) flavor symme- +try/breaking have been discussed in previous Ref. [81], we will focus on the hadronic amplitudes of the non-resonant +D → P1P2ℓ+νℓ decays in this subsection. + +6 +In terms of the SU(3) flavor symmetry, the quark current ¯qiγµ(1−γ5)c can be expressed as a SU(3) flavor anti-triplet +(¯3), and the effective Hamiltonian in Eq. (1) is transformed as [41] +Heff(c → qiℓ+νℓ) = GF +√ +2 H(¯3) ¯νℓγµ(1 − γ5)ℓ, +(20) +with H(¯3) = (0, Vcd, Vcs). The decay amplitude of the non-resonant D → P1P2ℓ+νℓ decay can be written as +A(D → P1P2ℓ+νℓ)N = GF +√ +2 H(D → P1P2)N ¯νℓγµ(1 − γ5)ℓ, +(21) +and the hadronic amplitude H(D → P1P2)N can be parameterized as +H(D → P1P2)N = c10DiP i +jP j +kH(¯3)k + c20DiP i +jH(¯3)jP k +k + c30DiH(¯3)iP j +kP k +j + c40DiH(¯3)iP k +k P j +j , +(22) +where ci0(i = 1, 2, 3, 4) are the nonperturbative coefficients under the SU(3) flavor symmetry. Feynman diagrams for +the non-resonant D → P1P2ℓ+νℓ decays are displayed in Fig. 1. +SU(3) flavor breaking effects come from different masses of u, d and s quarks, and they will become useful once we +have measurements of several D → P1P2ℓ+νℓ decays that are precise enough to see deviations from the SU(3) flavor +νℓ +ℓ+ +c +H(3)k +qk +¯qj +qj +¯qi +¯qi +( a ) +νℓ +ℓ+ +c +H(3)j +qj +¯qi +¯qi +¯qk +qk +( b ) +νℓ +ℓ+ +c +H(3)i +¯qi +¯qk +qk +qj +¯qj +( c ) +νℓ +ℓ+ +c +H(3)i +¯qi +¯qk +qk +¯qj +qj +( d ) +FIG. 1: Diagrams of the non-resonant D → P1P2ℓ+νℓ decays. + +7 +symmetry. The diagonalized mass matrix can be expressed as [59, 60] +� +� +� +� +� +mu +0 +0 +0 +md +0 +0 +0 +ms +� +� +� +� +� = 1 +3(mu + md + ms)I + 1 +2(mu − md)X + 1 +6(mu + md − 2ms)W, +(23) +with +X = +� +� +� +� +� +1 +0 +0 +0 −1 0 +0 +0 +0 +� +� +� +� +� , +W = +� +� +� +� +� +1 0 +0 +0 1 +0 +0 0 −2 +� +� +� +� +� . +(24) +Compared with s quark mass, the u and d quark masses are much smaller which can be ignored. The SU(3) flavor +breaking effects due to a non-zero s quark mass dominate the SU(3) breaking effects. When u and d quark mass +difference is ignored, the residual SU(3) flavor symmetry becomes the isospin symmetry and the term proportional to +X can be dropped. The identity I part contributes to the D → P1P2ℓ+νℓ decay amplitudes in a similar way as that +given in Eq. (21) which can be absorbed into the coefficients ci0. Only W part will contribute to the SU(3) breaking +effects. The SU(3) breaking contributions to the hadronic amplitudes due to the fact of ms ≫ mu,d are +∆H(D → P1P2)N = c11DiW i +aP a +j P j +kH(¯3)k + c12DiP i +jW j +aP a +k H(¯3)k + c13DiP i +jP j +kW k +a H(¯3)a ++ c21DiW i +aP a +j H(¯3)jP k +k + c22DiP i +jW j +aH(¯3)aP k +k ++ c31DiW i +aH(¯3)aP j +kP k +j + c32DiH(¯3)iP j +kW k +a P a +j ++ c41DiW i +aH(¯3)aP k +k P j +j , +(25) +where cij (i, j = 1, 2, 3, 4) are the nonperturbative SU(3) flavor breaking coefficients. +Full hadronic amplitudes of the different non-resonant D → P1P2ℓ+ν decays and their relations under the SU(3) +flavor symmetry/breaking are given in later Sec. III A. +III. +Numerical results of the D → P1P2ℓ+ν decays +The branching ratios with the non-resonant contributions, the light scalar meson resonant contributions and the +vector meson resonant contributions will be analyzed in this section. If not special specified, the theoretical input +parameters, such as the lifetimes and the masses, and the experimental data within the 2σ error bars from PDG [11] +will be used in our numerical analysis. +A. +Non-resonant D → P1P2ℓ+ν decays +The hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays including both the SU(3) flavor symmetry and +the SU(3) flavor breaking terms are summarized in the second column of Tab. I, in which we can see the relations +of different hadronic amplitudes. The following relations are hold in both the SU(3) flavor symmetry and the SU(3) + +8 +flavor breaking due to a strange quark mass. +H(D0 → π−K +0ℓ+νℓ)N = H(D+ → π+K−ℓ+νℓ)N = +√ +2H(D0 → π0K−ℓ+νℓ)N = − +√ +2H(D+ → π0K +0ℓ+νℓ)N, +H(D0 → η8K−ℓ+νℓ)N = H(D+ → η8K +0ℓ+νℓ)N, +H(D0 → η1K−ℓ+νℓ)N = H(D+ → η1K +0ℓ+νℓ)N, +H(D+ +s → K+K−ℓ+νℓ)N = H(D+ +s → K0K +0ℓ+νℓ)N, +H(D0 → K−K0ℓ+νℓ)N = H(D+ → K +0K0ℓ+νℓ)N − H(D+ → K+K−ℓ+νℓ)N, +H(D+ +s → K+π−ℓ+νℓ)N = − +√ +2H(D+ +s → K0π0ℓ+νℓ)N. +(26) +If assuming the SU(3) flavor breaking effects are small and can be ignored, more amplitude relations will be obtained. +Moreover, as shown in Fig. 1, the SU(3) flavor symmetry contributions of Fig. 1 (b-d) are suppressed by the Okubo- +Zweig-Iizuka (OZI) rule [82–84]. If ignoring both the OZI suppressed SU(3) flavor symmetry contributions and the +SU(3) flavor breaking contributions, almost all hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays can +be related by the coefficient c10. +Since the leptonic charged current ¯νℓγµ(1 − γ5)ℓ is the SU(3) flavor singlet, and it is completely generic between +different decay modes with certain ℓ = e or µ. The same relations as the hadronic amplitudes listed in Tab. I are +valid in the decay amplitudes of the D → P1P2ℓ+νℓ decays and the form factors of the D → P1P2 transitions. For the +non-resonant D → P1P2ℓ+νℓ decays, only B(D+ → π+K−µ+νµ)N has been measured, and B(D+ → π+K−e+νe)N +has been upper limited. Because the non-resonant D → P1P2ℓ+νℓ decays have not been measured enough to reveal +the OZI suppressed SU(3) flavor symmetry contributions and the SU(3) symmetry breaking effects, we ignore both +of them in our analysis, and then almost all hadronic amplitudes, form factors or decay amplitudes can be related +by the SU(3) flavor symmetry coefficient c10. The simple relations associated by the coefficient c10 for FA given in +Eq. (10) will be used to obtain our numerical results. Noted that, for consistency, only the SU(3) flavor symmetry +contributions will be considered in the light scalar meson resonant D → S(S → P1P2)ℓ+νℓ decays and the vector +meson resonant D → V (V → P1P2)ℓ+νℓ decays in later Sec. III B and Sec. III C, respectively. +The experimental data of B(D+ → π+K−µ+νµ)N within 2σ errors and the upper limit of B(D+ → π+K−e+νe)N at +90% confidence level from PDG [11] are listed in the second column of Tab. II, which will be used to determine c10 in +the non-resonant D+ → π+K−ℓ+νℓ decays, and then many other branching ratios of the non-resonant D → P1P2ℓ+νℓ +decays can be predicted by using the constrained c10 from the data of B(D+ → π+K−ℓ+νℓ)N listed in the second +column of Tab. II. Our predictions are listed in the third column of Tab. II for the c → sℓ+νℓ transitions and in the +second column of Tab. III for the c → dℓ+νℓ transitions. +From Tabs. +II-III, one can see that many branching ratios of the non-resonant D → P1P2ℓ+νℓ decays, such +as B(D0 → π−K +0ℓ+νℓ)N, B(D0 → π0K−ℓ+νℓ)N, B(D+ → π+K−ℓ+νℓ)N, B(D+ → π0K +0ℓ+νℓ)N, B(D+ +s +→ +K+K−ℓ+νℓ)N, B(D+ +s +→ K0K +0ℓ+νℓ)N, B(D+ → π+π−ℓ+νℓ)N and B(D+ → π0π0ℓ+νℓ)N, are on the orders of +O(10−3 −10−4), which could be measured by the BESIII, LHCb and BelleII experiments. Nevertheless, other decays, +for examples, the non-resonant D → ηPℓ+νℓ decays, are strongly suppressed by the narrow phase spaces, the mixing +angle θP or the CKM matrix element Vcd, their branching ratios are on the orders of O(10−5 − 10−7), and many of +them might be observed by the BESIII and BelleII experiments in the near future. + +9 +TABLE I: The hadronic amplitudes for the D → P1P2ℓ+νℓ decays. +C1 ≡ c10 + c11 + c12 − 2c13, C2 ≡ c20 + c21 − 2c22, +C3 ≡ c30 − 2c31, C4 ≡ c40 − 2c41, and [C +′,′′]R denotes the contributions come from the decays with R resonances. +Decay modes +Non-resonant hadronic amplitudes +Scalar resonant ones +Vector resonant ones +c → sℓ+νℓ: +D0 → π−K +0ℓ+νℓ +C1 +� +C′ +1 +� +K− +0 +� +C′′ +1 +� +K∗− +D0 → π0K−ℓ+νℓ +1 +√ +2 C1 +� 1 +√ +2 C′ +1 +� +K− +0 +� 1 +√ +2 C′′ +1 +� +K∗− +D0 → η8K−ℓ+νℓ +− 1 +√ +6 C1 + +√ +6c12 +· · · +· · · +D0 → η1K−ℓ+νℓ +2 +√ +3 +� +C1 + 3 +2 C2 +� +− +√ +3c12 +· · · +· · · +D+ → π+K−ℓ+νℓ +C1 +� +C′ +1 +� +K0 +0 +� +C′′ +1 +� +K∗0 +D+ → π0K +0ℓ+νℓ +− 1 +√ +2 C1 +� 1 +√ +2 C′ +1 +� +K0 +0 +� 1 +√ +2 C′′ +1 +� +K∗0 +D+ → η8K +0ℓ+νℓ +− 1 +√ +6 C1 + +√ +6c12 +· · · +· · · +D+ → η1K +0ℓ+νℓ +2 +√ +3 +� +C1 + 3 +2 C2 +� +− +√ +3c12 +· · · +· · · +D+ +s → K+K−ℓ+νℓ +C1 + 2C3 −3c11 +� +cos2θS C′ +1 +� +f0(980) +� +C′′ +1 +� +φ +D+ +s → K0K +0ℓ+νℓ +C1 + 2C3 −3c11 +� +cos2θS C′ +1 +� +f0(980) +� +C′′ +1 +� +φ +D+ +s → π0π0ℓ+νℓ +√ +2C3 + +√ +2c32 +� +sinθScosθSC′ +1 +� +f0(980) +� +−sinθScosθSC′ +1 +� +f0(500) +· · · +D+ +s → π+π−ℓ+νℓ +2C3 +�√ +2sinθScosθSC′ +1 +� +f0(980) +� +− +√ +2sinθScosθSC′ +1 +� +f0(500) +· · · +D+ +s → η8η8ℓ+νℓ +2 +√ +2 +3 +� +C1 + 3 +2 C3 +� +− +√ +2� +2c11 + 2c12 + c32 +� +· · · +· · · +D+ +s → η1η1ℓ+νℓ +√ +2 +3 (C1 + 3C2 + 3C3 + 9C4) − +√ +2(c11 + c12 + 3c21) +· · · +· · · +D+ +s → η8η1ℓ+νℓ +− 2 +√ +2 +3 +� +C1 + 3 +2 C2 +� ++2 +√ +2(c11 + c12 + 3 +2 c21 + c32) +· · · +· · · +c → dℓ+νℓ: +D0 → K−K0ℓ+νℓ +C1 −3(c12 − c13) +� +C′ +1 +� +a0(980) +· · · +D0 → π0π−ℓ+νℓ +· · · +· · · +� 1 +√ +2 C′′ +1 +� +ρ− +D0 → η8π−ℓ+νℓ +� 2 +3 C1 + +√ +6c13 +�� 2 +3 C′ +1 +� +a0(980) +� 1 +√ +6 C′′ +1 +� +ρ− +D0 → η1π−ℓ+νℓ +2 +√ +3 +� +C1 + 3 +2 C2 +� ++ +√ +3(2c13 + 3c22) +� 2 +√ +3 C′ +1 +� +a0(980) +� 1 +√ +3 C′′ +1 +� +ρ− +D+ → K +0K0ℓ+νℓ +C1 + 2C3 −3(c12 − c13 − 2c31) +� +1 +2 C′ +1 +� +a0(980)0 +� +1 +√ +2 sinθScosθSC′ +1 +� +f0(980) +· · · +D+ → K+K−ℓ+νℓ +2C3 +6c31 +� +− 1 +2 C′ +1 +� +a0(980)0 +� +1 +√ +2 sinθScosθSC′ +1 +� +f0(980) +· · · +D+ → π+π−ℓ+νℓ +C1 + 2C3 +3c13 + 6c31 +� +sin2θSC′ +1 +� +f0(980) +� +cos2θSC′ +1 +� +f0(500) +� 1 +2 C′′ +1 +� +ρ0,ω +D+ → π0π0ℓ+νℓ +1 +√ +2 (C1 + 2C3) + 1 +√ +2 (3c13 + 6c31 + 2c32) +� +1 +√ +2 sin2θSC′ +1 +� +f0(980) +� +1 +√ +2 cos2θSC′ +1 +� +f0(500) +· · · +D+ → η8π0ℓ+νℓ +− 1 +√ +3 +� +C1 + C2 +� +− +√ +3� +c13 + c22 +� +� +− +1 +√ +6 C′ +1 +� +a0(980) +· · · +D+ → η1π0ℓ+νℓ +− +� 2 +3 +� +C1 + C2 +� +− 1 +√ +6 +� +6c13 + 9c22 +� +� +− +1 +√ +3 C′ +1 +� +a0(980) +· · · +D+ → η8η8ℓ+νℓ +√ +2 +6 +� +C1 + 6C3 +� ++ 1 +√ +2 (c13 + 6c31 − 2c32) +· · · +· · · +D+ → η1η1ℓ+νℓ +√ +2 +3 (C1 + 3C2 + 3C3 + 9C4) + +√ +2(c13 + 3c22 + 3c31 + 9c41) +· · · +· · · +D+ → η8η1ℓ+νℓ +√ +2 +3 +� +C1 + 3 +2 C2 +� ++ +√ +2� +c13 + 3 +2 c22 + 2c32 +� +· · · +· · · +D+ +s → K+π−ℓ+νℓ +C1 −3c11 + 3c13 +� +C′ +1 +� +K0 +0 +� +C′′ +1 +� +K∗0 +D+ +s → K0π0ℓ+νℓ +− 1 +√ +2 C1 − 1 +√ +2 (−3c11 + 3c13) +� +− +1 +√ +2 C′ +1 +� +K0 +0 +� 1 +√ +2 C′′ +1 +� +K∗0 +D+ +s → η8K0ℓ+νℓ +− 1 +√ +6 C1 + 1 +√ +6 +� +3c11 + 6c12 − 3c13 +� +· · · +· · · +D+ +s → η1K0ℓ+νℓ +2 +√ +3 +� +C1 + 3 +2 C2 +� +− +√ +3� +2c11 + c12 − 2c13 + 3c21 − 3c22 +� +· · · +· · · + +10 +TABLE II: The experimental data and the SU(3) flavor symmetry predictions of the non-resonant branching ratios and the +total branching ratios of the D → P1P2ℓ+νℓ decays with the c → sℓ+νℓ transitions within the 2σ errors. The experimental +data are taken from PDG [11], ‘N’ denotes the non-resonant contributions, and ‘T’ denotes the total contributions including +the non-resonance, the light scalar meson resonances as well as the vector meson resonances. The same below. +Branching ratios +Exp. data with N +Ones with N +Exp. data with T +Ones with T +B(D0 → π−K +0e+νe)(×10−2) +· · · +0.076 ± 0.041 +1.44 ± 0.08 +1.57 ± 0.14 +B(D0 → π0K−e+νe)(×10−2) +· · · +0.039 ± 0.021 +1.6+2.6 +−1.0 +0.80 ± 0.07 +B(D0 → ηK−e+νe)(×10−6) +· · · +3.51 ± 3.51 +· · · +3.51 ± 3.51 +B(D0 → η′K−e+νe)(×10−6) +· · · +4.03 ± 2.17 +· · · +4.03 ± 2.17 +B(D+ → π+K−e+νe)(×10−2) +< 0.7 +0.20 ± 0.10 +4.02 ± 0.36 +4.06 ± 0.30 +B(D+ → π0K +0e+νe)(×10−2) +· · · +0.100 ± 0.052 +· · · +2.01 ± 0.15 +B(D+ → ηK +0e+νe)(×10−5) +· · · +0.89 ± 0.89 +· · · +0.89 ± 0.89 +B(D+ → η′K +0e+νe)(×10−5) +· · · +1.03 ± 0.55 +· · · +1.03 ± 0.55 +B(D+ +s → K+K−e+νe)(×10−2) +· · · +0.034 ± 0.018 +· · · +1.27 ± 0.13 +B(D+ +s → K0K +0e+νe)(×10−3) +· · · +0.33 ± 0.18 +· · · +8.58 ± 0.95 +B(D+ +s → π+π−e+νe)(×10−3) +· · · +· · · +· · · +1.47 ± 0.79 +B(D+ +s → π0π0e+νe)(×10−4) +· · · +· · · +· · · +8.58 ± 3.50 +B(D+ +s → ηηe+νe)(×10−4) +· · · +0.56 ± 0.49 +· · · +0.56 ± 0.49 +B(D+ +s → ηη′e+νe)(×10−6) +· · · +5.38 ± 3.19 +· · · +5.38 ± 3.19 +B(D0 → π−K +0µ+νµ)(×10−2) +· · · +0.073 ± 0.039 +· · · +1.47 ± 0.13 +B(D0 → π0K−µ+νµ)(×10−2) +· · · +0.038 ± 0.020 +· · · +0.75 ± 0.07 +B(D0 → ηK−µ+νµ)(×10−6) +· · · +3.18 ± 3.18 +· · · +3.18 ± 3.18 +B(D0 → η′K−µ+νµ)(×10−6) +· · · +2.76 ± 1.49 +· · · +2.76 ± 1.49 +B(D+ → π+K−µ+νµ)(×10−2) +0.19 ± 0.10 +0.19 ± 0.10 +3.65 ± 0.68 +3.80 ± 0.27 +B(D+ → π0K +0µ+νµ)(×10−2) +· · · +0.095 ± 0.050 +· · · +1.89 ± 0.13 +B(D+ → ηK +0µ+νµ)(×10−5) +· · · +0.81 ± 0.81 +· · · +0.81 ± 0.81 +B(D+ → η′K +0µ+νµ)(×10−5) +· · · +0.71 ± 0.38 +· · · +0.71 ± 0.38 +B(D+ +s → K+K−µ+νµ)(×10−2) +· · · +0.032 ± 0.017 +· · · +1.19 ± 0.12 +B(D+ +s → K0K +0µ+νµ)(×10−3) +· · · +0.30 ± 0.16 +· · · +8.02 ± 0.88 +B(D+ +s → π+π−µ+νµ)(×10−3) +· · · +· · · +· · · +1.25 ± 0.69 +B(D+ +s → π0π0µ+νµ)(×10−4) +· · · +· · · +· · · +7.34 ± 3.09 +B(D+ +s → ηηµ+νµ)(×10−4) +· · · +0.51 ± 0.45 +· · · +0.51 ± 0.45 +B(D+ +s → ηη′µ+νµ)(×10−6) +· · · +3.98 ± 2.36 +· · · +3.98 ± 2.36 + +11 +TABLE III: The experimental data and the SU(3) flavor symmetry predictions of the non-resonant branching ratios and the +total branching ratios of the D → P1P2ℓ+νℓ decays with the c → dℓ+νℓ transitions within the 2σ errors. +Branching ratios +Ones with N +Exp. data with T +Ones with T +B(D0 → K−K0e+νe)(×10−5) +0.83 ± 0.45 +· · · +1.25 ± 0.64 +B(D0 → π0π−e+νe)(×10−3) +0 +1.45 ± 0.14 +1.85 ± 0.11 +B(D0 → ηπ−e+νe)(×10−5) +4.34 ± 2.68 +· · · +16.38 ± 5.10 +B(D0 → η′π−e+νe)(×10−5) +0.39 ± 0.26 +· · · +0.57 ± 0.35 +B(D+ → K +0K0e+νe)(×10−5) +2.11 ± 1.13 +· · · +3.31 ± 1.69 +B(D+ → K+K−e+νe)(×10−5) +· · · +· · · +1.31 ± 0.63 +B(D+ → π+π−e+νe)(×10−3) +0.26 ± 0.14 +2.45 ± 0.20 +3.08 ± 0.51 +B(D+ → π0π0e+νe)(×10−4) +1.33 ± 0.71 +· · · +2.88 ± 1.75 +B(D+ → ηπ0e+νe)(×10−5) +5.68 ± 3.50 +· · · +9.68 ± 4.49 +B(D+ → η′π0e+νe)(×10−6) +5.21 ± 3.46 +· · · +8.28 ± 5.00 +B(D+ → ηηe+νe)(×10−6) +3.16 ± 2.26 +· · · +3.16 ± 2.26 +B(D+ → ηη′e+νe)(×10−8) +3.96 ± 2.37 +· · · +3.96 ± 2.37 +B(D+ +s → K+π−e+νe)(×10−3) +0.075 ± 0.041 +· · · +1.66 ± 0.17 +B(D+ +s → K0π0e+νe)(×10−4) +0.38 ± 0.21 +· · · +8.24 ± 0.85 +B(D+ +s → ηK0e+νe)(×10−5) +1.70 ± 1.06 +· · · +1.70 ± 1.06 +B(D+ +s → η′K0e+νe)(×10−7) +5.21 ± 3.47 +· · · +5.21 ± 3.47 +B(D0 → K−K0µ+νµ)(×10−5) +0.76 ± 0.43 +· · · +1.11 ± 0.57 +B(D0 → π0π−µ+νµ)(×10−3) +0 +· · · +1.76 ± 0.10 +B(D0 → ηπ−µ+νµ)(×10−5) +4.13 ± 2.55 +· · · +15.04 ± 4.76 +B(D0 → η′π−µ+νµ)(×10−5) +0.34 ± 0.23 +· · · +0.50 ± 0.31 +B(D+ → K +0K0µ+νµ)(×10−5) +1.93 ± 1.04 +· · · +2.94 ± 1.50 +B(D+ → K+K−µ+νµ)(×10−5) +· · · +· · · +1.09 ± 0.53 +B(D+ → π+π−µ+νµ)(×10−3) +0.25 ± 0.14 +· · · +2.92 ± 0.48 +B(D+ → π0π0µ+νµ)(×10−4) +1.29 ± 0.69 +· · · +2.68 ± 1.65 +B(D+ → ηπ0µ+νµ)(×10−5) +5.40 ± 3.33 +· · · +8.71 ± 4.16 +B(D+ → η′π0µ+νµ)(×10−6) +4.67 ± 3.10 +· · · +7.23 ± 4.37 +B(D+ → ηηµ+νµ)(×10−6) +2.83 ± 2.02 +· · · +2.83 ± 2.02 +B(D+ → ηη′µ+νµ)(×10−8) +2.43 ± 1.46 +· · · +2.43 ± 1.46 +B(D+ +s → K+π−µ+νµ)(×10−3) +0.072 ± 0.039 +· · · +1.58 ± 0.16 +B(D+ +s → K0π0µ+νµ)(×10−4) +0.36 ± 0.20 +· · · +7.81 ± 0.80 +B(D+ +s → ηK0µ+νµ)(×10−5) +1.57 ± 0.98 +· · · +1.57 ± 0.98 +B(D+ +s → η′K0µ+νµ)(×10−7) +4.08 ± 2.72 +· · · +4.08 ± 2.72 + +12 +B. +D → S(S → P1P2)ℓ+νℓ decays +We will analyze the D → P1P2ℓ+νℓ decays with the light scalar resonances in this subsection. As given in Eq. +(11), their branching ratios can be obtained by using B(D → Sℓ+νℓ) and B(S → P1P2). The detail analysis of +B(D → Sℓ+νℓ) by the SU(3) flavor symmetry can be found in Ref. [81]. +1. +Branching ratios of the S → P1P2 decays +As for the S → P1P2 decays, the partial decay widths can be written as [85] +Γ(S → P1P2) = +pc +8πm2 +S +g2 +S→P1P2, +(27) +where the center of mass momentum pc ≡ +� +λ(m2 +S,m2 +P1,m2 +P2) +2mS +, and gS→P1P2 is the strong coupling constant. With the +SU(3) flavor symmetry, the strong coupling constant can be parameterized as +g2q +S→P1P2 = g2Si +jP k +i P j +k +(28) +for the two quark scalar states, and +g4q +S→P1P2 = g4Sim +jn P j +i P n +m + g′ +4Sim +jmP n +i P j +n +(29) +for the four quark scalar states, where g2, g4 and g′ +4 are the nonperturbative parameters. The strong coupling constants +of these decays are listed in the second and third columns of Tab. IV for the two quark scalar states and the four +quark scalar states, respectively. +Since the width determination is very model dependent, there are not accurate values about the decay widths +of a0(980), f0(980) and f0(500) mesons in Ref. [11]. Therefore, it is difficult to obtain accurate B(S → P1P2) in +terms of Γ(S → P1P2)/ΓS, where ΓS is the decay width of scalar meson. We assume the light scalar mesons decay +dominantly into pairs of pseudoscalar mesons and all other decay channels are negligible, and then one can obtain +B(S → P1P2) without the decay width values of the light scalar mesons, for an example, B(f0(500) → π+π−) ≈ +Γ(f0(500)→π+π−) +Γ(f0(500)→π+π−)+Γ(f0(500)→π0π0). +In the two quark picture, the parameter g2 is canceled in the branching ratios. Therefore, B(K0 → πK, a0(980) → +KK, f0(500) → ππ) only depend on the masses of relevant mesons, B(a0(980) → η′π, η′π) depend on the meson masses +and the mixing angle θP , and B(f0(980) → ππ, KK) depend on the meson masses and the mixing angle θS. The +numerical results of B(S → P1P2) in the two quark picture are listed in the second column of Tab. V. One can see that +the branching ratios of the K0, a0(980), f0(500) decays are accurately predicted, nevertheless, B(f0(980) → ππ, KK) +are predicted with large error due to the indeterminate mixing angle θS. The three possible ranges for the mixing +angle θS, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ [69, 79], have been considered, and the predictions +of B(f0(980) → ππ, KK) are quite dependent on the mixing angle θS. +In the third column of Tab. +V, we also give the predictions with two quark picture of B(S → P1P2) further +constrained from the relevant experimental data of B(D → Sℓ+νℓ, S → P1P2) listed in later Tabs. VI-VII. The + +13 +TABLE IV: The strong coupling constants of the S → P1P2 decays by the SU(3) flavor symmetry. +strong couplings +ones for two quark state +ones for four quark state +gK− +0 →π0K− +1 +√ +2 g2 +− 1 +√ +2g4 +gK− +0 →π−K0 +g2 +g4 +gK0 +0→π+K− +g2 +g4 +gK0 +0→π0K0 +− 1 +√ +2 g2 +1 +√ +2g4 +ga0(980)−→ηπ− +2 g2 +� 1 +√ +6cosθP − +1 +√ +3sinθP +� +2 g′ +4 +� 1 +√ +6cosθP − +1 +√ +3sinθP +� +ga0(980)−→η′π− +2 g2 +� 1 +√ +6sinθP + +1 +√ +3cosθP +� +2 g′ +4 +� 1 +√ +6sinθP + +1 +√ +3cosθP +� +ga0(980)−→K0K− +g2 +g4 +ga0(980)0→ηπ0 +g2 +� 1 +√ +3cosθP − +� 2 +3sinθP +� +g′ +4 +� 1 +√ +6cosθP − +1 +√ +3sinθP +� +ga0(980)0→η′π0 +g2 +� 1 +√ +3sinθP + +� 2 +3cosθP +� +g′ +4 +� 1 +√ +6sinθP + +1 +√ +3cosθP +� +ga0(980)0→K+K− +1 +√ +2 g2 +1 +√ +2 g4 +ga0(980)0→K0K0 +− 1 +√ +2 g2 +− 1 +√ +2 g4 +gf0(980)→π+π− +√ +2 g2 sinθS +√ +2 g′ +4 cosφS + g4sinφS +gf0(980)→π0π0 +g2 sinθS +g′ +4 cosφS − +1 +√ +2g4sinφS +gf0(980)→K+K− +g2 cosθS +1 +√ +2g4cosφS +gf0(980)→K0K0 +g2 cosθS +1 +√ +2g4cosφS +gf0(500)→π+π− +√ +2 g2 cosθS +− +√ +2 g′ +4 sinφS + g4cosφS +gf0(500)→π0π0 +g2 cosθS +−g′ +4 sinφS − +1 +√ +2g4cosφS +predictions of B(f0(980) → P1P2) are quite accurate when θS is further constrained from [25◦, 40◦] to [25◦, 36◦], +from [140◦, 165◦] to [144◦, 151◦] and from |φS| ≤ 30◦ to 22◦ ≤ |φS| ≤ 30◦ by the relevant experimental data of +B(D → Sℓ+νℓ, S → P1P2) with 2σ errors. +Since θS in the two quark picture has been further constrained by +B(D → Sℓ+νℓ, S → P1P2), the predictions of B(f(980) → ππ, KK) are more accurate as listed in the third column +of Tab. V. Other B(S → P1P2) are not further constrained from the data of B(D → Sℓ+νℓ, S → P1P2), so we do not +list them in the third column of Tab. V. +In the four quark picture, the two nonperturbative parameters g4 and g′ +4 in the a0(980), f0(980), f0(500) decays, +and |g′ +4/g4| = 0.61 ± 0.13 are obtained by the data Γ(a0(980) → K ¯K)/Γ(a0(980) → ηπ) = 0.177 ± 0.048 from PDG +[11]. In this work, we treat g4 and g′ +4 as real number, then two possible cases (g′ +4/g4 > 0 and g′ +4/g4 < 0) are analyzed. +The numerical results with the four quark picture are listed in the last column of Tab. V. As for B(f0(980) → ππ) and +B(f0(500) → ππ), very large errors come from the mixing angles φS, and they are obviously different in the g′ +4/g4 > 0 +and g′ +4/g4 < 0 cases. In general, there is a relative strong phase between g′ +4 and g4, therefore, the common relevant +branching ratios are between ones in the g′ +4/g4 > 0 case and ones in the g′ +4/g4 < 0 case. In addition, B(K0 → P1P2) +are same in both the two quark and four quark pictures. + +14 +2. +Branching ratios of the D → S(S → P1P2)ℓ+νℓ decays +Then B(D → Sℓ+νℓ, S → P1P2) can be obtained in terms of B(S → P1P2) listed in Tab. V and the expressions +of B(D → Sℓ+νℓ) given in Ref. [81]. Using the experimental data of B(D+ +s → f0(980)e+νe) = (2.3 ± 0.8) × 10−3 +[11] as well as B(D → Sℓ+νℓ, S → P1P2) listed in the second columns of Tabs. VI-VII. The numerical results of +B(D → Sℓ+νℓ, S → P1P2) with 2σ errors for the two quark and four quark pictures are given in Tab. VI and Tab. +VII for the c → sℓ+νℓ and c → dℓ+νℓ transitions, respectively. Our comments on the results are as follows. +• The +experimental +lower +limits +of +B(D0 +→ +a0(980)−e+νe, +a0(980)− +→ +ηπ−) +and +B(D+ +→ +f0(500)e+νe, f0(500) → π+π−) have not been used to constrain the predictions of B(D → Sℓ+νℓ, S → P1P2), +since the two lower limits of the SU(3) flavor symmetry predictions are slightly lower than their experimental +data in both the two quark and four quark pictures. For B(D0 → a0(980)−e+νe, a0(980)− → ηπ−), one can see +that the prediction in the two quark picture agrees with experimental data within 2σ error bars, nevertheless, the +prediction in the four quark picture is smaller, which only agrees with experimental data within 3σ error bars. +As for B(D+ → f0(500)e+νe, f0(500) → π+π−), the prediction in the two quark picture is much smaller than +its experimental lower limit with 2σ error, nevertheless, the prediction with g′ +4 +g4 > 0 ( g′ +4 +g4 < 0 ) in the four quark +picture agrees with its data within 2σ (3σ) error bars. Therefore, in the later analysis of total contributions to +B(D → P1P2ℓ+νℓ), the predictions of B(D → Sℓ+νℓ, S → P1P2) with g′ +4 +g4 > 0 in the four quark picture will be +used. +• In the two quark picture, though the mixing angle θS only appears in the D → P1P2ℓ+νℓ decays with f0(980) +and f0(500) resonances, all other predictions of the branching ratios are slightly affected by the experimental +constraints. So we list all predictions in the three possible ranges of the mixing angle θS in the 3rd-5th columns +of Tabs. +VI-VII. One can see the all predictions included the decays with f0(980) and f0(500) resonances +are similar in the three possible ranges of the mixing angle θS. As mentioned before, θS is constrained from +[25◦, 40◦] to [25◦, 36◦], from [140◦, 165◦] to [144◦, 151◦] and from |φS| ≤ 30◦ to 22◦ ≤ |φS| ≤ 30◦ by the relevant +experimental data with 2σ errors. +• A lot of the branching ratio predictions are quite different between the two quark picture and the four quark +picture. Present datum of B(D+ → f0(500)e+νe, f0(500) → π+π−) favors the four quark picture of scalar +mesons. B(D → Sℓ+νℓ, S → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−3 −10−4). +Due to the CKM matrix element Vcd suppressed, B(D → Sℓ+νℓ, S → P1P2) with the c → dℓ+νℓ transitions are +predicted on the order of O(10−4 − 10−6). +• Some branching ratios of the D → S(S → P1P2)ℓ+νℓ decays have been obtained in Refs. [13, 61]. B(D+ → +Se+νe, S → π+π−) = (6.99 ± 2.46) × 10−4 [13], B(D+ → Sµ+νµ, S → π+π−) = (7.20 ± 2.52) × 10−4 [13], +B(D0 → a0(980)−ℓ+νℓ, a0(980)− → ηπ−) = (1.36 ± 0.21) × 10−4 [61]. Our predictions in the four quark picture +of B(D+ → Sℓ+νℓ, S → π+π−) are consistent with ones in Ref. [13], our predictions in the two quark picture +of B(D0 → a0(980)−ℓ+νℓ, a0(980)− → ηπ−) are consistent with ones in Ref. [61], nevertheless, our predictions +in the four quark picture are smaller than ones in Ref. [61]. + +15 +TABLE V: Branching ratios of the S → P1P2 decays within 2σ errors. The results are obtained by the SU(3) flavor symmetry +relations and Γ(a0(980) → K ¯K)/Γ(a0(980) → ηπ) = 0.177 ± 0.048 [11]. +†denotes the results with +g′ +4 +g4 > 0, and ♯denotes ones +with +g′ +4 +g4 < 0. +Branching ratios +ones with 2q state in S1 case +ones with 2q state in S2 case +ones with 4q state +B(K− +0 → π0K−) +0.34 ± 0.00 +0.34 ± 0.00 +B(K− +0 → π−K +0) +0.66 ± 0.00 +0.66 ± 0.00 +B(K +0 +0 → π+K−) +0.67 ± 0.00 +0.67 ± 0.00 +B(K +0 +0 → π0K +0) +0.33 ± 0.00 +0.33 ± 0.00 +B(a0(980)− → ηπ−) +0.64 ± 0.04 +0.86 ± 0.03 +B(a0(980)− → η′π−) +0.03 ± 0.01 +0.04 ± 0.01 +B(a0(980)− → K0K−) +0.33 ± 0.03 +0.10 ± 0.02 +B(a0(980)0 → ηπ0) +0.60 ± 0.04 +0.67 ± 0.06 +B(a0(980)0 → η′π0) +0.04 ± 0.01 +0.05 ± 0.02 +B(a0(980)0 → K+K−) +0.19 ± 0.02 +0.15 ± 0.03 +B(a0(980)0 → K0 ¯K0) +0.17 ± 0.01 +0.13 ± 0.03 +0.45 ± 0.09θS=[25◦,40◦] +0.43 ± 0.07θS=[25◦,35◦] +0.42 ± 0.16† +B(f0(980) → π+π−) +0.36 ± 0.17θS=[140◦,165◦] +0.41 ± 0.09θS=[144◦,158◦] +0.59 ± 0.13♯ +0.22 ± 0.22θS=[−30◦,30◦] +0.38 ± 0.06[22◦≤|θS|≤30◦] +0.22 ± 0.04θS=[25◦,40◦] +0.21 ± 0.03θS=[25◦,35◦] +0.34 ± 0.11† +B(f0(980) → π0π0) +0.18 ± 0.09θS=[140◦,165◦] +0.21 ± 0.04θS=[144◦,158◦] +0.20 ± 0.10♯ +0.11 ± 0.11θS=[−30◦,30◦] +0.19 ± 0.03[22◦≤|θS|≤30◦] +0.17 ± 0.07θS=[25◦,40◦] +0.19 ± 0.05θS=[25◦,35◦] +B(f0(980) → K+K−) +0.24 ± 0.14θS=[140◦,165◦] +0.20 ± 0.07θS=[144◦,158◦] +0.12 ± 0.04 +0.35 ± 0.17θS=[−30◦,30◦] +0.22 ± 0.04[22◦≤|θS|≤30◦] +0.16 ± 0.06θS=[25◦,40◦] +0.17 ± 0.05θS=[25◦,35◦] +B(f0(980) → K0 ¯K0) +0.22 ± 0.12θS=[140◦,165◦] +0.18 ± 0.06θS=[144◦,158◦] +0.11 ± 0.04 +0.32 ± 0.16θS=[−30◦,30◦] +0.20 ± 0.04[22◦≤|θS|≤30◦] +B(f0(500) → π+π−) +0.66 ± 0.00 +0.73 ± 0.09† +0.57 ± 0.12♯ +B(f0(500) → π0π0) +0.34 ± 0.00 +0.27 ± 0.09† +0.43 ± 0.12♯ + +16 +TABLE VI: The experimental data and the SU(3) flavor symmetry predictions of the D → S(S → P1P2)ℓ+νℓ decays with the c → sℓ+νℓ transitions within 2σ errors. +†denotes the results with +g′ +4 +g4 > 0, and ♯ denotes ones with +g′ +4 +g4 < 0. +Branching ratios +Exp. Data +Ones in the 2-quark picture with +Ones in the 4-quark picture +θS = [25◦, 35◦] +θS = [144◦, 158◦] +22◦ ≤ |θS| ≤ 30◦ +B(D0 → K− +0 e+νe, K− +0 → π−K +0)(×10−4) +· · · +19.99 ± 7.34 +19.86 ± 7.26 +19.74 ± 6.97 +8.37 ± 3.01 +B(D0 → K− +0 e+νe, K− +0 → π0K−)(×10−4) +· · · +10.18 ± 3.77 +10.12 ± 3.73 +10.05 ± 3.57 +4.19 ± 1.50 +B(D+ → K +0 +0e+νe, K +0 +0 → π+K−)(×10−3) +· · · +5.17 ± 1.92 +5.19 ± 1.85 +5.12 ± 1.86 +2.24 ± 0.83 +B(D+ → K +0 +0e+νe, K +0 +0 → π0K +0)(×10−3) +· · · +2.57 ± 0.96 +2.59 ± 0.92 +2.55 ± 0.92 +1.12 ± 0.42 +B(D+ +s → f0(980)e+νe, f0(980) → π+π−)(×10−3) +1.30 ± 0.63 [86] +1.19 ± 0.18 +1.17 ± 0.17 +1.18 ± 0.17 +1.22 ± 0.55†, +1.44 ± 0.49♯ +B(D+ +s → f0(980)e+νe, f0(980) → π0π0)(×10−4) +7.9 ± 2.9 [4] +5.95 ± 0.92 +5.89 ± 0.85 +5.90 ± 0.86 +7.91 ± 2.85†, +7.13 ± 2.10♯ +B(D+ +s → f0(980)e+νe, f0(980) → K+K−)(×10−4) +· · · +5.11 ± 2.34 +5.53 ± 2.78 +6.28 ± 2.07 +3.33 ± 1.53†, +3.07 ± 1.34♯ +B(D+ +s → f0(980)e+νe, f0(980) → K0K +0)(×10−4) +· · · +4.62 ± 2.12 +5.01 ± 2.52 +5.68 ± 1.87 +3.01 ± 1.39†, +2.78 ± 1.22♯ +B(D+ +s → f0(500)e+νe, f0(500) → π+π−)(×10−4) +· · · +9.91 ± 2.83 +9.67 ± 3.07 +9.44 ± 3.30 +2.49 ± 2.49†, +0.90 ± 0.90♯ +B(D+ +s → f0(500)e+νe, f0(500) → π0π0)(×10−5) +< 64 [4] +49.77 ± 14.23 +48.57 ± 15.43 +47.44 ± 16.56 +6.66 ± 6.66†, +0.78 ± 0.78♯ +B(D0 → K− +0 µ+νµ, K− +0 → π−K0)(×10−4) +· · · +17.27 ± 6.48 +17.16 ± 6.41 +17.04 ± 6.14 +7.19 ± 2.63 +B(D0 → K− +0 µ+νµ, K− +0 → π0K−)(×10−4) +· · · +8.63 ± 3.24 +8.58 ± 3.20 +8.52 ± 3.07 +3.59 ± 1.32 +B(D+ → K +0 +0µ+νµ, K +0 +0 → π+K−)(×10−3) +· · · +4.43 ± 1.68 +4.46 ± 1.62 +4.40 ± 1.62 +1.92 ± 0.73 +B(D+ → K +0 +0µ+νµ, K +0 +0 → π0K0)(×10−3) +· · · +2.22 ± 0.84 +2.23 ± 0.81 +2.20 ± 0.81 +0.96 ± 0.36 +B(D+ +s → f0(980)µ+νµ, f0(980) → π+π−)(×10−3) +· · · +1.01 ± 0.16 +1.00 ± 0.15 +1.00 ± 0.16 +1.02 ± 0.46†, +1.23 ± 0.42♯ +B(D+ +s → f0(980)µ+νµ, f0(980) → π0π0)(×10−4) +· · · +5.05 ± 0.83 +4.99 ± 0.77 +5.00 ± 0.78 +6.72 ± 2.48†, +6.04 ± 1.82♯ +B(D+ +s → f0(980)µ+νµ, f0(980) → K+K−)(×10−4) +· · · +4.31 ± 1.94 +4.70 ± 2.34 +5.34 ± 1.75 +2.79 ± 1.28†, +2.59 ± 1.14♯ +B(D+ +s → f0(980)µ+νµ, f0(980) → K0K +0)(×10−4) +· · · +3.90 ± 1.76 +4.25 ± 2.12 +4.83 ± 1.58 +2.52 ± 1.16†, +2.34 ± 1.03♯ +B(D+ +s → f0(500)µ+νµ, f0(500) → π+π−)(×10−4) +· · · +8.88 ± 2.62 +8.70 ± 2.86 +8.49 ± 3.05 +2.30 ± 2.30†, +0.83 ± 0.83♯ +B(D+ +s → f0(500)µ+νµ, f0(500) → π0π0)(×10−5) +· · · +44.67 ± 13.23 +43.85 ± 14.53 +42.77 ± 15.49 +6.16 ± 6.16†, +7.23 ± 7.23♯ + +17 +TABLE VII: The experimental data and the SU(3) flavor symmetry predictions of the D → S(S → P1P2)ℓ+νℓ decays with the c → dℓ+νℓ transitions within 2σ errors. +† denotes the results with +g′ +4 +g4 > 0, ♯ denotes ones with +g′ +4 +g4 < 0, and a denotes the experimental lower limits have not used to constrain the predictions. +Branching ratios +Exp. Data +Ones in the 2-quark picture with +Ones in the 4-quark picture +θS = [25◦, 35◦] +θS = [144◦, 158◦] +22◦ ≤ |θS| ≤ 30◦ +B(D0 → a0(980)−e+νe, a0(980)− → ηπ−)(×10−5) +13.3+6.8 +−6.0a +5.99 ± 2.69 +5.86 ± 2.48 +6.05 ± 2.57 +3.81 ± 0.98 +B(D0 → a0(980)−e+νe, a0(980)− → η′π−)(×10−6) +· · · +2.88 ± 1.71 +2.97 ± 1.77 +2.97 ± 1.73 +1.88 ± 0.98 +B(D0 → a0(980)−e+νe, a0(980)− → K0K−)(×10−6) +· · · +29.99 ± 13.81 +30.73 ± 13.81 +30.57 ± 13.70 +4.22 ± 1.93 +B(D+ → a0(980)0e+νe, a0(980)0 → ηπ0)(×10−5) +17+16 +−14 +7.35 ± 3.28 +7.25 ± 3.13 +7.32 ± 3.17 +4.00 ± 1.00 +B(D+ → a0(980)0e+νe, a0(980)0 → η′π0)(×10−6) +· · · +5.53 ± 3.26 +5.69 ± 3.32 +5.65 ± 3.20 +3.08 ± 1.56 +B(D+ → a0(980)0e+νe, a0(980)0 → K+K−)(×10−5) +· · · +2.28 ± 1.06 +2.30 ± 1.00 +2.29 ± 0.99 +0.88 ± 0.36 +B(D+ → a0(980)0e+νe, a0(980)0 → K0K +0)(×10−5) +· · · +1.99 ± 0.92 +2.01 ± 0.88 +2.00 ± 0.86 +0.77 ± 0.31 +B(D+ → f0(980)e+νe, f0(980) → π+π−)(×10−5) +< 2.8 [5] +1.15 ± 0.50 +1.10 ± 0.58 +0.96 ± 0.43 +1.65 ± 1.15†, +2.14 ± 0.65♯ +B(D+ → f0(980)e+νe, f0(980) → π0π0)(×10−6) +· · · +5.75 ± 2.53 +5.51 ± 2.92 +4.80 ± 2.18 +10.53 ± 3.67†, +10.10 ± 5.37♯ +B(D+ → f0(980)e+νe, f0(980) → K+K−)(×10−6) +· · · +5.07 ± 0.88 +5.06 ± 0.85 +5.01 ± 0.80 +4.35 ± 2.78†, +4.60 ± 2.76♯ +B(D+ → f0(980)e+νe, f0(980) → K0K +0)(×10−6) +· · · +5.07 ± 0.88 +5.06 ± 0.85 +5.01 ± 0.80 +4.35 ± 2.78†, +4.60 ± 2.76♯ +B(D+ → f0(500)e+νe, f0(500) → π+π−)(×10−4) +6.3 ± 1.0a +1.44 ± 0.64 +1.72 ± 0.92 +1.79 ± 0.85 +3.64 ± 2.57†, +2.95 ± 1.87♯ +B(D+ → f0(500)e+νe, f0(500) → π0π0)(×10−4) +· · · +0.72 ± 0.32 +0.87 ± 0.46 +0.91 ± 0.43 +1.45 ± 1.02†, +2.08 ± 1.57♯ +B(D+ +s → K0 +0e+νe, K0 +0 → π−K+)(×10−5) +· · · +22.34 ± 8.09 +22.13 ± 7.97 +22.34 ± 7.64 +9.54 ± 3.38 +B(D+ +s → K0 +0e+νe, K0 +0 → π0K0)(×10−5) +· · · +11.17 ± 4.04 +11.07 ± 3.99 +11.17 ± 3.82 +4.77 ± 1.69 +B(D0 → a0(980)−µ+νµ, a0(980)− → ηπ−)(×10−5) +· · · +4.95 ± 2.27 +4.84 ± 2.10 +5.00 ± 2.18 +3.14 ± 0.84 +B(D0 → a0(980)−µ+νµ, a0(980)− → η′π−)(×10−6) +· · · +2.39 ± 1.44 +2.46 ± 1.48 +2.45 ± 1.45 +1.56 ± 0.82 +B(D0 → a0(980)−µ+νµ, a0(980)− → K0K−)(×10−6) +· · · +24.78 ± 11.68 +25.37 ± 11.62 +25.20 ± 11.53 +3.51 ± 1.62 +B(D+ → a0(980)0µ+νµ, a0(980)0 → ηπ0)(×10−5) +· · · +6.09 ± 2.78 +6.00 ± 2.65 +6.06 ± 2.69 +3.30 ± 0.86 +B(D+ → a0(980)0µ+νµ, a0(980)0 → η′π0)(×10−6) +· · · +4.58 ± 2.74 +4.72 ± 2.79 +4.67 ± 2.69 +2.55 ± 1.31 +B(D+ → a0(980)0µ+νµ, a0(980)0 → K+K−)(×10−5) +· · · +1.89 ± 0.89 +1.91 ± 0.85 +1.89 ± 0.83 +0.73 ± 0.30 +B(D+ → a0(980)0µ+νµ, a0(980)0 → K0K +0)(×10−5) +· · · +1.65 ± 0.78 +1.66 ± 0.74 +1.65 ± 0.73 +0.64 ± 0.27 +B(D+ → f0(980)µ+νµ, f0(980) → π+π−)(×10−5) +· · · +0.94 ± 0.43 +0.91 ± 0.48 +0.79 ± 0.36 +1.37 ± 0.96†, +1.76 ± 0.55♯ +B(D+ → f0(980)µ+νµ, f0(980) → π0π0)(×10−6) +· · · +4.74 ± 2.14 +4.58 ± 2.43 +3.97 ± 1.82 +8.67 ± 3.13†, +8.32 ± 4.47♯ +B(D+ → f0(980)µ+νµ, f0(980) → K+K−)(×10−6) +· · · +4.21 ± 0.73 +4.19 ± 0.71 +4.15 ± 0.67 +3.55 ± 2.29†, +3.76 ± 2.26♯ +B(D+ → f0(980)µ+νµ, f0(980) → K0K +0)(×10−6) +· · · +4.21 ± 0.73 +4.19 ± 0.71 +4.15 ± 0.67 +3.55 ± 2.29†, +3.76 ± 2.26♯ +B(D+ → f0(500)µ+νµ, f0(980) → π+π−)(×10−4) +· · · +1.28 ± 0.59 +1.54 ± 0.84 +1.61 ± 0.79 +3.30 ± 2.39†, +2.68 ± 1.74♯ +B(D+ → f0(500)µ+νµ, f0(980) → π0π0)(×10−4) +· · · +0.64 ± 0.30 +0.78 ± 0.43 +0.81 ± 0.40 +1.32 ± 0.95†, +1.89 ± 1.46♯ +B(D+ +s → K0 +0µ+νµ, K0 +0 → π−K+)(×10−5) +· · · +19.61 ± 7.20 +19.43 ± 7.10 +19.60 ± 6.80 +8.38 ± 3.01 +B(D+ +s → K0 +0µ+νµ, K0 +0 → π0K0)(×10−5) +· · · +9.80 ± 3.60 +9.71 ± 3.55 +9.80 ± 3.40 +4.19 ± 1.50 + +18 +C. +D → V (V → P1P2)ℓ+νℓ decays +We will analyze the D → P1P2ℓ+νℓ decays with the vector resonances in this subsection. Since the light vector +mesons are understood well, the calculations of B(D → V ℓ+νℓ, V → P1P2) are much easier than ones of B(D → +Sℓ+νℓ, S → P1P2). From Eq. (11), their branching ratios of D → V (V → P1P2)ℓ+νℓ can be obtained by using +B(D → V ℓ+νℓ) and B(V → P1P2). The D → V ℓ+νℓ decays have been studied by the SU(3) flavor symmetry in Ref. +[81]. Many B(D → V ℓ+νℓ) have been accurately measured and have been listed in the second column of Tab. V in +Ref. [81]. The expressions of B(D → V ℓ+νℓ) within the C3 case in Ref. [81] will be taken for our analysis. +Following Ref. [85], B(V → P1P2) can be written as +B(V → P1P2) = τV p′3 +c +6πm2 +V +g2 +V →P1P2, +(30) +where p′ +c ≡ +� +λ(m2 +V ,m2 +P1,m2 +P2) +2mV +and gV →P1P2 are the strong coupling constants. Similar to g2q +S→P1P2 in Eq. (28), gV →P1P2 +can be parameterized by the SU(3) flavor symmetry +gV →P1P2 = gV V i +j P k +i P j +k, +(31) +where gV is the corresponding nonperturbative parameter. +At present, many involved B(V → P1P2) have been well measured [11] +B(K∗+ → πK) = (99.902 ± 0.018)%, +B(K∗0 → πK) = (99.754 ± 0.042)%, +B(ρ+ → π0π+) = 100%, +B(ρ0 → π+π−) = 100%, +B(φ → K+K−) = (49.1 ± 1.0)%, +B(ω → π+π−) = (1.53+0.22 +−0.26)%. +(32) +Using the following relations from Eq. (31) +√ +2gK∗−→π0K− = gK∗−→π−K0, +√ +2gK∗0→π0K0 = gK∗0→π−K+, +gρ−→π0π− = +√ +3gρ−→η8π− = +� +3/2gρ−→η1π−, +gφ→K+K− = gφ→K0K +0, +(33) +following B(V → P1P2) can be obtained +B(K∗0 → π0K0) = (33.02 ± 0.02)%, +B(K∗0 → π−K+) = (66.74 ± 0.04)%, +B(K∗+ → π0K+) = (33.62 ± 0.01)%, +B(K∗+ → π−K0) = (66.28 ± 0.01)%, +B(ρ+ → ηπ+) = (4.38 ± 0.66)%, +B(φ → K0K0) = (32.42 ± 1.04)%. +(34) +For D → V (V → P1P2)ℓ+νℓ decays, the branching ratios of D+ → K +∗0(K +∗0 → π+K−)e+νe and D+ → K +∗0(K +∗0 → +π+K−)µ+νµ have been measured, and the experimental data with 2σ errors are listed in the second column of Tab. +VIII. Using the experimental data of B(D+ → K +∗0ℓ+νℓ, K +∗0 → π+K−), B(V → P1P2) and B(D → V ℓ+νℓ), we +obtain the predictions of B(D → V ℓ+νℓ, V → P1P2) by the SU(3) flavor symmetry, which are given in the third +column of Tab. VIII. We can see that B(D → V ℓ+νℓ, V → P1P2) with the c → sℓ+νℓ transitions are predicted on +the order of O(10−2 − 10−3), and B(D → V ℓ+νℓ, V → P1P2) with the c → dℓ+νℓ transitions are predicted on the + +19 +TABLE VIII: The experimental data and the SU(3) flavor symmetry predictions of D → V (V → P1P2)ℓ+νℓ decays within 2σ +errors. +Branching ratios +Exp. Data +Our predictions +Previous ones +c → se+νe: +B(D0 → K∗−e+νe, K∗− → π−K +0)(×10−2) +. . . +1.42 ± 0.07 +. . . +B(D0 → K∗−e+νe, K∗− → π0K−)(×10−3) +. . . +7.18 ± 0.37 +7.17 [62] +B(D+ → K +∗0e+νe, K +∗0 → π+K−)(×10−2) +3.77 ± 0.34 +3.64 ± 0.11 +3.51 [62] +B(D+ → K +∗0e+νe, K +∗0 → π0K +0)(×10−2) +. . . +1.80 ± 0.06 +. . . +B(D+ +s → φe+νe, φ → K+K−)(×10−2) +. . . +1.20 ± 0.10 +. . . +B(D+ +s → φe+νe, φ → K0K +0)(×10−3) +. . . +7.94 ± 0.65 +. . . +c → sµ+νµ: +B(D0 → K∗−µ+νµ, K∗− → π−K +0)(×10−2) +. . . +1.33 ± 0.07 +. . . +B(D0 → K∗−µ+νµ, K∗− → π0K−)(×10−3) +. . . +6.76 ± 0.35 +7.17 [62] +B(D+ → K +∗0µ+νµ, K +∗0 → π+K−)(×10−2) +3.52 ± 0.20 +3.43 ± 0.11 +3.51 [62] +B(D+ → K +∗0µ+νµ, K +∗0 → π0K +0)(×10−2) +. . . +1.70 ± 0.05 +. . . +B(D+ +s → φµ+νµ, φ → K+K−)(×10−2) +. . . +1.13 ± 0.09 +. . . +B(D+ +s → φµ+νµ, φ → K0K +0)(×10−3) +. . . +7.46 ± 0.62 +. . . +c → de+νe: +B(D0 → ρ−e+νe, ρ− → π0π−)(×10−3) +. . . +1.85 ± 0.11 +1.63 [62] +B(D0 → ρ−e+νe, ρ− → ηπ−)(×10−5) +. . . +8.23 ± 1.59 +. . . +B(D+ → ρ0e+νe, ρ0 → π+π−)(×10−3) +. . . +2.40 ± 0.12 +1.57 ± 0.07 [13], +2.10 [62] +B(D+ → ωe+νe, ω → π+π−)(×10−5) +. . . +3.55 ± 0.82 +. . . +B(D+ +s → K∗0e+νe, K∗0 → π−K+)(×10−3) +. . . +1.49 ± 0.10 +. . . +B(D+ +s → K∗0e+νe, K∗0 → π0K0)(×10−4) +. . . +7.39 ± 0.51 +. . . +c → dµ+νµ: +B(D0 → ρ−µ+νµ, ρ− → π0π−)(×10−3) +. . . +1.76 ± 0.10 +. . . +B(D0 → ρ−µ+νµ, ρ− → ηπ−)(×10−5) +. . . +7.83 ± 1.51 +. . . +B(D+ → ρ0µ+νµ, ρ0 → π+π−)(×10−3) +. . . +2.29 ± 0.11 +1.57 ± 0.07 [13] +B(D+ → ωµ+νµ, ω → π+π−)(×10−5) +. . . +3.38 ± 0.78 +. . . +B(D+ +s → K∗0µ+νµ, K∗0 → π−K+)(×10−3) +. . . +1.42 ± 0.10 +. . . +B(D+ +s → K∗0µ+νµ, K∗0 → π0K0)(×10−4) +. . . +7.03 ± 0.48 +. . . + +20 +order of O(10−3 − 10−5). The predictions of B(D → V ℓ+νℓ, V → P1P2) are about one order larger than ones of the +corresponding B(D → Sℓ+νℓ, S → P1P2). +Previous predictions are also listed in the last column of Tab. VIII. Our predictions of B(D0 → K∗−ℓ+νℓ, K∗− → +π0K−) and B(D+ → K +∗0ℓ+νℓ, K +∗0 → π+K−) are in good agreement with ones in Ref. [62]. And our predictions of +B(D+ → ρ0ℓ+νℓ, ρ0 → π+π−) are slight larger than ones obtained by the light-front quark model and the light-cone +sum rules in Ref. [13]. +D. +Total branching ratios +As analyzed in above, some four-body semileptonic decays of D mesons receive the contributions of the non-resonant +states, the scalar resonant states and the vector resonant states, nevertheless, some decay modes only receive one or two +kinds of them. For clearly showing the resonant contributions, we also list the scalar and vector resonant amplitudes in +the third and last columns of Tab. I, respectively. The resonant amplitudes are obtained by multiplying the hadronic +helicity amplitudes H(D → Rℓ+νℓ) given in Ref. [81] and the strong coupling constants gR→P1P2 obtained in this +work. Noted that the resonant amplitudes listed in the last two columns of Tab. I only for clearly see the kinds of the +resonant contributions, and we do not using them to obtain the numerical total branching ratios B(D → P1P2ℓ+νℓ)T . +We have some comments for the contributions in Tab. I. For D(s) → ηKℓ+νℓ, η′Kℓ+νℓ, ηηℓ+νℓ, ηη′ℓ+νℓ decays, +since the both final state mesons are quite heavy, they only receive the non-resonant contributions. +The decays +D+ +s → π0π0ℓ+νℓ, D+ +s → π+π−ℓ+νℓ, D0 → K−K0ℓ+νℓ, D+ → K +0K0ℓ+νℓ, D+ → K+K−ℓ+νℓ, D+ → π0π0ℓ+νℓ and +D+ → η(′)π0ℓ+νℓ receive both the non-resonant contributions and the scalar resonant contributions, moreover, the +non-resonant contributions in the D+ +s → π0π0ℓ+νℓ, D+ +s → π+π−ℓ+νℓ and D+ → K+K−ℓ+νℓ decays are suppressed +by the OZI rule, and the main contributions of these decay branching ratios come from the scalar resonant states. All +other decay modes except the D0 → π0π−ℓ+νℓ decays receive all three kinds of the contributions, and their branching +ratios are dominant by the vector resonant states. Due to the quantum number constraint, the D0 → π0π−ℓ+νℓ +decays only receive the contributions of the vector resonant states. +In the last columns of Tabs. II-III, total branching ratio predictions of the D → P1P2ℓ+ν decays including the +possible non-resonant, scalar resonant and vector resonant contributions are listed. The present six experimental data +with 2σ errors are also listed in the forth column of Tab. II and in third column of Tab. III for convenient comparison. +One can see that, for B(D0 → π−K +−e+νe), B(D0 → π0K−e+νe), B(D+ → π+K−e+νe), B(D+ → π+K−µ+νµ) and +B(D+ → π+π−e+νe), our SU(3) flavor symmetry predictions are consistent with present data within 2σ error bars. +Our prediction of B(D0 → π0π−e+νe) is slightly larger than its experimental datum, nevertheless, the prediction will +be very close to the datum within 3σ error bars. +For some Cabibbo suppressed decays due to c → dℓ+νℓ transitions, such as the D0 → K−K0ℓ+νℓ, D0 → η′π−ℓ+νℓ, +D+ → K +0K0ℓ+νℓ, D+ → π0π0ℓ+νℓ, D+ → ηπ0ℓ+νℓ and D+ → η′π0ℓ+νℓ decays, they only receive both the non- +resonant contributions and the scalar resonant contributions, and we can see that both the non-resonant and the +scalar resonant contributions are important. The non-resonant contributions in the D+ → K+K−ℓ+νℓ decays are +suppressed by the OZI rule, and the scalar resonant contributions in the D+ → K+K−ℓ+νℓ decays are dominant. + +21 +IV. +Summary +Semileptonic decays of heavy mesons are quite interesting because of not only relatively simple theoretical description +but also the clean experimental signals. Some semileptonic decays D → P1P2ℓ+νℓ have been measured by BESIII, +CLEO and BABAR, etc. Using the present data of B(D → P1P2ℓ+νℓ) and the SU(3) flavor symmetry, we have +presented a theoretical analysis of the D → P1P2ℓ+νℓ decays with the non-resonant, the light scalar meson resonant +and the vector meson resonant contributions. +• Non-resonant D → P1P2ℓ+νℓ decays: The amplitude relations included the SU(3) flavor breaking effects have +been obtained. Almost all amplitudes can be related after ignoring the OZI suppressed and the SU(3) flavor +breaking contributions. Via the experimental data of the non-resonant branching ratios B(D+ → π+K−ℓ+νℓ)N, +we have predicted other non-resonant branching ratios. We have found that the branching ratios of the non- +resonant decays D0 → π−K +0ℓ+νℓ, π0K−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K +0ℓ+νℓ, π+π−ℓ+νℓ, π0π0ℓ+νℓ, and D+ +s → +K+K−ℓ+νℓ, K0K +0ℓ+νℓ are on the order of O(10−3 − 10−4), which might be measured by the BESIII, LHCb +and BelleII experiment, and some other decays might be measured at these experiments in near future. +• Decays with the light scalar meson resonances: Using the SU(3) flavor symmetry and the present +experimental data of B(D → Sℓ+νℓ), B(D → Sℓ+νℓ, S → P1P2) as well as B(S → P1P2), the not- +measured B(D → Sℓ+νℓ, S → P1P2) have been obtained by the SU(3) flavor symmetry. We have found that +B(D → Sℓ+νℓ, S → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−3 − 10−4), and +B(D → Sℓ+νℓ, S → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−4−10−6). The two +quark picture and the four quark picture for the scalar mesons have been analyzed in the D → S(S → P1P2)ℓ+νℓ +decays. Present experimental data might favorite the four quark picture for the scalar mesons. +• Decays with the vector meson resonances: +Using the experimental data of B(D+ → K +∗0e+νe, K +∗0 → +π+K−), B(D+ → K +∗0µ+νµ, K +∗0 → π+K−), many B(D → V ℓ+νℓ) and many B(V → P1P2), the not-measured +B(D → V ℓ+νℓ, V → P1P2) have been predicted by the SU(3) flavor symmetry. We have found that B(D → +V ℓ+νℓ, V → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−2 − 10−3), and B(D → +V ℓ+νℓ, V → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−3 − 10−5). +• Total branching ratios: Total branching ratio predictions including the possible non-resonant, light scalar +meson resonant and vector meson resonant contributions have been obtained. +The six total branching ra- +tios have been measured, and we did not use them to further constrain the predictions. +Our five predic- +tions are consistent with present data within 2σ errors, and the prediction of B(D0 → π0π−e+νe) will be +very close to the datum within 3σ error bars. We have found that the vector meson resonant contributions +are dominant in the D0 → π−K +0ℓ+νℓ, π0K−ℓ+νℓ, π0π−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K +0ℓ+νℓ, π+π−ℓ+νℓ, and +D+ +s → K+K−ℓ+νℓ, K0K +0ℓ+νℓ, K+π−ℓ+νℓ, K0π0ℓ+νℓ decays. 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D 92 (2015) no.1, 012009 [arXiv:1505.04205 [hep-ex]]. + diff --git a/FdAyT4oBgHgl3EQfSvci/content/tmp_files/load_file.txt b/FdAyT4oBgHgl3EQfSvci/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8b874941d9bce2a57cfcf5f36d3989517a64c4e --- /dev/null +++ b/FdAyT4oBgHgl3EQfSvci/content/tmp_files/load_file.txt @@ -0,0 +1,2700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf,len=2699 +page_content='Four-body Semileptonic Charm Decays D → P1P2ℓ+νℓ Based on SU(3) Flavor Analysis Ru-Min Wang1,†, Yi Qiao1, Yi-Jie Zhang1, Xiao-Dong Cheng2,§, Yuan-Guo Xu1,♯ 1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China 2College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang, Henan 464000, China †ruminwang@sina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='com §chengxd@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ccnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='cn ♯yuanguoxu@jxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='cn Motivated by the significant experimental progress in probing semileptonic decays D → P1P2ℓ+νℓ (ℓ = µ, e), we analyze the branching ratios of the D → P1P2ℓ+νℓ decays with the non- resonant, the light scalar meson resonant and the vector meson resonant contributions in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We obtain the hadronic amplitude relations between different decay modes by the SU(3) flavor analysis, and then predict relevant branching ratios of the D → P1P2ℓ+νℓ decays by the present ex- perimental data with 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Most of our predicted branching ratios are consistent with present experimental data within 2σ error bars, and others are consistent with the data within 3σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We find that the branching ratios of the non-resonant decays D0 → π−K 0ℓ+νℓ, π0K−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K 0ℓ+νℓ, π+π−ℓ+νℓ, π0π0ℓ+νℓ, and D+ s → K+K−ℓ+νℓ, K0K 0ℓ+νℓ are on the order of O(10−3 − 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The vector meson resonant contributions are dominant in the D0 → π−K 0ℓ+νℓ, π0K−ℓ+νℓ, π0π−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K 0ℓ+νℓ, π+π−ℓ+νℓ, and D+ s → K+K−ℓ+νℓ, K0K 0ℓ+νℓ, K+π−ℓ+νℓ, K0π0ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The non-resonant, the vector meson reso- nant and the scalar resonant contributions are all important in the D0 → ηπ−ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The D0 → K−K0ℓ+νℓ, η′π−ℓ+νℓ and D+ → K 0K0ℓ+νℓ, π0π0ℓ+νℓ, ηπ0ℓ+νℓ, η′π0ℓ+νℓ decays only receive both the non-resonant and the scalar resonant contributions, and both contributions are important in their branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' According to our predictions, many decay modes could be observed in the experiments like BESIII, LHCb and BelleII, and some decay modes might be measured in these experiments in near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00090v1 [hep-ph] 31 Dec 2022 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' INTRODUCTION Semileptonic heavy meson decays dominated by tree-level exchange of W-bosons in the SM are very important processes in testing the stand model and in searching for the new physics beyond the stand model, for example, the extraction of the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Four-body semileptonic exclusive decays D → P1P2ℓ+νℓ are generated by the c → s/dℓ+νℓ transitions, and they can receive contributions from the non- resonant, the light scalar meson resonant and the vector meson resonant contributions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Therefore, these decays are also a good laboratory for probing the internal structure of light hadrons [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Some non-resonant D → P1P2ℓ+νℓ decays, the light scalar meson resonant decays D → S(S → P1P2)ℓ+νℓ and the vector meson resonant decays D → S(S → P1P2)ℓ+νℓ have been observed by BESIII, BABAR, CLEO and MARKIII, etc [4–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Present experimental measurements give us an opportunity to additionally test theoretical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Experimental backgrounds of the semileptonic decays are cleaner than ones of the hadronic decays, and theoretical description of the semileptonic exclusive decays are relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Since leptons do not participate in the strong interaction, the weak and strong dynamics can be separated in these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' All the strong dynamics in the initial and final hadrons is included in the hadronic transition form factors, which are important for testing the theoretical calculations of the involved strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The form factors can be calculated, for examples, by the chiral perturbation theory [12], the unitarized chiral perturbation theory [13, 14], the light-cone sum rules [15–17] and the QCD factorization [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Nevertheless, due to our poor understanding of hadronic interactions, the evaluations of the form factors are difficult and often plugged with large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' One needs to find ways to minimize the uncertainties to extract useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the lack of reliable calculations, symmetries provide very important information for particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' SU(3) flavor symmetry is a symmetry in QCD for strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' From the perspective of the SU(3) flavor symmetry, the leptonic part of the D → P1P2ℓ+νℓ decay is SU(3) flavor singlet, which makes no difference between different decay modes with certain lepton (e or µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The different hadronic parts (the hadronic amplitudes or the hadronic form factors) of the D → P1P2ℓ+νℓ decays could be related by the SU(3) flavor symmetry without the detailed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Nevertheless, the size of the hadronic amplitudes or the form factors can not be determined by itself in the SU(3) flavor symmetry approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' However, if experimental data are enough, one may use the data to extract the hadronic amplitudes or the form factors, which can be viewed as predictions based on symmetry, has a smaller dependency on estimated form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Although the SU(3) flavor symmetry is only an approximate symmetry because up, down and strange quarks have different masses, it still provides some very useful information about the decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The SU(3) flavor symmetry has been widely used to study hadron decays, for instance, b-hadron decays [19–32], c-hadron decays [31–46] and light hadron decays [31, 47–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Although the SU(3) flavor symmetry works well in heavy hadron decays, the calculations of SU(3) flavor breaking effects would play a key role in the precise theoretical predictions of the observables and a precise test of the the unitarity of the CKM matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' If up and down quark masses are neglected, a non-zero strange quark mass breaks the SU(3) flavor symmetry down to the isospin symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' When up and down quark mass difference is kept, isospin symmetry is also broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Applications of the SU(3) flavor breaking approach on hadron decays can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 3 [53–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The SU(3) flavor breaking effects due to the fact of ms ≫ mu,d will be considered in our analysis of the non-resonant D → P1P2ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Four body semileptonic decay D → P1P2ℓ+νℓ have been studied, for instance, in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [13, 61–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In this work, we will study the D → P1P2ℓ+νℓ decays with the SU(3) flavor symmetry/breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In three cases of the non-resonant decays, the light scalar meson resonant decays and the vector meson resonant decays, we will firstly construct the hadronic amplitude relations between different decay modes, use the available data to extract the hadronic amplitudes, then predict the not-yet-measured modes for further tests in experiments, and finally analyze the contributions with the non-resonance, the light scalar meson resonances and the vector meson resonances in the branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II, the expressions of the branching ratios are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III, we will give our numerical results of the D → P1P2ℓ+ν decays with the non-resonant, the light scalar meson resonant and the vector meson resonant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Theoretical frame A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Decay branching ratios The effective Hamiltonian for c → qiℓ+νℓ transition can be written as Heff(c → qiℓ+νℓ) = GF √ 2 Vcqi ¯qiγµ(1 − γ5)c ¯νℓγµ(1 − γ5)ℓ, (1) where GF is the Fermi constant, Vcqi is the CKM matrix element, and qi = d, s for i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The decay amplitude of the D(p) → P1(k1)P2(k2)ℓ+(q1)νℓ(q2) decay can be divided into leptonic and hadronic parts A(D → P1P2ℓ+νℓ) = ⟨P1(k1)P2(k2)ℓ+(q1)νℓ(q2)|Heff(c → qiℓ+νℓ)|D(p)⟩ (2) = GF √ 2 VcqiLµHµ, (3) where Lµ = ¯νℓγµ(1 − γ5)ℓ is leptonic charged current, and Hµ = ⟨P1(k1)P2(k2)|¯s/ ¯dγµ(1 − γ5)c|D(p)⟩ is hadronic matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The leptonic part Lµ is calculable using the perturbation theory, while the hadronic part Hµ are encoded into the transition form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [18, 67], the D → P1P2 form factors are given as ⟨P1(k1)P2(k2)|¯s/ ¯dγµc|D(p)⟩ = iF⊥ 1 √ k2 qµ ⊥, (4) −⟨P1(k1)P2(k2)|¯s/ ¯dγµγ5c|D(p)⟩ = Ft qµ � q2 + F0 2 � q2 √ λ kµ 0 + F∥ 1 √ k2 ¯kµ ∥ , (5) with kµ 0 = kµ − k · q q2 qµ, (6) ¯kµ ∥ = ¯kµ − 4(k · q)(q · ¯k) λ kµ + 4k2(q · ¯k) λ qµ, (7) qµ ⊥ = 2ϵµαβγ qαkβ¯kγ √ λ , (8) 4 where k ≡ k1+k2, q ≡ q1+q2, ¯k ≡ k1−k2, ¯q ≡ q2−q1, and λ = λ(m2 D, q2, k2) with λ(a, b, c) = a2+b2+c2−2ab−2bc−2ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In terms of the form factors, the differential branching ratio of the non-resonant D → P1P2ℓ+νℓ decays can be written as [18] dB(D → P1P2ℓ+ν)N dq2 dk2 = 1 2τD|N|2βℓ(3 − βℓ)|FA|2, (9) with |N|2 = G2 F |Vcq|2 βℓq2� λ(m2 D, q2, k2) 3 · 210π5m3 D with βℓ = 1 − m2 ℓ q2 , |FA|2 = |F0|2 + 2 3(|F∥|2 + |F⊥|2) + 3m2 ℓ q2(3 − βℓ)|Ft|2, (10) where τM(mM) is lifetime(mass) of M particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In this work, we ignore the small contributions of |Ft|2 term, which is proportional to m2 ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The corresponding limits of integration are given by (mP1 + mP2)2 ≤ k2 ≤ (mDq − mℓ)2 and m2 ℓ ≤ q2 ≤ (mDq − √ k2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The calculations of the form factors F0, F∥, F⊥ and Ft are quite complicated, and their specific expressions in the QCD factorization limit can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Nevertheless, we will not use the specific expressions in this work, and we will relate the different hadronic decay amplitudes or the different form factors between different decay modes by the SU(3) flavor symmetry/breaking, which are discussed in later Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Except for the non-resonant D → P1P2ℓ+νℓ decays, the resonant D → R(R → P1P2)ℓ+νℓ decays with the scalar(R = S) resonance and the vector(R = V ) resonance are also studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the case of the de- cay widths of the resonances are very narrow, the resonant decay branching ratios respect a simple factorization relation B(D → Rℓ+νℓ, R → P1P2) = B(D → Rℓ+νℓ) × B(R → P1P2), (11) and this result is also a good approximation for wider resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (11) will be used in our analysis for the scalar resonant D → S(S → P1P2)ℓ+νℓ decays and the vector resonant D → V (V → P1P2)ℓ+νℓ decays in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III B and III C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Relevant B(D → Rℓ+νℓ) and B(R → P1P2) are also obtained by the SU(3) flavor symmetry in our later analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Meson multiplets Before giving the hadronic amplitudes based on the SU(3) flavor analysis, we will collect the representations for the multiplets of the SU(3) flavor group first in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Charmed mesons containing one heavy c quark are flavor SU(3) anti-triplets Di = � D0(c¯u), D+(c ¯d), D+ s (c¯s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (12) Light pseudoscalar meson (P) and vector meson (V ) octets and singlets under the SU(3) flavor symmetry of light u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' s quarks are [68] P = � � � � � π0 √ 2 + η8 √ 6 + η1 √ 3 π+ K+ π− − π0 √ 2 + η8 √ 6 + η1 √ 3 K0 K− K 0 − 2η8 √ 6 + η1 √ 3 � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (13) 5 V = � � � � � ρ0 √ 2 + ω √ 2 ρ+ K∗+ ρ− − ρ0 √ 2 + ω √ 2 K∗0 K∗− K ∗0 φ � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (14) where the η and η′ are mixtures of η1 = u¯u+d ¯d+s¯s √ 3 and η8 = u¯u+d ¯d−2s¯s √ 6 with the mixing angle θP � � η η′ � � = � � cosθP −sinθP sinθP cosθP � � � � η8 η1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (15) And θP = [−20◦, −10◦] from Particle Data Group (PDG) [11] will be used in our numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The structures of the light scalar mesons are not fully understood yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Many suggestions are discussed, such as ordinary two quark state, four quark state, meson-meson bound state, molecular state, glueball state or hybrid state, for examples, in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [69–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In this work, we will consider the two quark and the four quark scenarios for the scalar mesons below or near 1 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the two quark picture, the light scalar mesons can be written as [78] S = � � � � � a0 0 √ 2 + σ √ 2 a+ 0 K+ 0 a− 0 − a0 0 √ 2 + σ √ 2 K0 0 K− 0 K 0 0 f0 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (16) The two isoscalars f0(980) and f0(500) are obtained by the mixing of σ = u¯u+d ¯d √ 2 and f0 = s¯s, � � f0(980) f0(500) � � = � � cosθS sinθS −sinθS cosθS � � � � f0 σ � � , (17) where the three possible ranges of the mixing angle θS [69, 79], 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ will be analyzed in our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the four quark picture, the light scalar mesons are given as [11, 80] σ = u¯ud ¯d, f0 = (u¯u + d ¯d)s¯s/ √ 2, a0 0 = (u¯u − d ¯d)s¯s/ √ 2, a+ 0 = u ¯ds¯s, a− 0 = d¯us¯s, K+ 0 = u¯sd ¯d, K0 0 = d¯su¯u, K 0 0 = s ¯du¯u, K+ 0 = s¯ud ¯d, (18) and the two isoscalars are expressed as � � f0(980) f0(500) � � = � � cosφS sinφS −sinφS cosφS � � � � f0 σ � � , (19) where the constrained mixing angle φS = (174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2)◦ [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Non-resonant hadronic amplitudes Since the hadronic amplitudes of the semileptonic D → V/Sℓ+νℓ decays based on the SU(3) flavor symme- try/breaking have been discussed in previous Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81], we will focus on the hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 6 In terms of the SU(3) flavor symmetry, the quark current ¯qiγµ(1−γ5)c can be expressed as a SU(3) flavor anti-triplet (¯3), and the effective Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (1) is transformed as [41] Heff(c → qiℓ+νℓ) = GF √ 2 H(¯3) ¯νℓγµ(1 − γ5)ℓ, (20) with H(¯3) = (0, Vcd, Vcs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The decay amplitude of the non-resonant D → P1P2ℓ+νℓ decay can be written as A(D → P1P2ℓ+νℓ)N = GF √ 2 H(D → P1P2)N ¯νℓγµ(1 − γ5)ℓ, (21) and the hadronic amplitude H(D → P1P2)N can be parameterized as H(D → P1P2)N = c10DiP i jP j kH(¯3)k + c20DiP i jH(¯3)jP k k + c30DiH(¯3)iP j kP k j + c40DiH(¯3)iP k k P j j , (22) where ci0(i = 1, 2, 3, 4) are the nonperturbative coefficients under the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Feynman diagrams for the non-resonant D → P1P2ℓ+νℓ decays are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' SU(3) flavor breaking effects come from different masses of u, d and s quarks, and they will become useful once we have measurements of several D → P1P2ℓ+νℓ decays that are precise enough to see deviations from the SU(3) flavor νℓ ℓ+ c H(3)k qk ¯qj qj ¯qi ¯qi ( a ) νℓ ℓ+ c H(3)j qj ¯qi ¯qi ¯qk qk ( b ) νℓ ℓ+ c H(3)i ¯qi ¯qk qk qj ¯qj ( c ) νℓ ℓ+ c H(3)i ¯qi ¯qk qk ¯qj qj ( d ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1: Diagrams of the non-resonant D → P1P2ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The diagonalized mass matrix can be expressed as [59, 60] � � � � � mu 0 0 0 md 0 0 0 ms � � � � � = 1 3(mu + md + ms)I + 1 2(mu − md)X + 1 6(mu + md − 2ms)W, (23) with X = � � � � � 1 0 0 0 −1 0 0 0 0 � � � � � , W = � � � � � 1 0 0 0 1 0 0 0 −2 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (24) Compared with s quark mass, the u and d quark masses are much smaller which can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The SU(3) flavor breaking effects due to a non-zero s quark mass dominate the SU(3) breaking effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' When u and d quark mass difference is ignored, the residual SU(3) flavor symmetry becomes the isospin symmetry and the term proportional to X can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The identity I part contributes to the D → P1P2ℓ+νℓ decay amplitudes in a similar way as that given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (21) which can be absorbed into the coefficients ci0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Only W part will contribute to the SU(3) breaking effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The SU(3) breaking contributions to the hadronic amplitudes due to the fact of ms ≫ mu,d are ∆H(D → P1P2)N = c11DiW i aP a j P j kH(¯3)k + c12DiP i jW j aP a k H(¯3)k + c13DiP i jP j kW k a H(¯3)a + c21DiW i aP a j H(¯3)jP k k + c22DiP i jW j aH(¯3)aP k k + c31DiW i aH(¯3)aP j kP k j + c32DiH(¯3)iP j kW k a P a j + c41DiW i aH(¯3)aP k k P j j , (25) where cij (i, j = 1, 2, 3, 4) are the nonperturbative SU(3) flavor breaking coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Full hadronic amplitudes of the different non-resonant D → P1P2ℓ+ν decays and their relations under the SU(3) flavor symmetry/breaking are given in later Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Numerical results of the D → P1P2ℓ+ν decays The branching ratios with the non-resonant contributions, the light scalar meson resonant contributions and the vector meson resonant contributions will be analyzed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' If not special specified, the theoretical input parameters, such as the lifetimes and the masses, and the experimental data within the 2σ error bars from PDG [11] will be used in our numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Non-resonant D → P1P2ℓ+ν decays The hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays including both the SU(3) flavor symmetry and the SU(3) flavor breaking terms are summarized in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' I, in which we can see the relations of different hadronic amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The following relations are hold in both the SU(3) flavor symmetry and the SU(3) 8 flavor breaking due to a strange quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' H(D0 → π−K 0ℓ+νℓ)N = H(D+ → π+K−ℓ+νℓ)N = √ 2H(D0 → π0K−ℓ+νℓ)N = − √ 2H(D+ → π0K 0ℓ+νℓ)N, H(D0 → η8K−ℓ+νℓ)N = H(D+ → η8K 0ℓ+νℓ)N, H(D0 → η1K−ℓ+νℓ)N = H(D+ → η1K 0ℓ+νℓ)N, H(D+ s → K+K−ℓ+νℓ)N = H(D+ s → K0K 0ℓ+νℓ)N, H(D0 → K−K0ℓ+νℓ)N = H(D+ → K 0K0ℓ+νℓ)N − H(D+ → K+K−ℓ+νℓ)N, H(D+ s → K+π−ℓ+νℓ)N = − √ 2H(D+ s → K0π0ℓ+νℓ)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (26) If assuming the SU(3) flavor breaking effects are small and can be ignored, more amplitude relations will be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Moreover, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1, the SU(3) flavor symmetry contributions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1 (b-d) are suppressed by the Okubo- Zweig-Iizuka (OZI) rule [82–84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' If ignoring both the OZI suppressed SU(3) flavor symmetry contributions and the SU(3) flavor breaking contributions, almost all hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays can be related by the coefficient c10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Since the leptonic charged current ¯νℓγµ(1 − γ5)ℓ is the SU(3) flavor singlet, and it is completely generic between different decay modes with certain ℓ = e or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The same relations as the hadronic amplitudes listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' I are valid in the decay amplitudes of the D → P1P2ℓ+νℓ decays and the form factors of the D → P1P2 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' For the non-resonant D → P1P2ℓ+νℓ decays, only B(D+ → π+K−µ+νµ)N has been measured, and B(D+ → π+K−e+νe)N has been upper limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Because the non-resonant D → P1P2ℓ+νℓ decays have not been measured enough to reveal the OZI suppressed SU(3) flavor symmetry contributions and the SU(3) symmetry breaking effects, we ignore both of them in our analysis, and then almost all hadronic amplitudes, form factors or decay amplitudes can be related by the SU(3) flavor symmetry coefficient c10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The simple relations associated by the coefficient c10 for FA given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (10) will be used to obtain our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Noted that, for consistency, only the SU(3) flavor symmetry contributions will be considered in the light scalar meson resonant D → S(S → P1P2)ℓ+νℓ decays and the vector meson resonant D → V (V → P1P2)ℓ+νℓ decays in later Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III B and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The experimental data of B(D+ → π+K−µ+νµ)N within 2σ errors and the upper limit of B(D+ → π+K−e+νe)N at 90% confidence level from PDG [11] are listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II, which will be used to determine c10 in the non-resonant D+ → π+K−ℓ+νℓ decays, and then many other branching ratios of the non-resonant D → P1P2ℓ+νℓ decays can be predicted by using the constrained c10 from the data of B(D+ → π+K−ℓ+νℓ)N listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our predictions are listed in the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II for the c → sℓ+νℓ transitions and in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III for the c → dℓ+νℓ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' From Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II-III, one can see that many branching ratios of the non-resonant D → P1P2ℓ+νℓ decays, such as B(D0 → π−K 0ℓ+νℓ)N, B(D0 → π0K−ℓ+νℓ)N, B(D+ → π+K−ℓ+νℓ)N, B(D+ → π0K 0ℓ+νℓ)N, B(D+ s → K+K−ℓ+νℓ)N, B(D+ s → K0K 0ℓ+νℓ)N, B(D+ → π+π−ℓ+νℓ)N and B(D+ → π0π0ℓ+νℓ)N, are on the orders of O(10−3 −10−4), which could be measured by the BESIII, LHCb and BelleII experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Nevertheless, other decays, for examples, the non-resonant D → ηPℓ+νℓ decays, are strongly suppressed by the narrow phase spaces, the mixing angle θP or the CKM matrix element Vcd, their branching ratios are on the orders of O(10−5 − 10−7), and many of them might be observed by the BESIII and BelleII experiments in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 9 TABLE I: The hadronic amplitudes for the D → P1P2ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' C1 ≡ c10 + c11 + c12 − 2c13, C2 ≡ c20 + c21 − 2c22, C3 ≡ c30 − 2c31, C4 ≡ c40 − 2c41, and [C ′,′′]R denotes the contributions come from the decays with R resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='Decay modes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='Non-resonant hadronic amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='Scalar resonant ones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='Vector resonant ones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='c → sℓ+νℓ: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → π−K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K∗− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → η8K−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 C1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → η1K−ℓ+νℓ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → π+K−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K∗0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → π0K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K∗0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η8K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 C1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η1K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3c12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → K+K−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 2C3 −3c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='cos2θS C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → K0K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 2C3 −3c11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='cos2θS C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → π0π0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2C3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2c32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='sinθScosθSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='−sinθScosθSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(500) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → π+π−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2sinθScosθSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2sinθScosθSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(500) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → η8η8ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2c11 + 2c12 + c32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → η1η1ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 (C1 + 3C2 + 3C3 + 9C4) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2(c11 + c12 + 3c21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → η8η1ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2(c11 + c12 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 c21 + c32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='c → dℓ+νℓ: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → K−K0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 −3(c12 − c13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → π0π−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ρ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → η8π−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 C1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='�� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ρ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D0 → η1π−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3(2c13 + 3c22) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ρ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0K0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 2C3 −3(c12 − c13 − 2c31) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980)0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 sinθScosθSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → K+K−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2C3 +6c31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980)0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 sinθScosθSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → π+π−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 2C3 +3c13 + 6c31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='sin2θSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='cos2θSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(500) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ρ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → π0π0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 (C1 + 2C3) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 (3c13 + 6c31 + 2c32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 sin2θSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 cos2θSC′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='f0(500) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η8π0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='c13 + c22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η1π0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6c13 + 9c22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='a0(980) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η8η8ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 6C3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 (c13 + 6c31 − 2c32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η1η1ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 (C1 + 3C2 + 3C3 + 9C4) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2(c13 + 3c22 + 3c31 + 9c41) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ → η8η1ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='c13 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 c22 + 2c32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → K+π−ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 −3c11 + 3c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K∗0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → K0π0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 (−3c11 + 3c13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='K∗0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → η8K0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 C1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3c11 + 6c12 − 3c13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='D+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='s → η1K0ℓ+νℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='C1 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 C2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2c11 + c12 − 2c13 + 3c21 − 3c22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='TABLE II: The experimental data and the SU(3) flavor symmetry predictions of the non-resonant branching ratios and the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='total branching ratios of the D → P1P2ℓ+νℓ decays with the c → sℓ+νℓ transitions within the 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The experimental data are taken from PDG [11], ‘N’ denotes the non-resonant contributions, and ‘T’ denotes the total contributions including the non-resonance, the light scalar meson resonances as well as the vector meson resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The same below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' data with N Ones with N Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' data with T Ones with T B(D0 → π−K 0e+νe)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 B(D0 → π0K−e+νe)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='039 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='021 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 B(D0 → ηK−e+νe)(×10−6) · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 B(D0 → η′K−e+νe)(×10−6) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 B(D+ → π+K−e+νe)(×10−2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 B(D+ → π0K 0e+νe)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='100 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='052 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15 B(D+ → ηK 0e+νe)(×10−5) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 B(D+ → η′K 0e+νe)(×10−5) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 B(D+ s → K+K−e+νe)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='034 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='018 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 B(D+ s → K0K 0e+νe)(×10−3) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='95 B(D+ s → π+π−e+νe)(×10−3) · · · · · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='79 B(D+ s → π0π0e+νe)(×10−4) · · · · · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 B(D+ s → ηηe+νe)(×10−4) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 B(D+ s → ηη′e+νe)(×10−6) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 B(D0 → π−K 0µ+νµ)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='073 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='039 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 B(D0 → π0K−µ+νµ)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='020 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 B(D0 → ηK−µ+νµ)(×10−6) · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 B(D0 → η′K−µ+νµ)(×10−6) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 B(D+ → π+K−µ+νµ)(×10−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='27 B(D+ → π0K 0µ+νµ)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='095 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='050 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 B(D+ → ηK 0µ+νµ)(×10−5) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 B(D+ → η′K 0µ+νµ)(×10−5) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 B(D+ s → K+K−µ+νµ)(×10−2) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='017 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 B(D+ s → K0K 0µ+νµ)(×10−3) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 B(D+ s → π+π−µ+νµ)(×10−3) · · · · · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 B(D+ s → π0π0µ+νµ)(×10−4) · · · · · · 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09 B(D+ s → ηηµ+νµ)(×10−4) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 B(D+ s → ηη′µ+νµ)(×10−6) · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='98 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='98 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 11 TABLE III: The experimental data and the SU(3) flavor symmetry predictions of the non-resonant branching ratios and the total branching ratios of the D → P1P2ℓ+νℓ decays with the c → dℓ+νℓ transitions within the 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios Ones with N Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' data with T Ones with T B(D0 → K−K0e+νe)(×10−5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 B(D0 → π0π−e+νe)(×10−3) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 B(D0 → ηπ−e+νe)(×10−5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 · · 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 B(D0 → η′π−e+νe)(×10−5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='35 B(D+ → K 0K0e+νe)(×10−5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='31 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 B(D+ → K+K−e+νe)(×10−5) · · · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='63 B(D+ → π+π−e+νe)(×10−3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 B(D+ → π0π0e+νe)(×10−4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='75 B(D+ → ηπ0e+νe)(×10−5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 · · 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 B(D+ → η′π0e+νe)(×10−6) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 B(D+ → ηηe+νe)(×10−6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26 · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26 B(D+ → ηη′e+νe)(×10−8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='96 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='96 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 B(D+ s → K+π−e+νe)(×10−3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='041 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 B(D+ s → K0π0e+νe)(×10−4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 B(D+ s → ηK0e+νe)(×10−5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='70 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='70 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 B(D+ s → η′K0e+νe)(×10−7) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='47 · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='47 B(D0 → K−K0µ+νµ)(×10−5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 B(D0 → π0π−µ+νµ)(×10−3) 0 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 B(D0 → ηπ−µ+νµ)(×10−5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 · · 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 B(D0 → η′π−µ+νµ)(×10−5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='31 B(D+ → K 0K0µ+νµ)(×10−5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='93 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='94 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 B(D+ → K+K−µ+νµ)(×10−5) · · · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 B(D+ → π+π−µ+νµ)(×10−3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48 B(D+ → π0π0µ+νµ)(×10−4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 B(D+ → ηπ0µ+νµ)(×10−5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='40 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 B(D+ → η′π0µ+νµ)(×10−6) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 · · 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 B(D+ → ηηµ+νµ)(×10−6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 B(D+ → ηη′µ+νµ)(×10−8) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 B(D+ s → K+π−µ+νµ)(×10−3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='072 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='039 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 B(D+ s → K0π0µ+νµ)(×10−4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 · · 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 B(D+ s → ηK0µ+νµ)(×10−5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='98 · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='98 B(D+ s → η′K0µ+νµ)(×10−7) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='08 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='72 · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='08 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='72 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' D → S(S → P1P2)ℓ+νℓ decays We will analyze the D → P1P2ℓ+νℓ decays with the light scalar resonances in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' As given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (11), their branching ratios can be obtained by using B(D → Sℓ+νℓ) and B(S → P1P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The detail analysis of B(D → Sℓ+νℓ) by the SU(3) flavor symmetry can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios of the S → P1P2 decays As for the S → P1P2 decays, the partial decay widths can be written as [85] Γ(S → P1P2) = pc 8πm2 S g2 S→P1P2, (27) where the center of mass momentum pc ≡ � λ(m2 S,m2 P1,m2 P2) 2mS , and gS→P1P2 is the strong coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' With the SU(3) flavor symmetry, the strong coupling constant can be parameterized as g2q S→P1P2 = g2Si jP k i P j k (28) for the two quark scalar states, and g4q S→P1P2 = g4Sim jn P j i P n m + g′ 4Sim jmP n i P j n (29) for the four quark scalar states, where g2, g4 and g′ 4 are the nonperturbative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The strong coupling constants of these decays are listed in the second and third columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' IV for the two quark scalar states and the four quark scalar states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Since the width determination is very model dependent, there are not accurate values about the decay widths of a0(980), f0(980) and f0(500) mesons in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Therefore, it is difficult to obtain accurate B(S → P1P2) in terms of Γ(S → P1P2)/ΓS, where ΓS is the decay width of scalar meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We assume the light scalar mesons decay dominantly into pairs of pseudoscalar mesons and all other decay channels are negligible, and then one can obtain B(S → P1P2) without the decay width values of the light scalar mesons, for an example, B(f0(500) → π+π−) ≈ Γ(f0(500)→π+π−) Γ(f0(500)→π+π−)+Γ(f0(500)→π0π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the two quark picture, the parameter g2 is canceled in the branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Therefore, B(K0 → πK, a0(980) → KK, f0(500) → ππ) only depend on the masses of relevant mesons, B(a0(980) → η′π, η′π) depend on the meson masses and the mixing angle θP , and B(f0(980) → ππ, KK) depend on the meson masses and the mixing angle θS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The numerical results of B(S → P1P2) in the two quark picture are listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' One can see that the branching ratios of the K0, a0(980), f0(500) decays are accurately predicted, nevertheless, B(f0(980) → ππ, KK) are predicted with large error due to the indeterminate mixing angle θS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The three possible ranges for the mixing angle θS, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ [69, 79], have been considered, and the predictions of B(f0(980) → ππ, KK) are quite dependent on the mixing angle θS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V, we also give the predictions with two quark picture of B(S → P1P2) further constrained from the relevant experimental data of B(D → Sℓ+νℓ, S → P1P2) listed in later Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VI-VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The 13 TABLE IV: The strong coupling constants of the S → P1P2 decays by the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='strong couplings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ones for two quark state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ones for four quark state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gK− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 →π0K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gK− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0 →π−K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gK0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0→π+K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gK0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0→π0K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)−→ηπ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6cosθP − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3sinθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6cosθP − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3sinθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)−→η′π− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6sinθP + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3cosθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6sinθP + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3cosθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)−→K0K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)0→ηπ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3cosθP − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3sinθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6cosθP − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3sinθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)0→η′π0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3sinθP + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3cosθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='6sinθP + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3cosθP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)0→K+K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='ga0(980)0→K0K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gf0(980)→π+π− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 sinθS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 cosφS + g4sinφS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gf0(980)→π0π0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 sinθS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 cosφS − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2g4sinφS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gf0(980)→K+K− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 cosθS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2g4cosφS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gf0(980)→K0K0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 cosθS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2g4cosφS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gf0(500)→π+π− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g2 cosθS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2 g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 sinφS + g4cosφS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='gf0(500)→π0π0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='g2 cosθS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='−g′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='4 sinφS − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='2g4cosφS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='predictions of B(f0(980) → P1P2) are quite accurate when θS is further constrained from [25◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 40◦] to [25◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 36◦],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' from [140◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 165◦] to [144◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 151◦] and from |φS| ≤ 30◦ to 22◦ ≤ |φS| ≤ 30◦ by the relevant experimental data of B(D → Sℓ+νℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' S → P1P2) with 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Since θS in the two quark picture has been further constrained by B(D → Sℓ+νℓ, S → P1P2), the predictions of B(f(980) → ππ, KK) are more accurate as listed in the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Other B(S → P1P2) are not further constrained from the data of B(D → Sℓ+νℓ, S → P1P2), so we do not list them in the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the four quark picture, the two nonperturbative parameters g4 and g′ 4 in the a0(980), f0(980), f0(500) decays, and |g′ 4/g4| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 are obtained by the data Γ(a0(980) → K ¯K)/Γ(a0(980) → ηπ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='177 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='048 from PDG [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In this work, we treat g4 and g′ 4 as real number, then two possible cases (g′ 4/g4 > 0 and g′ 4/g4 < 0) are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The numerical results with the four quark picture are listed in the last column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' As for B(f0(980) → ππ) and B(f0(500) → ππ), very large errors come from the mixing angles φS, and they are obviously different in the g′ 4/g4 > 0 and g′ 4/g4 < 0 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In general, there is a relative strong phase between g′ 4 and g4, therefore, the common relevant branching ratios are between ones in the g′ 4/g4 > 0 case and ones in the g′ 4/g4 < 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In addition, B(K0 → P1P2) are same in both the two quark and four quark pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios of the D → S(S → P1P2)ℓ+νℓ decays Then B(D → Sℓ+νℓ, S → P1P2) can be obtained in terms of B(S → P1P2) listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V and the expressions of B(D → Sℓ+νℓ) given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Using the experimental data of B(D+ s → f0(980)e+νe) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='8) × 10−3 [11] as well as B(D → Sℓ+νℓ, S → P1P2) listed in the second columns of Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VI-VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The numerical results of B(D → Sℓ+νℓ, S → P1P2) with 2σ errors for the two quark and four quark pictures are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VI and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VII for the c → sℓ+νℓ and c → dℓ+νℓ transitions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our comments on the results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The experimental lower limits of B(D0 → a0(980)−e+νe, a0(980)− → ηπ−) and B(D+ → f0(500)e+νe, f0(500) → π+π−) have not been used to constrain the predictions of B(D → Sℓ+νℓ, S → P1P2), since the two lower limits of the SU(3) flavor symmetry predictions are slightly lower than their experimental data in both the two quark and four quark pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' For B(D0 → a0(980)−e+νe, a0(980)− → ηπ−), one can see that the prediction in the two quark picture agrees with experimental data within 2σ error bars, nevertheless, the prediction in the four quark picture is smaller, which only agrees with experimental data within 3σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' As for B(D+ → f0(500)e+νe, f0(500) → π+π−), the prediction in the two quark picture is much smaller than its experimental lower limit with 2σ error, nevertheless, the prediction with g′ 4 g4 > 0 ( g′ 4 g4 < 0 ) in the four quark picture agrees with its data within 2σ (3σ) error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Therefore, in the later analysis of total contributions to B(D → P1P2ℓ+νℓ), the predictions of B(D → Sℓ+νℓ, S → P1P2) with g′ 4 g4 > 0 in the four quark picture will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the two quark picture, though the mixing angle θS only appears in the D → P1P2ℓ+νℓ decays with f0(980) and f0(500) resonances, all other predictions of the branching ratios are slightly affected by the experimental constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' So we list all predictions in the three possible ranges of the mixing angle θS in the 3rd-5th columns of Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VI-VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' One can see the all predictions included the decays with f0(980) and f0(500) resonances are similar in the three possible ranges of the mixing angle θS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' As mentioned before, θS is constrained from [25◦, 40◦] to [25◦, 36◦], from [140◦, 165◦] to [144◦, 151◦] and from |φS| ≤ 30◦ to 22◦ ≤ |φS| ≤ 30◦ by the relevant experimental data with 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' A lot of the branching ratio predictions are quite different between the two quark picture and the four quark picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Present datum of B(D+ → f0(500)e+νe, f0(500) → π+π−) favors the four quark picture of scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D → Sℓ+νℓ, S → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−3 −10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Due to the CKM matrix element Vcd suppressed, B(D → Sℓ+νℓ, S → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−4 − 10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Some branching ratios of the D → S(S → P1P2)ℓ+νℓ decays have been obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [13, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ → Se+νe, S → π+π−) = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46) × 10−4 [13], B(D+ → Sµ+νµ, S → π+π−) = (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='52) × 10−4 [13], B(D0 → a0(980)−ℓ+νℓ, a0(980)− → ηπ−) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21) × 10−4 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our predictions in the four quark picture of B(D+ → Sℓ+νℓ, S → π+π−) are consistent with ones in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [13], our predictions in the two quark picture of B(D0 → a0(980)−ℓ+νℓ, a0(980)− → ηπ−) are consistent with ones in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [61], nevertheless, our predictions in the four quark picture are smaller than ones in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 15 TABLE V: Branching ratios of the S → P1P2 decays within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The results are obtained by the SU(3) flavor symmetry relations and Γ(a0(980) → K ¯K)/Γ(a0(980) → ηπ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='177 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='048 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' †denotes the results with g′ 4 g4 > 0, and ♯denotes ones with g′ 4 g4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios ones with 2q state in S1 case ones with 2q state in S2 case ones with 4q state B(K− 0 → π0K−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 B(K− 0 → π−K 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 B(K 0 0 → π+K−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 B(K 0 0 → π0K 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 B(a0(980)− → ηπ−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 B(a0(980)− → η′π−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 B(a0(980)− → K0K−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 B(a0(980)0 → ηπ0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 B(a0(980)0 → η′π0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 B(a0(980)0 → K+K−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 B(a0(980)0 → K0 ¯K0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09θS=[25◦,40◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07θS=[25◦,35◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16† B(f0(980) → π+π−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17θS=[140◦,165◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09θS=[144◦,158◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13♯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22θS=[−30◦,30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06[22◦≤|θS|≤30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04θS=[25◦,40◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03θS=[25◦,35◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11† B(f0(980) → π0π0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09θS=[140◦,165◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04θS=[144◦,158◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10♯ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11θS=[−30◦,30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03[22◦≤|θS|≤30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07θS=[25◦,40◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05θS=[25◦,35◦] B(f0(980) → K+K−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14θS=[140◦,165◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07θS=[144◦,158◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17θS=[−30◦,30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04[22◦≤|θS|≤30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06θS=[25◦,40◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05θS=[25◦,35◦] B(f0(980) → K0 ¯K0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12θS=[140◦,165◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06θS=[144◦,158◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16θS=[−30◦,30◦] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04[22◦≤|θS|≤30◦] B(f0(500) → π+π−) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12♯ B(f0(500) → π0π0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12♯ 16 TABLE VI: The experimental data and the SU(3) flavor symmetry predictions of the D → S(S → P1P2)ℓ+νℓ decays with the c → sℓ+νℓ transitions within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' †denotes the results with g′ 4 g4 > 0, and ♯ denotes ones with g′ 4 g4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Data Ones in the 2-quark picture with Ones in the 4-quark picture θS = [25◦, 35◦] θS = [144◦, 158◦] 22◦ ≤ |θS| ≤ 30◦ B(D0 → K− 0 e+νe, K− 0 → π−K 0)(×10−4) · · 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='74 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 B(D0 → K− 0 e+νe, K− 0 → π0K−)(×10−4) · · 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 B(D+ → K 0 0e+νe, K 0 0 → π+K−)(×10−3) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 B(D+ → K 0 0e+νe, K 0 0 → π0K 0)(×10−3) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='42 B(D+ s → f0(980)e+νe, f0(980) → π+π−)(×10−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='63 [86] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55†, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49♯ B(D+ s → f0(980)e+νe, f0(980) → π0π0)(×10−4) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='9 [4] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='91 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85†, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10♯ B(D+ s → f0(980)e+νe, f0(980) → K+K−)(×10−4) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53†, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34♯ B(D+ s → f0(980)e+νe, f0(980) → K0K 0)(×10−4) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='39†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22♯ B(D+ s → f0(500)e+νe, f0(500) → π+π−)(×10−4) · · 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='91 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='44 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49†, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='90♯ B(D+ s → f0(500)e+νe, f0(500) → π0π0)(×10−5) < 64 [4] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='44 ± 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66†, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78♯ B(D0 → K− 0 µ+νµ, K− 0 → π−K0)(×10−4) · · 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='27 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='41 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='63 B(D0 → K− 0 µ+νµ, K− 0 → π0K−)(×10−4) · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='63 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='52 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='59 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 B(D+ → K 0 0µ+νµ, K 0 0 → π+K−)(×10−3) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='40 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 B(D+ → K 0 0µ+νµ, K 0 0 → π0K0)(×10−3) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 B(D+ s → f0(980)µ+νµ, f0(980) → π+π−)(×10−3) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46†, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='42♯ B(D+ s → f0(980)µ+νµ, f0(980) → π0π0)(×10−4) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='72 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48†, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='82♯ B(D+ s → f0(980)µ+νµ, f0(980) → K+K−)(×10−4) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='31 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='70 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='79 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='59 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14♯ B(D+ s → f0(980)µ+νµ, f0(980) → K0K 0)(×10−4) · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='90 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='25 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='52 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03♯ B(D+ s → f0(500)µ+νµ, f0(500) → π+π−)(×10−4) · · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='70 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30†, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83♯ B(D+ s → f0(500)µ+νµ, f0(500) → π0π0)(×10−5) · · 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='16†, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23♯ 17 TABLE VII: The experimental data and the SU(3) flavor symmetry predictions of the D → S(S → P1P2)ℓ+νℓ decays with the c → dℓ+νℓ transitions within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' † denotes the results with g′ 4 g4 > 0, ♯ denotes ones with g′ 4 g4 < 0, and a denotes the experimental lower limits have not used to constrain the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Data Ones in the 2-quark picture with Ones in the 4-quark picture θS = [25◦, 35◦] θS = [144◦, 158◦] 22◦ ≤ |θS| ≤ 30◦ B(D0 → a0(980)−e+νe, a0(980)− → ηπ−)(×10−5) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='8 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='98 B(D0 → a0(980)−e+νe, a0(980)− → η′π−)(×10−6) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='98 B(D0 → a0(980)−e+νe, a0(980)− → K0K−)(×10−6) · · 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='93 B(D+ → a0(980)0e+νe, a0(980)0 → ηπ0)(×10−5) 17+16 −14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='35 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='25 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 B(D+ → a0(980)0e+νe, a0(980)0 → η′π0)(×10−6) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='08 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='56 B(D+ → a0(980)0e+νe, a0(980)0 → K+K−)(×10−5) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 B(D+ → a0(980)0e+νe, a0(980)0 → K0K 0)(×10−5) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='31 B(D+ → f0(980)e+νe, f0(980) → π+π−)(×10−5) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='8 [5] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65♯ B(D+ → f0(980)e+νe, f0(980) → π0π0)(×10−6) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='75 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67†, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37♯ B(D+ → f0(980)e+νe, f0(980) → K+K−)(×10−6) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='35 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78†, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='60 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76♯ B(D+ → f0(980)e+νe, f0(980) → K0K 0)(×10−6) · · 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='88 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='35 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78†, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='60 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76♯ B(D+ → f0(500)e+νe, f0(500) → π+π−)(×10−4) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='87♯ B(D+ → f0(500)e+νe, f0(500) → π0π0)(×10−4) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='08 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57♯ B(D+ s → K0 0e+νe, K0 0 → π−K+)(×10−5) · · 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='97 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='54 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 B(D+ s → K0 0e+νe, K0 0 → π0K0)(×10−5) · · 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='99 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 B(D0 → a0(980)−µ+νµ, a0(980)− → ηπ−)(×10−5) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='95 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='84 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='84 B(D0 → a0(980)−µ+νµ, a0(980)− → η′π−)(×10−6) · · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='39 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='82 B(D0 → a0(980)−µ+νµ, a0(980)− → K0K−)(×10−6) · · 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 B(D+ → a0(980)0µ+νµ, a0(980)0 → ηπ0)(×10−5) · · 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='00 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='86 B(D+ → a0(980)0µ+νµ, a0(980)0 → η′π0)(×10−6) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='72 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='79 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='31 B(D+ → a0(980)0µ+νµ, a0(980)0 → K+K−)(×10−5) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 B(D+ → a0(980)0µ+νµ, a0(980)0 → K0K 0)(×10−5) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='27 B(D+ → f0(980)µ+νµ, f0(980) → π+π−)(×10−5) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='96†, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55♯ B(D+ → f0(980)µ+νµ, f0(980) → π0π0)(×10−6) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='74 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='58 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='82 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13†, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='47♯ B(D+ → f0(980)µ+νµ, f0(980) → K+K−)(×10−6) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='29†, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26♯ B(D+ → f0(980)µ+νµ, f0(980) → K0K 0)(×10−6) · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='29†, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26♯ B(D+ → f0(500)µ+νµ, f0(980) → π+π−)(×10−4) · · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='39†, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='68 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='74♯ B(D+ → f0(500)µ+νµ, f0(980) → π0π0)(×10−4) · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='95†, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46♯ B(D+ s → K0 0µ+νµ, K0 0 → π−K+)(×10−5) · · 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='61 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='60 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01 B(D+ s → K0 0µ+νµ, K0 0 → π0K0)(×10−5) · · 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='60 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='71 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='19 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='50 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' D → V (V → P1P2)ℓ+νℓ decays We will analyze the D → P1P2ℓ+νℓ decays with the vector resonances in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Since the light vector mesons are understood well, the calculations of B(D → V ℓ+νℓ, V → P1P2) are much easier than ones of B(D → Sℓ+νℓ, S → P1P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (11), their branching ratios of D → V (V → P1P2)ℓ+νℓ can be obtained by using B(D → V ℓ+νℓ) and B(V → P1P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The D → V ℓ+νℓ decays have been studied by the SU(3) flavor symmetry in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Many B(D → V ℓ+νℓ) have been accurately measured and have been listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' V in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The expressions of B(D → V ℓ+νℓ) within the C3 case in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81] will be taken for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [85], B(V → P1P2) can be written as B(V → P1P2) = τV p′3 c 6πm2 V g2 V →P1P2, (30) where p′ c ≡ � λ(m2 V ,m2 P1,m2 P2) 2mV and gV →P1P2 are the strong coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Similar to g2q S→P1P2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (28), gV →P1P2 can be parameterized by the SU(3) flavor symmetry gV →P1P2 = gV V i j P k i P j k, (31) where gV is the corresponding nonperturbative parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' At present, many involved B(V → P1P2) have been well measured [11] B(K∗+ → πK) = (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='902 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='018)%, B(K∗0 → πK) = (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='754 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='042)%, B(ρ+ → π0π+) = 100%, B(ρ0 → π+π−) = 100%, B(φ → K+K−) = (49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='0)%, B(ω → π+π−) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='53+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='26)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (32) Using the following relations from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (31) √ 2gK∗−→π0K− = gK∗−→π−K0, √ 2gK∗0→π0K0 = gK∗0→π−K+, gρ−→π0π− = √ 3gρ−→η8π− = � 3/2gρ−→η1π−, gφ→K+K− = gφ→K0K 0, (33) following B(V → P1P2) can be obtained B(K∗0 → π0K0) = (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='02)%, B(K∗0 → π−K+) = (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04)%, B(K∗+ → π0K+) = (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01)%, B(K∗+ → π−K0) = (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='01)%, B(ρ+ → ηπ+) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='66)%, B(φ → K0K0) = (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='42 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='04)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' (34) For D → V (V → P1P2)ℓ+νℓ decays, the branching ratios of D+ → K ∗0(K ∗0 → π+K−)e+νe and D+ → K ∗0(K ∗0 → π+K−)µ+νµ have been measured, and the experimental data with 2σ errors are listed in the second column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Using the experimental data of B(D+ → K ∗0ℓ+νℓ, K ∗0 → π+K−), B(V → P1P2) and B(D → V ℓ+νℓ), we obtain the predictions of B(D → V ℓ+νℓ, V → P1P2) by the SU(3) flavor symmetry, which are given in the third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We can see that B(D → V ℓ+νℓ, V → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−2 − 10−3), and B(D → V ℓ+νℓ, V → P1P2) with the c → dℓ+νℓ transitions are predicted on the 19 TABLE VIII: The experimental data and the SU(3) flavor symmetry predictions of D → V (V → P1P2)ℓ+νℓ decays within 2σ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Branching ratios Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Data Our predictions Previous ones c → se+νe: B(D0 → K∗−e+νe, K∗− → π−K 0)(×10−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D0 → K∗−e+νe, K∗− → π0K−)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='37 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 [62] B(D+ → K ∗0e+νe, K ∗0 → π+K−)(×10−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 [62] B(D+ → K ∗0e+νe, K ∗0 → π0K 0)(×10−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → φe+νe, φ → K+K−)(×10−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → φe+νe, φ → K0K 0)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='65 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' c → sµ+νµ: B(D0 → K∗−µ+νµ, K∗− → π−K 0)(×10−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D0 → K∗−µ+νµ, K∗− → π0K−)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='17 [62] B(D+ → K ∗0µ+νµ, K ∗0 → π+K−)(×10−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 [62] B(D+ → K ∗0µ+νµ, K ∗0 → π0K 0)(×10−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → φµ+νµ, φ → K+K−)(×10−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='09 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → φµ+νµ, φ → K0K 0)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='62 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' c → de+νe: B(D0 → ρ−e+νe, ρ− → π0π−)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='63 [62] B(D0 → ρ−e+νe, ρ− → ηπ−)(×10−5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='59 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ → ρ0e+νe, ρ0 → π+π−)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 [13], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 [62] B(D+ → ωe+νe, ω → π+π−)(×10−5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='82 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → K∗0e+νe, K∗0 → π−K+)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → K∗0e+νe, K∗0 → π0K0)(×10−4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' c → dµ+νµ: B(D0 → ρ−µ+νµ, ρ− → π0π−)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D0 → ρ−µ+νµ, ρ− → ηπ−)(×10−5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='83 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='51 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ → ρ0µ+νµ, ρ0 → π+π−)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='07 [13] B(D+ → ωµ+νµ, ω → π+π−)(×10−5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='78 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → K∗0µ+νµ, K∗0 → π−K+)(×10−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' B(D+ s → K∗0µ+νµ, K∗0 → π0K0)(×10−4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content='48 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 20 order of O(10−3 − 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The predictions of B(D → V ℓ+νℓ, V → P1P2) are about one order larger than ones of the corresponding B(D → Sℓ+νℓ, S → P1P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Previous predictions are also listed in the last column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our predictions of B(D0 → K∗−ℓ+νℓ, K∗− → π0K−) and B(D+ → K ∗0ℓ+νℓ, K ∗0 → π+K−) are in good agreement with ones in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' And our predictions of B(D+ → ρ0ℓ+νℓ, ρ0 → π+π−) are slight larger than ones obtained by the light-front quark model and the light-cone sum rules in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Total branching ratios As analyzed in above, some four-body semileptonic decays of D mesons receive the contributions of the non-resonant states, the scalar resonant states and the vector resonant states, nevertheless, some decay modes only receive one or two kinds of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' For clearly showing the resonant contributions, we also list the scalar and vector resonant amplitudes in the third and last columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The resonant amplitudes are obtained by multiplying the hadronic helicity amplitudes H(D → Rℓ+νℓ) given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' [81] and the strong coupling constants gR→P1P2 obtained in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Noted that the resonant amplitudes listed in the last two columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' I only for clearly see the kinds of the resonant contributions, and we do not using them to obtain the numerical total branching ratios B(D → P1P2ℓ+νℓ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We have some comments for the contributions in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' For D(s) → ηKℓ+νℓ, η′Kℓ+νℓ, ηηℓ+νℓ, ηη′ℓ+νℓ decays, since the both final state mesons are quite heavy, they only receive the non-resonant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The decays D+ s → π0π0ℓ+νℓ, D+ s → π+π−ℓ+νℓ, D0 → K−K0ℓ+νℓ, D+ → K 0K0ℓ+νℓ, D+ → K+K−ℓ+νℓ, D+ → π0π0ℓ+νℓ and D+ → η(′)π0ℓ+νℓ receive both the non-resonant contributions and the scalar resonant contributions, moreover, the non-resonant contributions in the D+ s → π0π0ℓ+νℓ, D+ s → π+π−ℓ+νℓ and D+ → K+K−ℓ+νℓ decays are suppressed by the OZI rule, and the main contributions of these decay branching ratios come from the scalar resonant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' All other decay modes except the D0 → π0π−ℓ+νℓ decays receive all three kinds of the contributions, and their branching ratios are dominant by the vector resonant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Due to the quantum number constraint, the D0 → π0π−ℓ+νℓ decays only receive the contributions of the vector resonant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' In the last columns of Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II-III, total branching ratio predictions of the D → P1P2ℓ+ν decays including the possible non-resonant, scalar resonant and vector resonant contributions are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The present six experimental data with 2σ errors are also listed in the forth column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' II and in third column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' III for convenient comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' One can see that, for B(D0 → π−K −e+νe), B(D0 → π0K−e+νe), B(D+ → π+K−e+νe), B(D+ → π+K−µ+νµ) and B(D+ → π+π−e+νe), our SU(3) flavor symmetry predictions are consistent with present data within 2σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our prediction of B(D0 → π0π−e+νe) is slightly larger than its experimental datum, nevertheless, the prediction will be very close to the datum within 3σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' For some Cabibbo suppressed decays due to c → dℓ+νℓ transitions, such as the D0 → K−K0ℓ+νℓ, D0 → η′π−ℓ+νℓ, D+ → K 0K0ℓ+νℓ, D+ → π0π0ℓ+νℓ, D+ → ηπ0ℓ+νℓ and D+ → η′π0ℓ+νℓ decays, they only receive both the non- resonant contributions and the scalar resonant contributions, and we can see that both the non-resonant and the scalar resonant contributions are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The non-resonant contributions in the D+ → K+K−ℓ+νℓ decays are suppressed by the OZI rule, and the scalar resonant contributions in the D+ → K+K−ℓ+νℓ decays are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 21 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Summary Semileptonic decays of heavy mesons are quite interesting because of not only relatively simple theoretical description but also the clean experimental signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Some semileptonic decays D → P1P2ℓ+νℓ have been measured by BESIII, CLEO and BABAR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Using the present data of B(D → P1P2ℓ+νℓ) and the SU(3) flavor symmetry, we have presented a theoretical analysis of the D → P1P2ℓ+νℓ decays with the non-resonant, the light scalar meson resonant and the vector meson resonant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Non-resonant D → P1P2ℓ+νℓ decays: The amplitude relations included the SU(3) flavor breaking effects have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Almost all amplitudes can be related after ignoring the OZI suppressed and the SU(3) flavor breaking contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Via the experimental data of the non-resonant branching ratios B(D+ → π+K−ℓ+νℓ)N, we have predicted other non-resonant branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We have found that the branching ratios of the non- resonant decays D0 → π−K 0ℓ+νℓ, π0K−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K 0ℓ+νℓ, π+π−ℓ+νℓ, π0π0ℓ+νℓ, and D+ s → K+K−ℓ+νℓ, K0K 0ℓ+νℓ are on the order of O(10−3 − 10−4), which might be measured by the BESIII, LHCb and BelleII experiment, and some other decays might be measured at these experiments in near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Decays with the light scalar meson resonances: Using the SU(3) flavor symmetry and the present experimental data of B(D → Sℓ+νℓ), B(D → Sℓ+νℓ, S → P1P2) as well as B(S → P1P2), the not- measured B(D → Sℓ+νℓ, S → P1P2) have been obtained by the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We have found that B(D → Sℓ+νℓ, S → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−3 − 10−4), and B(D → Sℓ+νℓ, S → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−4−10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The two quark picture and the four quark picture for the scalar mesons have been analyzed in the D → S(S → P1P2)ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Present experimental data might favorite the four quark picture for the scalar mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Decays with the vector meson resonances: Using the experimental data of B(D+ → K ∗0e+νe, K ∗0 → π+K−), B(D+ → K ∗0µ+νµ, K ∗0 → π+K−), many B(D → V ℓ+νℓ) and many B(V → P1P2), the not-measured B(D → V ℓ+νℓ, V → P1P2) have been predicted by the SU(3) flavor symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We have found that B(D → V ℓ+νℓ, V → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−2 − 10−3), and B(D → V ℓ+νℓ, V → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−3 − 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Total branching ratios: Total branching ratio predictions including the possible non-resonant, light scalar meson resonant and vector meson resonant contributions have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' The six total branching ra- tios have been measured, and we did not use them to further constrain the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Our five predic- tions are consistent with present data within 2σ errors, and the prediction of B(D0 → π0π−e+νe) will be very close to the datum within 3σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' We have found that the vector meson resonant contributions are dominant in the D0 → π−K 0ℓ+νℓ, π0K−ℓ+νℓ, π0π−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K 0ℓ+νℓ, π+π−ℓ+νℓ, and D+ s → K+K−ℓ+νℓ, K0K 0ℓ+νℓ, K+π−ℓ+νℓ, K0π0ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' All three kinds of contributions are important in D0 → ηπ−ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Both the non-resonant and the scalar resonant contributions are important in D0 → K−K0ℓ+νℓ, η′π−ℓ+νℓ and D+ → K 0K0ℓ+νℓ, π0π0ℓ+νℓ, ηπ0ℓ+νℓ, η′π0ℓ+νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' 22 Although SU(3) flavor symmetry is approximate, it can still provide very useful information about these decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' According to our rough predictions, many decay modes could be observed at BESIII, LHCb and BelleII, and some decay modes might be measured in near future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQfSvci/content/2301.00090v1.pdf'} +page_content=' Therefore, the SU(3) flavor symmetry will be further tested by these semileptonic decays in future experiments.' metadata={'source': 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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf,len=548 +page_content='ITA-ELECTION-2022: A multi-platform dataset of social media conversations around the 2022 Italian general election Francesco Pierri, Geng Liu, Stefano Ceri Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy E-mail: {name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='surname}@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='it Abstract Online social media play a major role in shaping public dis- course and opinion, especially during political events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We present the first public multi-platform dataset of Italian- language political conversations, focused on the 2022 Italian general election taking place on September 25th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Leveraging public APIs and a keyword-based search, we collected mil- lions of posts published by users, pages and groups on Face- book, Instagram and Twitter, along with metadata of TikTok and YouTube videos shared on these platforms, over a period of four months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We augmented the dataset with a collection of political ads sponsored on Meta platforms, and a list of so- cial media handles associated with political representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Our data resource will allow researchers and academics to further our understanding of the role of social media in the democratic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Introduction Online social media provide researchers and academics with unprecedented opportunities to observe a wide range of po- litical and societal phenomena (Rossi, Righetti, and Marino 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' They also play a critical role in shaping public opin- ion during political events (Vitak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2011), and represent a rich source of data to study the interplay between politi- cal actors’ campaigns (Sahly, Shao, and Kwon 2019), media outlets’ agenda settings (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2016), and users’ news consumption (Allcott and Gentzkow 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' In Italy, as of 20221, YouTube is the platform used by the largest amount of internet users (88%), followed by Meta platforms (64%) and TikTok (54%), whereas Twitter only accounts for approximately 7%2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' However, previous studies of online social media during Italian elections and referen- dum mostly focused on Twitter (Rossi, Righetti, and Marino 2021), due to the large availability of data via its APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' In this work, instead, we present a public data resource of polit- ical conversations and user-generated content shared around the 2022 Italian general election, which allows researchers and academics to study multiple social platforms simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Copyright © 2022, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='com/statistics/1311549/top-social-platforms- italy/ 2datareportal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='com/reports/digital-2022-italy The 2022 Italian general election was the first ever to take place in autumn, as a consequence of the fall of the govern- ment of national unity led by Mario Draghi in July3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' The election had a record-low voter turnout and it was won by the right-wing coalition of Giorgia Meloni with over 43% of the vote share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Among the opponents, the Centre-left coali- tion led by Enrico Letta obtained approximately 25% of the voters, the populist Movimento 5 Stelle led by former PM Giuseppe Conte reached less than 16%, and the liberal and centrist Third Pole, which included former PM Matteo Renzi, obtained almost 8% of the vote share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We present ITA-ELECTION-2022, the first public multi-platform dataset of Italian-language political conver- sations taking place on online social media, with a focus on the 2022 Italian general election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We collected millions of social media posts from Facebook, Instagram and Twit- ter, as well as advertisements sponsored on Meta platforms and metadata for TikTok and YouTube videos shared on the aforementioned platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We finally augment the dataset with a collection of social media handles associated with Italian political representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' To collect the data, we em- ployed a snowball sampling procedure and curated a list of relevant terms to accordingly perform a keyword-based search during a period of four months (July 2022 - October 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We provide public access to the data via GitHub and DataVerse repositories, as detailed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' The outline of this paper is the following: in the next sec- tion we review existing public data resources related to the present work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' then, we describe the data collection proce- dure(s) carried out to build the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' next, we describe a few potential applications of the collected data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' finally, we discuss limitations, draw conclusions and provide some eth- ical remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Related Work There are several public datasets that allow to study social media conversations around political issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We focus our literature review on the Italian context, and then describe a few datasets related to other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We also refer the in- terested reader to (Rossi, Righetti, and Marino 2021) for an overview of studies that describe the interplay between so- cial media and Italian politics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 3en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='org/wiki/2022 Italian general election arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='05119v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='SI] 12 Jan 2023 (Basile, Lai, and Sanguinetti 2018) collect tweets in the Italian language continuously from 2012 to 2018, extracting a number of smaller datasets enriched with different kinds of annotations for linguistic purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' They provide access to tweet IDs and annotations in a public repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Pierri, Artoni, and Ceri 2020) analyze the prevalence of Italian disinformation spreading on Twitter in the five months preceding the 2019 European Parliament election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' They collect over 300 k tweets sharing thousands of news articles originating from websites flagged as unreliable by journalists and fact-checkers, providing public access to tweet IDs and lists of websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' The same authors provide a similar dataset collected in a different period of 2019, and that contains tweets sharing links to mainstream and tradi- tional news websites, both in the Italian and French language (Pierri 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Di Giovanni and Brambilla 2021) study the polarization around the 2020 Italian constitutional referendum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' They col- lect a dataset of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='2 M tweets discussing the event – and provide access to their IDs –, with the goal of designing a hashtag-based semi-automatic approach to label Twitter users’ stance towards the referendum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Following the COVID-19 pandemic, several researchers collected social media data to study conversations around the crisis, with a particular focus on the impact of vaccine misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Crupi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2022) study the evolution of Italian Twitter conversations around vaccines during the pe- riod 2019-2021, whereas (Di Giovanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2022) collect tweets in multiple languages (French, German and Italian) during the first year of world vaccination programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Both contributions give public access to tweet IDs, with the lat- ter providing also a set of labeled pro/anti-vaccines tweets and hashtags that can be used for training machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Calisir and Brambilla 2020) provide a dataset of tweets discussing Brexit for a period of 45 months, from January 2016 until September 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' The data, which comprises 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='8 million tweets and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='97 million users, is enriched with meta- data such as the bot score of users, sentiment score of tweets, and political stance labels predicted by a classifier developed by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' There is a large number of datasets that focus on the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' elections (both presidential and midterms), and we provide here a non-exhaustive list of available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2011) mapped candidates from the 2010 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Midterm election with their Twitter accounts and a random sample of their followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Bovet and Makse 2019) collected over 171 M tweets in the English language, mentioning Donald Trump and Hillary Clinton during the 2016 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Presiden- tial election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2019) and (Yang, Hui, and Menczer 2022) collected tweets discussing the 2018 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Midterm election, both using a hashtag-based search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' tweets shar- ing the hashtag ”#ivoted” on election day) and querying Twitter APIs with general keywords related to the midterm election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' (Chen, Deb, and Ferrara 2022) provide a longitudi- nal dataset of over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='2 billion U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' politics- and election- related tweets shared around the period of the 2020 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Presidential election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Related to the same election, (Abilov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' 2021) released a multi-modal dataset of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='6 M tweets Figure 1: Example of an ad run on Meta platforms along with the information provided by Meta Ad Library API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='6 M retweets from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='6 M users related to voter fraud claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' They augmented the data with cluster labels, users’ suspension status, and perceptual hashes of tweeted images as well as aggregate data from external links and YouTube videos shared on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Data Collection This section describes the data collection procedure(s) car- ried out to gather data from different social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We remark that we employed the same list of keywords related to the Italian election, which we obtained through a snowball sampling approach using Twitter data only, to query different APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Our dataset conforms with FAIR prin- ciples: it is Findable, Accessible and Reusable as it is pub- licly accessible in an online Github4 and DataVerse reposi- tory5, where we provide the means to recreate it almost com- pletely (see limitations discussed next).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' It is also Interopera- ble as the data files are released in “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='csv” and “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='txt” formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' We summarize some statistics of the dataset in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content=' Twitter We collected all tweets in the Italian language related to the election by using tweepy Python library to query Twitter v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='1 Filter streaming API endpoint6 in the period September 4github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='com/frapierri/ita-election-2022 5doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='7910/DVN/EALXH2 6developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE4T4oBgHgl3EQfgw1Q/content/2301.05119v1.pdf'} +page_content='com/en/docs/twitter-api/v1/tweets/filter- realtime/overview Inactive 8Sep2022-23Sep2022 Platforms Categories :Estimatedaudiencesize:100K-500kpeople 自Amount spent (EUR): 0. +ii) The observability Gramian Q of the FOM is required +for the construction of the ROM. +iii) The ROM is asymptotically stable. +Here and in the following, ∥ · ∥L2 +T refers to the L2-norm +on the time-interval [0, T ]. The availability of an a pri- +ori bound as in Assumption 1 is of high practical value +on its own, especially for the selection of an appropriate +ROM dimension n. But this a priori result yields a worst- +case estimation on the error, since it does not take into +account any specific knowledge on the simulation setup. +The bound we propose in this paper can be considered +an a posteriori adaption of the a priori bound. The main +idea is to filter out a signal w and to apply the pes- +simistic a priori bound to the remainder u−w only. The +approximation w is chosen such that a more explicit er- +ror analysis becomes feasible that leads to less overesti- +mation. We further split the error part related to w into +its long-time behavior, also denoted as steady-state, and +a term that is equivalent to the output energy a certain +initial state has in the error system (3). The steady-state +can be determined in an offline-online efficient way, and +the other error term can be sharply bounded using the +approach from [9]. +The peculiarity of our proposed error bound is that it +takes into account information of the input u, i.e., is +an a posteriori result, and, simultaneously, exploits the +features inherent in balancing-related MOR, cf., Assump- +tion 1. Consequently, it outperforms other rigorous a +posteriori bounds, e.g., the residual based bounds de- +rived in [7]. Let us mention that effective error estimates +have been proposed in literature [4,5], but those are typ- +ically not rigorous and thus may lead to an underestima- +tion of the actual error, which is problematic for certain +applications. +The structure of the paper revolves around the deriva- +tion of our error bound. In Section 2, we briefly outline +the standard a priori bound for balancing-related MOR, +and the error bound from [9] for inhomogeneous initial +conditions. Those are ingredients of our bound. Addi- +tionally, we make use of the notion of steady-states for a +certain class of approximations on the input in our error +analysis; the respective results are stated in Sections 3-4. +Our a posteriori bound is proven in Section 5. Its effec- +tivity is numerically demonstrated in Section 6 at the +example of two academic benchmarks. +2 +Results for balancing-related MOR +The observability and controllability of a state in the +FOM can be quantified in terms of the observability +Gramian Q and controllability Gramian P, defined as +Q = +� ∞ +0 +etAT cT cetAdt, +P = +� ∞ +0 +etAbbT etAT dt. +These matrices are well-defined for asymptotically sta- +ble systems. They play a crucial role in the analysis and +implementation of BT and SPA, see [1]. Notably, these +balancing-related MOR methods allow for a rigorous er- +ror analysis that is based on the Gramians. In this sec- +tion, we outline the error bounds from literature which +also play a role in our refined error analysis. These re- +sults require, respectively justify, Assumption 1 in our +approach. +2.1 +A priori error bound +Asymptotically stable systems can be transformed in a +so-called balanced form, in which the Gramians Q and +P are simultaneously diagonalized and equal. The di- +agonal entries σ1 ≥ σ2 ≥ . . . ≥ σN ≥ 0 of the result- +ing diagonal matrix are called the Hankel Singular Val- +ues (HSVs) and represent a measure for observability and +controllability of the balanced states. In both BT and +SPA, the ROM is composed of the states related to the +largest HSVs, see [6] for details on the methods. Given +the ROM dimension n is chosen such that σn > σn+1 and +zero initial conditions are considered (x0 = 0), the re- +duction error can be shown to fulfill the a priori error +bound ∥E∥L2∞ ≤ α∥u∥L2∞ with α = 2 �N +j=n+1 σj com- +posed of the neglected HSVs, see [3]. As we require a +bound on a finite time interval [0, T ] (Assumption 1-i)), +we apply the former bound on the extension of the input +u : [0, T ] → R to the infinite interval [0, ∞), obtained by +2 + +setting u(t) = 0 for t > T . This yields +∥E∥L2 +T ≤ ∥E∥L2 +∞ ≤ α∥u∥L2 +T . +2.2 +Error originating from initial conditions +The case with trivial input (u(t) = 0 for t ≥ 0) allows +for a simple error analysis based on the notion of ob- +servability. In this setting, the output of the error sys- +tem (3) is solely determined by the initial conditions +˘x(0) = ˘x0 = [xT +0 , xrT +0 ]T . It reads y˘x0 = ˘cet ˘A˘x0. As +shown in [9], a bound for its norm follows directly from +applying the notion of observability to the error systems, +i.e., +∥y˘x0∥L2 +T ≤ ∥y˘x0∥L2∞ = +� +˘xT +0 ˘Q˘x0, +(4) +˘Q = +� ∞ +0 +et ˘A +T +˘cT ˘cet ˘Adt = +� +Q +S +ST Qr +� +. +Hereby, ˘Q is the observability Gramian of the error sys- +tem, composed of the Gramians Q and Qr of the FOM and +ROM and a matrix S ∈ RN,n that can be determined by +solving a sparse-dense Sylvester equation [9]. Note that +Q is already required for constructing the ROM with BT +or SPA and thus is available for the error bound without +additional costs (Assumption 1-ii)). Only the matrices +Qr and S have to be determined, and this has a com- +parably low computational cost. Once this is done, the +evaluation of (4) for any initial condition only requires +matrix-vector multiplications, and thus can be used as +an online-efficient error bound. +Remark 2. The inequality (4) holds for any T > 0 and +becomes an equality for T → ∞. On the other hand, +when using time-limited model order reduction, e.g., +[12], ˘Q can be replaced by the time-limited Gramian, +and the inequality then also becomes an equality for a +finite time T . +Remark 3. In a large-scale setting it may be necessary +to substitute the Gramians with low-rank approxima- +tions for computational reasons, see [3]. In the presence +of low-rank errors the results of this section cannot be +shown rigorously, but they can still serve as a basis for +error estimation. +3 +Linear signal generator driven systems +Consider the linear asymptotically stable system +˙˘x(t) = ˘A˘x(t) + ˘bw(t), +˘x(0) = ˘x0, +˘y(t) = ˘c˘x(t) + ˘dw(t), +(5) +with system matrices as in (3). We assume it to be driven +by a linear signal generator, i.e., w to be described a +linear autonomous differential equation +w(t) = cξξ(t), +˙ξ(t) = Aξξ(t), +ξ(0) = ξ0 ∈ Cq, (6) +with Aξ ∈ Cq,q and cξ ∈ C1,q. We employ the notion of +steady-states from [2,8] and the representation of the so- +lution to (5) induced by it. Steady-state refers in this con- +text to the long-time behavior the solution approaches +independently of the choice of initial condition ˘x0. Note +that the impact of the initial condition fades away over +time due to the asymptotic stability of the system. We +assume ˘A and Aξ to not have any common eigenvalues, +which implies that the Sylvester equation +˘AΠ + ˘bcξ = ΠAξ +(7) +is uniquely solvable for Π ∈ RN+n,q. It follows that +� ˘A ˘bcξ +0 Aξ +� � +Π +Iq +� += +� ˘AΠ + (− ˘AΠ + ΠAξ) +Aξ +� += +� +Π +Iq +� +Aξ, +with Iq ∈ Rq,q denoting the unit matrix. Using the latter +relation, it can be shown that the steady-state of the +signal generator driven system reads ˘xst(t) = Πξ(t), t ≥ +0. This is the specific solution of (5)-(6) with ˘x0 = Πξ0, +since +d +dt +� +˘xst(t) +ξ(t) +� += +� ˘A ˘bcξ +0 Aξ +� � +Π +Iq +� +ξ(t) += +� +Π +Iq +� +Aξξ(t) = +� +Π +Iq +� +d +dtξ(t). +Thus, we can assign exactly one steady-state to a sig- +nal w given by a linear signal generator, and the linear +mapping +˘F : w(·) �→ ˘F(w(·)) = (˘cΠ + ˘dcξ)ξ(·) +is well-defined. Finally note that the output response of +(5)-(6) with a general choice of ˘x0 reads +˘y(t) = +(˘cΠ + ˘dcξ)ξ(t) +� +�� +� +steady-state response ˘ +F(w) ++ +˘ce +˘At(˘x0 − Πξ0) +� +�� +� +transient response +, +and the transient response decays exponentially. An il- +lustration of the convergenceto the steady-state is shown +in Fig. 1, using zero initial conditions. +3 + +0 +2 +4 +Time t +-3 +0 +3 +Output y +Fig. 1. Illustration +of the convergence +to the steady-state +response +for +sig- +nal w(t) = cos(5t) +and +the +model +given +by +the +Beam +benchmark +(cf., Section 6.1 and +[3, Section 24]). +4 +Steady-state response to a Fourier series +For a prescribed order K ≥ 0, the truncated Fourier +series of the function u : [0, T ] → R is defined by +wK(t) = λ0 + +K +� +ℓ=1 +λℓcos +� +2πℓ t +T +� ++ λK+ℓ sin +� +2πℓ t +T +� +=: +2K +� +ℓ=0 +λℓgℓ(t), +t ∈ [0, T ], +(8) +with Fourier coefficients λℓ = � T +0 gℓ(t)u(t)dt/∥gℓ∥2 +LT +2 , +ℓ = 0, . . . , 2K. The truncated Fourier series wK and the +steady-state response ˘F(wK) it infers on a linear system, +cf. Section 3, are examined in more detail in this section. +A computationally convenient representation of ˘F(wK) +is key for an efficient implementation of our error bound. +Let i denote the imaginary unit (i2 = −1), and let α ∈ R. +An exponential pulse t �→ eiαt can be described by a +signal generator as in (6), whereby a scalar-valued state +ξα : [0, T ] �→ C can be used. The related Sylvester equa- +tion (7) simplifies to a linear equation, i.e., +eiαt = ξα(t), +˙ξα(t) = iαξα(t), +ξα(0) = 1, +Πα = (iαI − ˘A)−1˘b ∈ CN. +(9) +The steady-state response of this pulse is given by +˘F(eiα·) = (˘cΠα + ˘d)eiα·. Using that the cosine and +sine functions are the real and imaginary parts of the +exponential pulse, i.e., for any t ≥ 0 it holds +cos(αt) = Re(eiαt), +sin(αt) = Im(eiαt), +we can derive a convenient representation of ˘F(gℓ) for +ℓ = 1, . . . , 2K. For the cosine functions that is +˘F(cos(α ·)) = ˘cRe +� +Πα +� +Re(eiα·) + i Im(eiα·) +�� ++ ˘d cos(α ·) += (˘cRe(Πα) + ˘d) cos(α ·) − (˘cIm(Πα)) sin(α ·), +and, by a similar calculation, it follows +˘F(sin(α ·)) = (˘cIm(Πα)) cos(α ·) + (˘cRe(Πα) + ˘d) sin(α ·). +All in all, by employing the linearity of ˘F and exploit- +ing that {g0, . . . , g2K} and { ˘F(g0), . . . , ˘F(g2K)} are or- +thogonal sets of functions with respect to the L2 +T -scalar +product, we conclude the following lemma. +Lemma 4. The steady-state response of system (5) with +w = wK as in (8), i.e. ˘F(wK) = �2K +ℓ=0 λℓ ˘F(gℓ), fulfills +(∥ ˘F(wK)∥L2 +T )2 = T +���˘cΠ0 + ˘d +��� +2 +λ2 +0 ++ T +2 +K +� +ℓ=1 +���˘cΠℓ + ˘d +��� +2 +(λ2 +ℓ + λ2 +K+ℓ), +whereby Πℓ for ℓ = 0, . . . , K is given by (9). Further, the +steady-state has the initial conditions +˘xst,0 := λ0Π0 + +K +� +ℓ=1 +λℓRe(Πℓ) + λK+ℓIm(Πℓ). +Remark 5. The quantities ˘cΠℓ, ℓ = 0, . . . , K, are the +so-callled moments of the error system (3) at the fre- +quencies sℓ = iℓ, cf. [1]. In other words, they are the +differences in the moments of the FOM and the ROM. +5 +A posteriori error bound +The main result of this paper relies on the Fourier series +approximation wK of the given input u, and a specific +splitting of the reduction error. The error is decomposed +into the steady-state and transient response induced by +wK and the initial conditions, and the response originat- +ing from the remainder of the input u − wK. Different +techniques are used to bound these three terms individ- +ually. +Theorem 6 (A posteriori error bound). Let Assump- +tion 1 hold, considering the square-integrable input +u : [0, T ] → R and initial conditions x0 and xr0. Let ˘Q +be as in (4) and K ∈ N. Let wK, ˘F(wK) and ˘xst,0 be as +in Lemma 4. +Then the reduction error E = y − yr is bounded by +∥E∥L2 +T ≤ γK,u,x0 +γK,u,x0 = ∥ ˘F(wK)∥L2 +T + +� +˘xT +c ˘Q˘xc + α∥u − wK∥L2 +T , +where ˘xc = [xT +0 , xrT +0 ]T − ˘xst,0 ∈ RN+n. +Proof. We employ the linearity of the systems to split +the reduction error into three sub parts. Each of them +has a representation as output of the error system +(3) with a certain choice of input ˘u and initial con- +dition ˘x(0). We split the reduction error according to +E = ˘yst + ˘y˘xc + ˘yrest, with +4 + +• ˘yst obtained by input ˘u = w and ˘x(0) = ˘xst,0; +• ˘y˘xc obtained for trivial input ˘u ≡ 0 and ˘x(0) = ˘xc; +• ˘yrest obtained by input ˘u = u − wK and ˘x(0) = 0. +Clearly, ∥E∥L2 +T ≤ ∥˘yst∥L2 +T + ∥˘y˘xc∥L2 +T + ∥˘yrest∥L2 +T holds +by the triangle inequality. The claimed error bound +γK,u,x0 is obtained using Lemma 4 to bound the steady- +state ˘yst, formula (4) to bound the transient response +˘y˘xc, and the a priori error result from Assumption 1 for +the rest ˘yrest. +Remark 7. The order K of the Fourier series wK is +the only parameter to be chosen in our error bound. We +propose K ≈ 10 as a guide number. In Section 6.2 an +illustrative study of its influence is made. +Let us emphasize that our error bound allows for an +efficient offline-online decomposition. Almost all re- +quired quantities except for the Fourier coefficients +λℓ are independent of the input u and the initial +conditions x0 and can therefore be determined in +the offline phase. Moreover, it can be used that +∥u − wK∥L2 +T = +� +∥u∥2 +L2 +T − ∥wK∥2 +L2 +T holds due to the or- +thogonality of wK and u. The evaluation of the Fourier +coefficients is thus the main step of the online phase, +and its computational cost does not scale with the +dimension of the FOM. +6 +Numerical validation +The efficiency of our error bound is showcased with two +academic benchmark examples. We draw comparisons +to the well-known a priori bound (Section 2) and illus- +trate the influence of the order K used in the underlying +Fourier series, cf. Remark 7. +The FOMs (Beam and CD Player) used in our numerical +tests are from the SLICOT benchmark collection [3, Sec- +tion 24]. All numerical results have been generated using +MATLAB Version 9.1.0 (R2016b) on an Intel Core i5-7500 +CPU with 16.0GB RAM. The simulations are based on +the MATLAB built-in integrator ode15s with tolerances +set to ’AbsTol = 10−10’ and ’RelTol= 10−7’. Time inte- +grals were approximated by quadrature with the trape- +zoidal rule on a uniform time mesh with 2 000 points. +Finally, the simulation time T = 2π is used for all tests. +6.1 +Beam (study in ROM dimension n) +The Beam benchmark is a single-input single-output +model with N = 349. We simulate this FOM with zero ini- +tial conditions and the input u(t) = 4 sin3(2.7 t) + e0.2 t, +t ∈ [0, T ], and compare it to simulations with ROMs +obtained by either BT or SPA and varying dimensions +n ∈ [2, 30]. As shown in Fig. 2, our proposed bound +(with parameter K = 10) improves the error estimation +Error vs. bounds (varying n) +ROM by BT +ROM by SPA +2 +10 +20 +30 +10-2 +101 +104 +2 +10 +20 +30 +10-2 +101 +104 +ROM dimension n +ROM dimension n +Fig. 2. Beam. Reduction error versus the proposed a pos- +teriori bound γK,u,x0 (with x0 = 0 and K = 10) and the +standard a priori bound. +of the a priori bound by about one order in average. The +improvement is more pronounced for the reduction with +SPA, which also shows a smaller reduction error. In con- +trast to that, the a priori bound is the same for BT and +SPA, i.e., not adapted to the setting. +6.2 +CD Player (study in order K of Fourier series) +We consider the CD Player from the SLICOT benchmark +collection, which is a multi-input multi-output model +with N = 120. Since the paper is restricted to the single- +input single-output case, we replace the input matrix by +its first column and the output matrix by its second row +for our numerical tests. This FOM is reduced by BT using +n = 8. We focus here on the influence of the parame- +ter K on the performance of our bound. For the com- +parisons, we adapt the a priori bound according to [9], +where errors related to initial conditions are bounded +separately from the input using (4), which yields the +bound ∆x0 := ˘xT +0 ˘Q˘x0 with ˘x0 given as the initial con- +ditions of the error system. +For the simulations, we choose x0 = [1, . . . , 1]T/100, and +the input +u(t) = +� +t +− 8 exp(−t/2) +t ≤ π +(π − t) − 8 exp(−t/2) +t > π , +which has a discontinuity at t = π, see Fig. 3 for a plot in +time. The nonzero initial conditions and the input dis- +continuity result in two peaks in the output response, +cf. Fig. 4. +Moreover, the discontinuity implies a slow +decay of the Fourier series approximation of u. This is +illustrated in Fig. 5 for K ∈ [0, 15], where also the re- +sulting a posteriori bound in comparison to the a priori +bound and the reduction error is shown. It is seen that +the Fourier series approximation wK has still an error +of about 10% for K = 15. The error is overestimated +by less than one order by our a posteriori error bound +with K ≥ 10, which is a significant improvement com- +pared to the almost two order of magnitude observed for +5 + +0 +2 +-10 +-5 +0 +Fig. 3. CD Player. +Plot in time for used +input +u +and +two +Fourier +series +ap- +proximations of dif- +ferent order. +Time t +0 +2 +-30 +0 +30 +60 +Fig. 4. CD Player. +Plot in time of out- +puts y and yr for the +FOM and ROM, respec- +tively. +Time t +Fourier series quality +Error vs. bounds +0 +10 +3% +10% +30% +100% +0 +5 +10 +15 +100 +101 +102 +103 +Order K of wK +Order K of wK +Fig. 5. CD Player, parameter study in K. Left: Relative er- +ror of Fourier series approximation. Right: Reduction error +versus the proposed a posteriori bound γK,u,x0 and the stan- +dard a priori bound (using ∆x0 as in [9] for the initial con- +ditions). ROM obtained by BT with n = 8. +the a priori bound. This small example showcases that +our a posteriori bound is still effective even if the under- +lying Fourier series approximation only has a mediocre +approximation quality. +Conclusion +We proposed an a posteriori extension of the well-konwn +a priori error bound for balancing-related model reduc- +tion. It alleviates the worst-case type error estimation +that is inherent in the a priori error bound. The idea is to +filter out an approximation on the input signal, for which +a more explicit error analysis can be done in an efficient +manner. In this paper, we used a truncated Fourier se- +ries approximation of the input and derived an efficient +offline-online decomposition for it. +On a final note, we would like to point towards pos- +sible extensions of our result. For certain applications, +it could be interesting to study other input approxima- +tions, e.g., using signal generators related to a known +frequency range of interest. The extension of our bound +for time-limited balanced truncation is straight forward, +using the results from [12], cf., Remark 2. Multi-input +multi-output systems could also be considered, but this +requires a separate approximation of each of the input +components. +References +[1] +A. Antoulas. +Approximation of Large-Scale Dynamical +Systems, volume 6 of Adv. Des. Control. SIAM, Philadelphia, +PA, 2005. +[2] +A. Astolfi. Model reduction by moment matching for linear +and nonlinear systems. +IEEE Transactions on Automatic +Control, 55(10):2321–2336, 2010. +[3] +P. Benner, V. Mehrmann, and D. C. Sorensen, editors. +Dimension Reduction of Large-Scale Systems. Lecture Notes +in Computational Science and Engineering. Springer, 1 +edition, 2005. +[4] +L. +Feng, +A. +C. +Antoulas, +and +P. +Benner. +Some +a +posteriori error bounds for reduced-order modelling of (non-) +parametrized linear systems. ESAIM: Math. Model. Numer. +Anal., 51(6):2127–2158, 2017. +[5] +L. +Feng +and +P. +Benner. +A +new +error +estimator +for +reduced-order +modeling +of +linear +parametric +systems. +IEEE Transactions on Microwave Theory and Techniques, +67(12):4848–4859, 2019. +[6] +M. Green and D. J. Limebeer. Linear robust control. CRRC, +2012. +[7] +B. Haasdonk and M. Ohlberger. +Efficient reduced models +and a posteriori error estimation for parametrized dynamical +systems by offline/online decomposition. +Math. Comput. +Model. Dyn. Syst., 17(2):145–161, 2011. +[8] +A. Isidori and C. I. Byrnes. +Steady-state behaviors in +nonlinear systems with an application to robust disturbance +rejection. Annu. Rev. Control, 32(1):1–16, 2008. +[9] +B. Liljegren-Sailer. +Effective error estimation for model +reduction with inhomogeneous initial conditions. +arXiv e- +prints 2201.06631, 2022. +[10] Y. +Liu +and +B. +D. +Anderson. +Singular +perturbation +approximation of balanced systems. +Internat. J. Control, +50(4):1379–1405, 1989. +[11] B. Moore. Principal component analysis in linear systems: +Controllability, observability, and model reduction. +IEEE +Trans. Automat. Control, 26(1):17–32, 1981. +[12] M. Redmann. An L2 +T -error bound for time-limited balanced +truncation. Systems Control Lett., 136:104620, 2020. +6 + diff --git a/IdAzT4oBgHgl3EQfHvvy/content/tmp_files/load_file.txt b/IdAzT4oBgHgl3EQfHvvy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ccdd8466abb36f13ead39a608bc56698d6b7b832 --- /dev/null +++ b/IdAzT4oBgHgl3EQfHvvy/content/tmp_files/load_file.txt @@ -0,0 +1,348 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf,len=347 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='01052v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='DS] 3 Jan 2023 A time domain a posteriori error bound for balancing-related model order reduction ⋆ Bj¨orn Liljegren-Sailer a aUniversit¨at Trier, FB IV - Mathematik, Lehrstuhl Modellierung und Numerik, D-54286 Trier, Germany Abstract The aim in model order reduction is to approximate an input-output map described by a large-scale dynamical system with a low-dimensional and cheaper-to-evaluate reduced order model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' While high fidelity can be achieved by a variety of methods, only a few of them allow for rigorous error control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In this paper, we propose a rigorous error bound for the reduction of linear systems with balancing-related reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' More specifically, we consider the simulation over a finite time interval and provide an a posteriori adaption of the standard a priori bound for Balanced Truncation and Balanced Singular Perturbation Approximation in that setting, which improves the error estimation while still yielding a rigorous bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Our result is based on an error splitting induced by a Fourier series approximation of the input and a subsequent refined error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We make use of system-theoretic concepts, such as the notion of signal generator driven systems, steady-states and observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Our bound is also applicable in the presence of nonzero initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Numerical evidence for the sharpness of the bound is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Key words: error bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' a posteriori;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' balanced truncation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' balancing-related;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' model order reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 1 Introduction Consider the linear time-invariant system ˙x(t) = Ax(t) + bu(t), x(0) = x0 ∈ RN, y(t) = cx(t) + du(t) (1) for t ∈ [0, T ] with state x : [0, T ] → RN, initial condi- tions x0 and an asymptotically stable state equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', A ∈ RN,N Hurwitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Moreover, let b ∈ RN,1, c ∈ R1,N and d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' When the state dimension N is large com- pared to the dimension of the input u and output y – for ease of presentation we assume the scalar-valued case u, y : [0, T ] → R – the computational costs for evaluat- ing (1) many times can become very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' This model is referred to as full order model (FOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We are interested in a lower-dimensional reduced order model (ROM) that is cheaper to evaluate and still sufficiently accurate, in the sense that it reproduces a similar output yr ≈ y for ⋆ Corresponding author B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Liljegren-Sailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' +49 651 201-3468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Email address: bjoern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='sailer@uni-trier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='de (Bj¨orn Liljegren-Sailer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' the inputs u of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The considered ROM reads ˙xr(t) = Arxr(t) + bru(t), xr(0) = xr0 ∈ Rn, yr(t) = crxr(t) + dru(t), (2) with reduced state xr : [0, T ] → Rn, n ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' A vari- ety of model order reduction methods (MOR) have been proposed, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', [1,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' For example, in the projection- based approaches one seeks for appropriate reduction bases V, W ∈ RN,n with WT V = In (unit matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The ROM is then defined by xr0 = WT x0, Ar = WT AV, br = WT b, cr = cV and dr = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The reduction error E(t) := y(t) − yr(t), t ∈ [0, T ], is not known in practice, so it is of high interest to have bounds or estimates on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' For its analysis it is useful to introduce the error system ˙˘x(t) = � A Ar � � �� � =: ˘A ˘x(t) + � b br � � �� � =:˘b ˘u(t), ˘x(0) = ˘x0, ˘y(t) = � c, −cr � � �� � =:˘c ˘x(t) + (d − dr) � �� � =: ˘d ˘u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' (3) By construction, ˘y(t) = E(t) for t ∈ [0, T ] if ˘u = u and ˘x0 = [xT 0 , xrT 0 ]T are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Thus, (3) is just a concise representation of the error dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In this paper, a rigorous a posteriori bound for the L2-error in finite time [0, T ] is derived from system- theoretic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' It is most naturally applied in combination with balancing-related MOR methods, such as Balanced Truncation (BT) and Balanced Singular Perturbation Approximation (SPA) [6,11,10], since it exploits and relies on the key features of these methods, summarized in the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The FOM is considered for square- integrable input u : [0, T ] → R and reduced by a method, for which the following holds: i) An a priori bound is accessible for the ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' That is, assuming zero initial conditions (x0 = 0), it holds ∥E∥L2 T := �� T 0 |E(t)|2dt ≤ α∥u∥L2 T for a known constant α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' ii) The observability Gramian Q of the FOM is required for the construction of the ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' iii) The ROM is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Here and in the following, ∥ · ∥L2 T refers to the L2-norm on the time-interval [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The availability of an a pri- ori bound as in Assumption 1 is of high practical value on its own, especially for the selection of an appropriate ROM dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' But this a priori result yields a worst- case estimation on the error, since it does not take into account any specific knowledge on the simulation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The bound we propose in this paper can be considered an a posteriori adaption of the a priori bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The main idea is to filter out a signal w and to apply the pes- simistic a priori bound to the remainder u−w only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The approximation w is chosen such that a more explicit er- ror analysis becomes feasible that leads to less overesti- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We further split the error part related to w into its long-time behavior, also denoted as steady-state, and a term that is equivalent to the output energy a certain initial state has in the error system (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The steady-state can be determined in an offline-online efficient way, and the other error term can be sharply bounded using the approach from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The peculiarity of our proposed error bound is that it takes into account information of the input u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', is an a posteriori result, and, simultaneously, exploits the features inherent in balancing-related MOR, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', Assump- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Consequently, it outperforms other rigorous a posteriori bounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', the residual based bounds de- rived in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Let us mention that effective error estimates have been proposed in literature [4,5], but those are typ- ically not rigorous and thus may lead to an underestima- tion of the actual error, which is problematic for certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The structure of the paper revolves around the deriva- tion of our error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In Section 2, we briefly outline the standard a priori bound for balancing-related MOR, and the error bound from [9] for inhomogeneous initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Those are ingredients of our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Addi- tionally, we make use of the notion of steady-states for a certain class of approximations on the input in our error analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' the respective results are stated in Sections 3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Our a posteriori bound is proven in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Its effec- tivity is numerically demonstrated in Section 6 at the example of two academic benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 2 Results for balancing-related MOR The observability and controllability of a state in the FOM can be quantified in terms of the observability Gramian Q and controllability Gramian P, defined as Q = � ∞ 0 etAT cT cetAdt, P = � ∞ 0 etAbbT etAT dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' These matrices are well-defined for asymptotically sta- ble systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' They play a crucial role in the analysis and implementation of BT and SPA, see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Notably, these balancing-related MOR methods allow for a rigorous er- ror analysis that is based on the Gramians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In this sec- tion, we outline the error bounds from literature which also play a role in our refined error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' These re- sults require, respectively justify, Assumption 1 in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='1 A priori error bound Asymptotically stable systems can be transformed in a so-called balanced form, in which the Gramians Q and P are simultaneously diagonalized and equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The di- agonal entries σ1 ≥ σ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' ≥ σN ≥ 0 of the result- ing diagonal matrix are called the Hankel Singular Val- ues (HSVs) and represent a measure for observability and controllability of the balanced states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In both BT and SPA, the ROM is composed of the states related to the largest HSVs, see [6] for details on the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Given the ROM dimension n is chosen such that σn > σn+1 and zero initial conditions are considered (x0 = 0), the re- duction error can be shown to fulfill the a priori error bound ∥E∥L2∞ ≤ α∥u∥L2∞ with α = 2 �N j=n+1 σj com- posed of the neglected HSVs, see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' As we require a bound on a finite time interval [0, T ] (Assumption 1-i)), we apply the former bound on the extension of the input u : [0, T ] → R to the infinite interval [0, ∞), obtained by 2 setting u(t) = 0 for t > T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' This yields ∥E∥L2 T ≤ ∥E∥L2 ∞ ≤ α∥u∥L2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='2 Error originating from initial conditions The case with trivial input (u(t) = 0 for t ≥ 0) allows for a simple error analysis based on the notion of ob- servability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In this setting, the output of the error sys- tem (3) is solely determined by the initial conditions ˘x(0) = ˘x0 = [xT 0 , xrT 0 ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' It reads y˘x0 = ˘cet ˘A˘x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' As shown in [9], a bound for its norm follows directly from applying the notion of observability to the error systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', ∥y˘x0∥L2 T ≤ ∥y˘x0∥L2∞ = � ˘xT 0 ˘Q˘x0, (4) ˘Q = � ∞ 0 et ˘A T ˘cT ˘cet ˘Adt = � Q S ST Qr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Hereby, ˘Q is the observability Gramian of the error sys- tem, composed of the Gramians Q and Qr of the FOM and ROM and a matrix S ∈ RN,n that can be determined by solving a sparse-dense Sylvester equation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Note that Q is already required for constructing the ROM with BT or SPA and thus is available for the error bound without additional costs (Assumption 1-ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Only the matrices Qr and S have to be determined, and this has a com- parably low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Once this is done, the evaluation of (4) for any initial condition only requires matrix-vector multiplications, and thus can be used as an online-efficient error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The inequality (4) holds for any T > 0 and becomes an equality for T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' On the other hand, when using time-limited model order reduction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', [12], ˘Q can be replaced by the time-limited Gramian, and the inequality then also becomes an equality for a finite time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In a large-scale setting it may be necessary to substitute the Gramians with low-rank approxima- tions for computational reasons, see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In the presence of low-rank errors the results of this section cannot be shown rigorously, but they can still serve as a basis for error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 3 Linear signal generator driven systems Consider the linear asymptotically stable system ˙˘x(t) = ˘A˘x(t) + ˘bw(t), ˘x(0) = ˘x0, ˘y(t) = ˘c˘x(t) + ˘dw(t), (5) with system matrices as in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We assume it to be driven by a linear signal generator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', w to be described a linear autonomous differential equation w(t) = cξξ(t), ˙ξ(t) = Aξξ(t), ξ(0) = ξ0 ∈ Cq, (6) with Aξ ∈ Cq,q and cξ ∈ C1,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We employ the notion of steady-states from [2,8] and the representation of the so- lution to (5) induced by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Steady-state refers in this con- text to the long-time behavior the solution approaches independently of the choice of initial condition ˘x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Note that the impact of the initial condition fades away over time due to the asymptotic stability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We assume ˘A and Aξ to not have any common eigenvalues, which implies that the Sylvester equation ˘AΠ + ˘bcξ = ΠAξ (7) is uniquely solvable for Π ∈ RN+n,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' It follows that � ˘A ˘bcξ 0 Aξ � � Π Iq � = � ˘AΠ + (− ˘AΠ + ΠAξ) Aξ � = � Π Iq � Aξ, with Iq ∈ Rq,q denoting the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Using the latter relation, it can be shown that the steady-state of the signal generator driven system reads ˘xst(t) = Πξ(t), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' This is the specific solution of (5)-(6) with ˘x0 = Πξ0, since d dt � ˘xst(t) ξ(t) � = � ˘A ˘bcξ 0 Aξ � � Π Iq � ξ(t) = � Π Iq � Aξξ(t) = � Π Iq � d dtξ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Thus, we can assign exactly one steady-state to a sig- nal w given by a linear signal generator, and the linear mapping ˘F : w(·) �→ ˘F(w(·)) = (˘cΠ + ˘dcξ)ξ(·) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Finally note that the output response of (5)-(6) with a general choice of ˘x0 reads ˘y(t) = (˘cΠ + ˘dcξ)ξ(t) � �� � steady-state response ˘ F(w) + ˘ce ˘At(˘x0 − Πξ0) � �� � transient response , and the transient response decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' An il- lustration of the convergenceto the steady-state is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 1, using zero initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 3 0 2 4 Time t 3 0 3 Output y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Illustration of the convergence to the steady-state response for sig- nal w(t) = cos(5t) and the model given by the Beam benchmark (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='1 and [3, Section 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 4 Steady-state response to a Fourier series For a prescribed order K ≥ 0, the truncated Fourier series of the function u : [0, T ] → R is defined by wK(t) = λ0 + K � ℓ=1 λℓcos � 2πℓ t T � + λK+ℓ sin � 2πℓ t T � =: 2K � ℓ=0 λℓgℓ(t), t ∈ [0, T ], (8) with Fourier coefficients λℓ = � T 0 gℓ(t)u(t)dt/∥gℓ∥2 LT 2 , ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The truncated Fourier series wK and the steady-state response ˘F(wK) it infers on a linear system, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Section 3, are examined in more detail in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' A computationally convenient representation of ˘F(wK) is key for an efficient implementation of our error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Let i denote the imaginary unit (i2 = −1), and let α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' An exponential pulse t �→ eiαt can be described by a signal generator as in (6), whereby a scalar-valued state ξα : [0, T ] �→ C can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The related Sylvester equa- tion (7) simplifies to a linear equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', eiαt = ξα(t), ˙ξα(t) = iαξα(t), ξα(0) = 1, Πα = (iαI − ˘A)−1˘b ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' (9) The steady-state response of this pulse is given by ˘F(eiα·) = (˘cΠα + ˘d)eiα·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Using that the cosine and sine functions are the real and imaginary parts of the exponential pulse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', for any t ≥ 0 it holds cos(αt) = Re(eiαt), sin(αt) = Im(eiαt), we can derive a convenient representation of ˘F(gℓ) for ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' For the cosine functions that is ˘F(cos(α ·)) = ˘cRe � Πα � Re(eiα·) + i Im(eiα·) �� + ˘d cos(α ·) = (˘cRe(Πα) + ˘d) cos(α ·) − (˘cIm(Πα)) sin(α ·), and, by a similar calculation, it follows ˘F(sin(α ·)) = (˘cIm(Πα)) cos(α ·) + (˘cRe(Πα) + ˘d) sin(α ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' All in all, by employing the linearity of ˘F and exploit- ing that {g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , g2K} and { ˘F(g0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , ˘F(g2K)} are or- thogonal sets of functions with respect to the L2 T -scalar product, we conclude the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The steady-state response of system (5) with w = wK as in (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' ˘F(wK) = �2K ℓ=0 λℓ ˘F(gℓ), fulfills (∥ ˘F(wK)∥L2 T )2 = T ���˘cΠ0 + ˘d ��� 2 λ2 0 + T 2 K � ℓ=1 ���˘cΠℓ + ˘d ��� 2 (λ2 ℓ + λ2 K+ℓ), whereby Πℓ for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , K is given by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Further, the steady-state has the initial conditions ˘xst,0 := λ0Π0 + K � ℓ=1 λℓRe(Πℓ) + λK+ℓIm(Πℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The quantities ˘cΠℓ, ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , K, are the so-callled moments of the error system (3) at the fre- quencies sℓ = iℓ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In other words, they are the differences in the moments of the FOM and the ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 5 A posteriori error bound The main result of this paper relies on the Fourier series approximation wK of the given input u, and a specific splitting of the reduction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The error is decomposed into the steady-state and transient response induced by wK and the initial conditions, and the response originat- ing from the remainder of the input u − wK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Different techniques are used to bound these three terms individ- ually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Theorem 6 (A posteriori error bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Let Assump- tion 1 hold, considering the square-integrable input u : [0, T ] → R and initial conditions x0 and xr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Let ˘Q be as in (4) and K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Let wK, ˘F(wK) and ˘xst,0 be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Then the reduction error E = y − yr is bounded by ∥E∥L2 T ≤ γK,u,x0 γK,u,x0 = ∥ ˘F(wK)∥L2 T + � ˘xT c ˘Q˘xc + α∥u − wK∥L2 T , where ˘xc = [xT 0 , xrT 0 ]T − ˘xst,0 ∈ RN+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We employ the linearity of the systems to split the reduction error into three sub parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Each of them has a representation as output of the error system (3) with a certain choice of input ˘u and initial con- dition ˘x(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We split the reduction error according to E = ˘yst + ˘y˘xc + ˘yrest, with 4 ˘yst obtained by input ˘u = w and ˘x(0) = ˘xst,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' ˘y˘xc obtained for trivial input ˘u ≡ 0 and ˘x(0) = ˘xc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' ˘yrest obtained by input ˘u = u − wK and ˘x(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Clearly, ∥E∥L2 T ≤ ∥˘yst∥L2 T + ∥˘y˘xc∥L2 T + ∥˘yrest∥L2 T holds by the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The claimed error bound γK,u,x0 is obtained using Lemma 4 to bound the steady- state ˘yst, formula (4) to bound the transient response ˘y˘xc, and the a priori error result from Assumption 1 for the rest ˘yrest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The order K of the Fourier series wK is the only parameter to be chosen in our error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We propose K ≈ 10 as a guide number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='2 an illustrative study of its influence is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Let us emphasize that our error bound allows for an efficient offline-online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Almost all re- quired quantities except for the Fourier coefficients λℓ are independent of the input u and the initial conditions x0 and can therefore be determined in the offline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Moreover, it can be used that ∥u − wK∥L2 T = � ∥u∥2 L2 T − ∥wK∥2 L2 T holds due to the or- thogonality of wK and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The evaluation of the Fourier coefficients is thus the main step of the online phase, and its computational cost does not scale with the dimension of the FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 6 Numerical validation The efficiency of our error bound is showcased with two academic benchmark examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We draw comparisons to the well-known a priori bound (Section 2) and illus- trate the influence of the order K used in the underlying Fourier series, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The FOMs (Beam and CD Player) used in our numerical tests are from the SLICOT benchmark collection [3, Sec- tion 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' All numerical results have been generated using MATLAB Version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='0 (R2016b) on an Intel Core i5-7500 CPU with 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='0GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The simulations are based on the MATLAB built-in integrator ode15s with tolerances set to ’AbsTol = 10−10’ and ’RelTol= 10−7’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Time inte- grals were approximated by quadrature with the trape- zoidal rule on a uniform time mesh with 2 000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Finally, the simulation time T = 2π is used for all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='1 Beam (study in ROM dimension n) The Beam benchmark is a single-input single-output model with N = 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We simulate this FOM with zero ini- tial conditions and the input u(t) = 4 sin3(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='7 t) + e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='2 t, t ∈ [0, T ], and compare it to simulations with ROMs obtained by either BT or SPA and varying dimensions n ∈ [2, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 2, our proposed bound (with parameter K = 10) improves the error estimation Error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' bounds (varying n) ROM by BT ROM by SPA 2 10 20 30 10-2 101 104 2 10 20 30 10-2 101 104 ROM dimension n ROM dimension n Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Reduction error versus the proposed a pos- teriori bound γK,u,x0 (with x0 = 0 and K = 10) and the standard a priori bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' of the a priori bound by about one order in average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The improvement is more pronounced for the reduction with SPA, which also shows a smaller reduction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In con- trast to that, the a priori bound is the same for BT and SPA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', not adapted to the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='2 CD Player (study in order K of Fourier series) We consider the CD Player from the SLICOT benchmark collection, which is a multi-input multi-output model with N = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Since the paper is restricted to the single- input single-output case, we replace the input matrix by its first column and the output matrix by its second row for our numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' This FOM is reduced by BT using n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' We focus here on the influence of the parame- ter K on the performance of our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' For the com- parisons, we adapt the a priori bound according to [9], where errors related to initial conditions are bounded separately from the input using (4), which yields the bound ∆x0 := ˘xT 0 ˘Q˘x0 with ˘x0 given as the initial con- ditions of the error system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' For the simulations, we choose x0 = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' , 1]T/100, and the input u(t) = � t − 8 exp(−t/2) t ≤ π (π − t) − 8 exp(−t/2) t > π , which has a discontinuity at t = π, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 3 for a plot in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The nonzero initial conditions and the input dis- continuity result in two peaks in the output response, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Moreover, the discontinuity implies a slow decay of the Fourier series approximation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 5 for K ∈ [0, 15], where also the re- sulting a posteriori bound in comparison to the a priori bound and the reduction error is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' It is seen that the Fourier series approximation wK has still an error of about 10% for K = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The error is overestimated by less than one order by our a posteriori error bound with K ≥ 10, which is a significant improvement com- pared to the almost two order of magnitude observed for 5 0 2 10 5 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' CD Player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Plot in time for used input u and two Fourier series ap- proximations of dif- ferent order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Time t 0 2 30 0 30 60 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' CD Player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Plot in time of out- puts y and yr for the FOM and ROM, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Time t Fourier series quality Error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' bounds 0 10 3% 10% 30% 100% 0 5 10 15 100 101 102 103 Order K of wK Order K of wK Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' CD Player, parameter study in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Left: Relative er- ror of Fourier series approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Right: Reduction error versus the proposed a posteriori bound γK,u,x0 and the stan- dard a priori bound (using ∆x0 as in [9] for the initial con- ditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' ROM obtained by BT with n = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' the a priori bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' This small example showcases that our a posteriori bound is still effective even if the under- lying Fourier series approximation only has a mediocre approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Conclusion We proposed an a posteriori extension of the well-konwn a priori error bound for balancing-related model reduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' It alleviates the worst-case type error estimation that is inherent in the a priori error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The idea is to filter out an approximation on the input signal, for which a more explicit error analysis can be done in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' In this paper, we used a truncated Fourier se- ries approximation of the input and derived an efficient offline-online decomposition for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' On a final note, we would like to point towards pos- sible extensions of our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' For certain applications, it could be interesting to study other input approxima- tions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', using signal generators related to a known frequency range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' The extension of our bound for time-limited balanced truncation is straight forward, using the results from [12], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=', Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Multi-input multi-output systems could also be considered, but this requires a separate approximation of each of the input components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Antoulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Approximation of Large-Scale Dynamical Systems, volume 6 of Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfHvvy/content/2301.01052v1.pdf'} 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b/M9E2T4oBgHgl3EQfqgii/content/tmp_files/2301.04040v1.pdf.txt @@ -0,0 +1,7322 @@ +arXiv:2301.04040v1 [math.DG] 10 Jan 2023 +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION +FORMS +QIAOCHU MA +Abstract. This paper aims to study the asymptotic expansion of analytic torsion +forms associated with a certain series of flat bundles {Fp}p∈N∗. We prove the existence +of the full expansion and give a formula for the sub-leading term, while Bismut-Ma- +Zhang [9] have studied the first order expansion and expressed the leading term as the +integral of a locally computable differential form. +Introduction +0.1. Backgrounds. In the 1930s, the Reidemeister torsion was introduced by Reide- +meister [39] and Franz [19] in their study of lens spaces. The Reidemeister torsion was the +first homeomorphic invariant which is not homopoty. Let (F, ∇F) be a unitary flat vector +bundle over a compact manifold X. We assume that H•(X, F) = 0. The Reidemeister +torsion is a real number obtained by a simplicial complex with values in F associated +with a triangulation of X, which turns out to be independent of the triangulation. +Ray and Singer asked if there is an analytic interpretation of the Reidemeister torsion. +They defined their analytic torsion [38] as an alternating product of the regularized de- +terminant of Hodge Laplacian and conjectured that it coincides with the Reidemeister +torsion for unitary flat bundles. This conjecture was proved by Cheeger [16] and M¨uller +[32] independently. Bismut-Zhang [12] and M¨uller [33] simultaneously considered gen- +eralizations of this result. M¨uller [33] extended it to the case when F is unimodular, +X is oriented and odd-dimensional. Bismut-Zhang [12] generalized it to any flat vector +bundle with a Hermitian metric. +In this paper, we study the asymptotic property of analytic torsions. Let {Fp}p∈N∗ be +a certain family of flat vector bundles over M, we obtain the full asymptotic expansion +of the analytic torsion of Fp when p → +∞. +Let us now give some background on the results of this paper. +In [16, Example 1.3], Cheeger observed the relation between the Reidemeister torsion of +a simplicial complex and the size of its torsion subgroup of homology. By using this rela- +tion and discussing the asymptotics of analytic torsions of quotients of symmetric spaces +by a decreasing sequence of lattices in an underlying Lie group, Bergeron-Venkatesh [3] +studied the growth of the size of torsion elements in the homology of an arithmetic group. +This was the first application of analytic torsion in arithmetic. +In [34], M¨uller considered the asymptotics of analytic torsions for a family of flat +bundles over a compact 3-dimensional hyperbolic manifold as a real analogue of Bismut- +Vasserot’s result on the holomorphic torsion [11]. Let Γ\H3 be a compact 3-dimensional +hyperbolic manifold with constant sectional curvatures −4. +For p ∈ N∗, put Fp = +SympC2, where C2 is the flat bundle on Γ\H3 associated with the tautological represen- +tation of SL2(C) on C2 and Symp denotes the p-th symmetric power. Let T (Γ\H3, ∇Fp) +1 + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +2 +be the the analytic torsion of Fp. Using Selberg’s trace formula, M¨uller [34] obtained +lim +p→+∞ p−2T (Γ\H3, ∇Fp) = 2 +πVol(Γ\H3). +(0.1) +In [8] and [9], Bismut-Ma-Zhang gave a general construction of a family of flat vector +bundles {Fp}p∈N∗ on any compact manifold and they expressed the asymptotics of ana- +lytic torsions as an integral of a local computable differential form on the base manifold. +Indeed, they worked in a general setting of the analytic torsion forms of Bismut-Lott [7]. +Our main result is to extend Bismut-Ma-Zhang’s work to get a full expansion of +torsions and give an explicit formula for the sub-leading term. Now we explain in detail. +0.2. The existence of full expansion of torsions. In [7], Bismut-Lott constructed +the analytic torsion form as a family extension of the Ray-Singer torsion. Let π: M → S +be a fibration of manifolds with compact fiber X of dimension m. We set a metric gTX +on the relative tangent bundle TX and a horizontal bundle T HM ⊂ TM such that +TM = T HM ⊕ TX. +(0.2) +For any flat vector bundle (F, ∇F) over M with a Hermitian metric gF. Bismut-Lott’s +torsion form is a differential form T (T HM, gTX, ∇F, gF) ∈ Ωeven(S) and its 0-degree +component T0(T HM, gTX, ∇F, gF) is the function which to b ∈ S assigns the Ray-Singer +torsion of the fibre (Xb, gTXb) over b, computed using the flat bundle (F|Xb, ∇F|Xb, gF |Xb). +Let N be a compact K¨ahler manifold with dimC N = n, L a positive holomorphic +line bundle and ξ a holomorphic vector bundle on N. +Let G be a Lie group acting +holomorphically on N and this action can be lifted to L and ξ. Let PG → M be a flat +principal G-bundle. Set N = PG ×G N, we have a natural projection q: N → M and the +real relative tangent bundle TRN = ker q∗. For p ∈ N∗, let Lp be the p-th tensor power +of L. Let (F, ∇F) be another flat vector bundle on M with metric gF. Put +Fp = +� +PG ×G H(0,0)(N, Lp ⊗ ξ) +� +⊗ F, +(0.3) +where G acts naturally on H(0,0)(N, Lp⊗ξ). We summarize geometric settings as follows: +We have a flat connection ∇Fp induced by the flat connection on PG and ∇F. We also +TRN +Lp ⊗ ξ = PG ×G (Lp ⊗ ξ) +N +N = PG ×G N +PG ×G H(0,0)(N, Lp ⊗ ξ) +X +M +S +R•q∗ +q +π +Figure 1. +denote PG ×G L (resp. +PG ×G ξ) by L (resp. +ξ). +Given a metric gTRN on TRN, it +induces a volume form dvN ∈ Ω2n(N ) along the fibre together with PG. Let gL (resp. +gξ) be a metic on L (resp. ξ) over N . Then the L2-metric on H(0,0)(N, Lp ⊗ ξ) given by +(dvN, gL, gξ) and gF define a metric gFp on Fp. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +3 +In [9, Definition 9.13], Bismut-Ma-Zhang introduced a non-degeneracy condition for +gL (see § 3.5). Under this condition, Bismut-Ma-Zhang [9, § 4.3, § 9.10] proved that, for +p ∈ N∗ large enough, we have H•(M, Fp) = 0. In the rest of this section, we always +assume that L verifies the non-degeneracy condition. +For a ∈ R, let ψa be the automorphism of Λ• (T ∗S) such that, if α ∈ Λk(T ∗S), then +ψaα = akα. Let o(TX) be the orientation line of TX. Our main result is the following +theorem (see Theorem 5.1). +Theorem 0.1. There are locally computable differential forms W L,ξ +i +∈ Ω•(M, o(TX)) +such that for any k, ℓ ∈ N, there exists C > 0 such that as p → +∞, we have +���p−n−1ψ1/√pT +� +T HM, gTX, ∇Fp, gFp� +− +k +� +i=0 +p−i +� +X +W L,ξ +i +��� +C ℓ(S) ⩽ Cp−k−1. +(0.4) +We note that if dim X = m is odd, T +� +T HM, gTX, ∇Fp, gFp� +∈ Ωeven(S)/dΩodd(S) is +a topological invariant for p ∈ N∗ large, by the anomaly formula of Bismut-Lott [7, +Theorem 3.24], it is independent on (T HM, gTX, gFp). Hence each W L,ξ +i +is a topological +invariant for i ∈ N. In [9, Theorem 9.32], Bismut-Ma-Zhang established Theorem 0.1 +for k = 0 and gave an explicit formula for W L,ξ +0 +. +Let � +M → S be a fibre bundle with fibre � +X, and let Γ be a discrete group acting +fibrewise freely and properly discontinuously on � +M such that Γ\� +M = M. We have the +Γ-torsion form T Γ� +T HM, gTX, ∇Fp, gFp� +∈ Ωeven(S) as in (5.14) (see also [1], [24] and +[31]). Then T Γ� +T HM, gTX, ∇Fp, gFp� +has the same asymptotic expansion as in (0.4), +indeed, we have the following stronger result (see Theorem 5.2). +Theorem 0.2. The asymptotics of the two torsions differ only by an exponentially decay +term: there is a > 0 such that for ℓ ∈ N, there is C > 0 that as p → +∞, we have +���T Γ� +T HM, gTX, ∇Fp, gFp� +− T +� +T HM, gTX, ∇Fp, gFp���� +C ℓ(S) ⩽ Ce−ap. +(0.5) +Theorem 0.2 was first established by Bismut-Ma-Zhang [9, § 7.6] when S = {pt}, a +point. If S = {pt}, X = Γ\G/K, a compact locally symmetric manifold, and Fp is +induced by multiples of the highest weight λ ∈ u∗ of an irreducible U-representation, +where U is a compact form of G with lie algebra u, Bismut-Ma-Zhang [9, Proposition +8.12] showed that the nondegeneracy condition is the same to WU · λ ∩ t∗ = ∅, where +WU is the Weyl group of U and t is the Lie algebra of a maximal torus in K, which +is exactly the strong acyclic condition θΛ ̸= Λ [14, Propostion II.6.12], M¨uller-Pfaff +[35, Propositions 1.2, 1.3] gave a new proof of (0.5) and showed that T Γ� +gTX, ∇Fp, gFp� +is a polynomial of p ∈ N∗, see also Liu [22, Theorem 7.4.3] for another proof of this +polynomial property. +0.3. An explicit formula for W L,ξ +1 +for reductive G. Now we explain another main +result, an explicit formula for W L,ξ +1 +when G is a connected reductive linear Lie group, +rkξ = 1 and F = C trivial. +Let G be a connected reductive linear Lie group with Lie algebra g. Let K ⊂ G be a +maximal compact subgroup of G with Lie algebra k. We have the Cartan decomposition +g = p ⊕ k. Let U be a compact form of G with Lie algebra u = √−1p ⊕ k. +Recall that N is a compact complex manifold with dimC N = n and (L, gL) is a +positive line bundle on N with the first Chern form c1(L, gL). We assume further that + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +4 +U acts holomorphically on N and this action lifts to a holomorphic unitary action on L, +then c1(L, gL) is a U-invariant form. Let µL : N → u∗ be the associated moment map +obtained from the Kostant formula (6.7). +Let θg be the g-valued flat connection form on PG. Let PK be a reduction of PG to +K-principal bundle. Set gr = PK ×K g. By projection of θg on p and k with respect to +the Cartan decomposition, we get θg = θp + θk. +For A ∈ u, let RL(A) be the Duistermaat-Heckman integral +RL(A) = +� +N +exp +� +2πi⟨µL, A⟩ + c1(L, gL) +� +, +(0.6) +then we naturally extend RL(·) to a holomorphic function on gC ∼= u ⊗R C. +Let Sg be the symmetric algebra of g and we denote by Sg its formal completion. +Canonically, Sg can be identified with the algebra of real differential operators with +constant coefficients on g. Then Sg naturally acts on RL(·) (see also [9, § 1.4]). +Let {ei}m +i=1 be a local orthonormal frame of TX with dual frame {ei}m +i=1. Let � +TX be +another copy of TX and let �θp be the restriction of θp on � +TX. Set +|�θp|2 = +m +� +i=1 +��θp(ei) +�2 ∈ C ∞(M, S2gr), +�θp,2 = 1 +2�ei ∧ �ej[�θp(�ei), �θp(�ej)] ∈ C ∞(M, Λ2(� +T ∗X) ⊗ gr). +(0.7) +Let ∇gr,u be the connection on gr induced by θk. For t ⩾ 0, let σt be a section of +Λ•(T ∗M)�⊗Λ•(� +T ∗X) ⊗ Sgr given by +σt = −1 +4 +� +RTXei, ej +� +�ei ∧ �ej − θp,2 + +√ +t∇ +� +T ∗X⊗gr,u�θp + t|�θp|2 + t�θp,2. +(0.8) +Denote by +� ˆB : Λ•(TM)�⊗Λ•(� +TX) → Λ•(TM) the Berezin integral (see (4.58)). Let ϕ +be the endomorphism of Λ•(T ∗M) ⊗ C which maps α ∈ Λk(T ∗M) ⊗ C to (2πi)−k/2α. +If ξ is a trivial line bundle, we denote the form W L,ξ +i +in (0.4) by W L +i . Bismut-Ma-Zhang +[9, Definition 2.11] gave the following formula for W L +0 : +W L +0 = − +√ +2πiϕ +� +∞ +0 +� +� +B θp ∧ �θp +2 +exp(−σt)RL(0) dt +√ +t ∈ Ω•(M, o(TX)), +(0.9) +where the operator +θp∧ � +θp +2 +exp(−σt) ∈ Λ•(T ∗M)�⊗Λ•(� +T ∗X) ⊗ Sgr acts on the function +RL(·) and we evaluate at 0. +Now we assume that (ξ, gξ) is a holomorphic Hermitian line bundle on N. +Let +(TN, gTN) be the holomorphic tangent bundle of N, where gTN is induced by c1(L, gL): +if A ∈ TN, gTN = −ic1(L, gL)(A, A). We assume that the action of U on N lifts to +holomorphic unitary actions on (ξ, gξ) and (TN, gTN) with moment maps µξ and µdet TN. +Formally, for p ∈ N∗, let ξ +1 +p be the p-th root of L and (det TN) +1 +2p be the 2p-th root of +det TN at the level of cohomology and moment map (see (6.16)), then R +L⊗ξ +1 +p ⊗(det TN) +1 +2p +and W L⊗ξ +1 +p ⊗(det TN) +1 +2p +0 +are well defined even if ξ +1 +p ⊗ (det TN) +1 +2p may not be. +We have the following formula for W L,ξ +1 +(see Theorem 6.4). + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +5 +Theorem 0.3. For k = 1 in (0.4), as p → +∞ we have +p−n−1ψ1/√pT +� +T HM, gTX, ∇Fp, gFp� += +� +X +W L⊗ξ +1 +p ⊗(det TN) +1 +2p +0 ++ O +� +p−2� +∈ Ω•(S). +(0.10) +In other words, +W L,ξ +0 ++ p−1W L,ξ +1 += W L⊗ξ +1 +p ⊗(det TN) +1 +2p +0 ++ O +� +p−2� +. +(0.11) +In § 6.6 we discuss a special case of Theorem 0.3 when N is a coadjoint orbit of u∗, +and we use this to compute asymptotics of torsion for some concrete examples in § 6.7. +Liu considered the asymptotics of equivariant torsions for compact locally symmet- +ric spaces [23] and asymptotic torsions for compact locally symmetric orbifolds [22]. +There are analogous results under holomorphic settings, Bismut-Vasserot [10] studied +the asymptotics of holomorphic torsion associated with increasing powers of a positive +line bundle over a K¨ahler manifold and they gave a formula for the leading term. This +asymptotic expansion played an important role in the arithmetic Hilbert-Samuel the- +orem of Gillet–Soul´e [21]. They also extended their results in [11], where the powers +of the positive line bundle are replaced by the symmetric powers of a Griffiths positive +holomorphic vector bundle. In [18], Finski generalized [10] to obtain a full asymptotic +expansion and gave a formula for the second term of the expansion. Puchol [36] studied +asymptotics of holomorphic torsion forms, which extended [10, 11] to family versions. +Savale [40, 41] obtained the asymptotics of the eta invariants using semiclassical analysis. +0.4. Main techniques. Now we briefly describe the techniques that we will use in the +proof of Theorems 0.1-0.3 as well as the main points in this paper. +0.4.1. Toeplitz operators. The theory of Toeplitz operators is recalled (see § 2). Espe- +cially, one of the key results is the growth of the exponential of a Toeplitz operator (see +Theorem 2.7), which is used to get uniform estimations for operators. +0.4.2. The spectral gap. Under the nondegeneracy condition (see § 3.5), Bismut-Ma- +Zhang [9, § 9.10] showed that the Hodge-de Rham Laplacian (see (1.13)) has a spectral +gap: DFp,2 +X +⩾ Cp2 for p ∈ N∗ large, this condition is vitally important in analysis, for +instance, it ensures that W L,ξ +i +in (0.4) is well defined (see Theorem 5.1). +0.4.3. Analytic localization method. The proof of Theorems 0.1 and 0.3 relies on a re- +finement of Bismut-Ma-Zhang’s argument [9]. The strategy to get a full expansion is to +study the local asymptotics of certain heat kernels. We will use the analytic localiza- +tion method of Bismut-Lebeau [6]. Bismut’s family local index theory in the context of +families [4, 5] also plays an important role, in particular, we apply the rescaling for two +Clifford variables as in [12, Chapter 4d)]. +Following [6, Chapter 11] and [9, § 9.12], in § 4.2. we prove that the limit of the trace +of certain heat kernels used in the definition of analytic torsion forms (see (4.4)) can be +localized, by using the finite propagation speed of the wave operator (see [25, Appendix +D]). And as analysis carried through in § 4.3-§ 4.9, we use the techniques in [25, Chapter +4] to get estimations for high order expansions of resolvents and heat kernels. +Compared with [9] and [36], to get the leading term, one only needs to get a uniform +bound and the pointwise convergence for the heat traces, then apply the dominated con- +vergence, while to give the full expansion as in (0.4), we need to give precise estimations +for each term in the local expansion of the heat kernel as well as the remainder. To do + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +6 +so, the most important trick is to preserve the spectral gap property for the resulting +Laplacian in the localization and rescaling procedures. Hence we should carefully choose +parameters at each step of the analysis, especially for the weighted norm (see (4.78)) +and the corresponding elliptic regularity (see Theorem 4.18). +0.5. The organization of the paper. This paper is organized as follows. In § 1, we +review the main results of Bismut-Lott [7] of analytic torsion forms. In § 2, we recall some +results on the Toeplitz operator following [25, § 7] and analyze the asymptotic behavior +of the exponential of a Toeplitz operator. In § 3, we recall Bismut-Ma-Zhang’s definition +[9, § 9.7] for a series of flat bundles {Fp}p∈N∗ over M, which is the main geometric object +in the whole paper. In § 4, we give the full expansion of the odd superconnection form +h +� +A′, gΩ•(X,Fp)� +as p → +∞. In § 5, we state and prove the main theorem, which gives +the existence of the full expansion of the analytic torsions and the Γ-torsions associated +with Fp. In § 6, we consider the case where G is a reductive Lie group and give an +explicit formula for the sub-leading term in the asymptotics of torsion obtained in § 5. +In the whole paper, we apply the superconnection formalism of Quillen (see [37] and +[4, § 1]). If E = E+ ⊕ E− is a Z2-vector space, and τ = ±1 defines the Z2-grading, if +A ∈ End(E), we denote by Trs[A] the supertrace: +Trs[A] = Tr[τA]. +(0.12) +For a multi-index α = (α1, · · · , αk) ∈ Nk and a multi-variable Z = (Z1, · · · , Zk), set +|α| = +k +� +i=0 +αi, +Zα = Zα1 +1 · · ·Zαk +k , +∂α +∂Zα = ∂α1 +∂Zα1 · · · ∂αk +∂Zαk . +(0.13) +We also use the Einstein summation convention, if in a term the same index appears +twice, that term is assumed to be summed over all possible values of that index. +Acknowledgment. This work is the main result of our PhD. thesis, which was done at +Universit´e Paris Cit´e. I am deeply grateful to my thesis supervisor Prof. Xiaonan Ma +for his patient guidance and numerous suggestions. This project has received funding +from the European Union’s Horizon 2020 research and innovation programme under the +Marie Sk�lodowska-Curie grant agreement No 754362. +1. The analytic torsion forms +In this section, we will summarize the main results on the analytic torsion forms +following Bismut-Lott [7], which generalize the classical Ray-Singer analytic torsion [38]. +This section is organized as follows. +In § 1.1, we introduce the smooth fibration +π: M → S and define some associated tensors. In § 1.2, given a flat complex vector +bundle (F, ∇F) with a Hermitian metric gF, we define a natural unitary connection +∇F,u and the associated odd characteristic form. In § 1.3, we reinterpret the de Rham +operator dF as a flat superconnection on the bundle Ω•(X, F) over S. In § 1.4, we get +a transgression formula for the odd closed forms h(A′, gΩ(X,F ) +t +) ∈ Ω•(S) and define the +analytic torsion form. In § 1.5, we recall Bismut-Lott’s Lichnerowicz formula associated +with the odd closed form. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +7 +1.1. A smooth fibration. Let π: M → S be a fibration of smooth manifolds with +compact fibre X of dimension m. Let TX ⊂ TM be the tangent bundle to the fibers +X. Let T HM ⊂ TM be a horizontal subbundle with (0.2) and P TX : TM → TX the +projection map. Since T HM ∼= π∗TS, for U ∈ TS, let UH ∈ T HM be the horizontal lift +of U, such that π∗UH = U. Let T H(·, ·) ∈ Λ2(TS) ⊗ TX be the curvature of (π, T HM): +T H(UH, V H) = −P TX[UH, V H], +for U, V ∈ TS. +(1.1) +We also have an identification of bundles through the horizontal lift +π∗Λ (T ∗S) �⊗Λ (T ∗X) ∼= Λ (T ∗M) . +(1.2) +Let gTX and gTS be Riemannian metrics on TX and TS respectively. We equip TM +with the metric gTM = gTX ⊕π∗gTS and the corresponding Levi-Civita connection ∇TM. +Let ∇TX be the connection on TX defined by in [5, Definition 1.6]: +∇TX = P TX∇TMP TX, +(1.3) +and denote its curvature by RTX. Let ∇TS be the Levi-Civita connection of (TS, gTS). +Let ∇TM,⊕ be the connection on TM given by ∇TM,⊕ = π∗∇TS ⊕ ∇TX. Put +S(·) = ∇TM +· +− ∇TM,⊕ +· +∈ Ω1(M, End(TM)). +(1.4) +1.2. A flat bundle and its odd forms. Let (F, ∇F) be a complex flat vector bundle +on M with a Hermitian metric gF. Following [12, Definition 4.1], set +ω +� +∇F, gF� += +� +gF�−1∇FgF ∈ Ω1(M, End(F)), +(1.5) +and it takes values in self-adjoint elements of End(F). +By [12, Definitions 4.2, Proposition 4.3], we have the following unitary conneciton +∇F,u on F with its curvature: +∇F,u = ∇F + 1 +2ω +� +∇F, gF� +, +RF,u = −1 +4ω +� +∇F, gF�2. +(1.6) +For x ∈ R, set +h (x) = x exp +� +x2� +, +(1.7) +Set ϕ the endomorphism of Λ•(T ∗M) ⊗R C sends α ∈ Λk(T ∗M) ⊗R C to (2πi)−k/2α. Put +h +� +∇F, gF� += (2πi)1/2 ϕTr +� +h +� +ω +� +∇F, gF� +/2 +�� +. +(1.8) +Theorem 1.1 ([7, Theorems 1.8, 1.9 and 1.11]). The odd form h +� +∇F, gF� +is real and +closed and its cohomology class does not depend on gF. +1.3. A superconnection and odd forms. We make the same assumption as in § 1.1. +Let Ω•(X, F) be a formal smooth infinite-dimensional Z-graded vector bundle over S +whose fibre over b ∈ S is C∞(Xb, Λ•(T ∗X) ⊗ F|b). By (1.2) we have an identification of +Z-graded vector spaces +Ω• (M, F) ∼= Ω•(S, Ω•(X, F)). +(1.9) +The exterior differential operator dF acting on Ω• (M, F), then by (1.9), it can be +considered as a flat superconnection on Ω• (X, F), we also denote it by A′ = dF. +If U ∈ TS, the Lie derivative operator LUH acts naturally on Ω•(X, F). Let ∇Ω•(X,F ) +be the connection on Ω• (X, F) such that if U ∈ TM and s ∈ Ω• (X, F), +∇Ω•(X,F ) +UH +s = LUHs. +(1.10) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +8 +We set iT H = 1 +2f α∧f β∧iT H(fα,fβ) (see (1.1)), which acts naturally on Λ•(T ∗S)�⊗Λ•(T ∗X)⊗ +F. Let dF +X be the fibrewise exterior differential operator on Ω•(X, F). +Proposition 1.2 ([7, Proposition 3.4]). The superconnection A′(= dF) satisfies +A′ = dF +X + ∇Ω•(X,F ) + iT H. +(1.11) +Let gΩ•(X,F ) be L2-metric on Ω•(X, F) induced by (gTX, gF). Then let dF,∗ +X +be the +fibrewise formal adjoint operator of dF +X. +Let ∇Ω•(X,F ),∗ be the adjoint connection of +∇Ω•(X,F ). By identifying TX and T ∗X through gTX, we can consider T H as a section of +Λ2(T ∗S)�⊗T ∗X. Then T H∧ acts naturally on Λ•(T ∗S)�⊗Λ•(T ∗X) ⊗ F. +Let A′′ be the adjoint of A′ with respect to the metric gΩ•(X,F ) in the sense of [7, § 1.4], +which is also a flat superconnection. By [7, Proposition 3.7], we have +A′′ = dF,∗ +X + ∇Ω•(X,F ),∗ − T H ∧ . +(1.12) +Set +A = 1 +2 (A′ + A′′) , +B = 1 +2 (A′′ − A′) , +DF +X = dF +X + dF,∗ +X , +(1.13) +then A is a superconnection on Ω•(X, F), B is a section of (π∗Λ• (T ∗S) �⊗End (Ω•(X, F)))odd +and DF +X is the fiberwise Dirac operator. +Let ϕ be the action on Λ(T ∗S) ⊗ C as in (1.8). +Definition 1.3. For B given in (1.13), we set the following odd form similar to (1.8): +h +� +A′, gΩ•(X,F )� += (2πi)1/2 ϕTrs [h(B)] ∈ Ω•(S). +(1.14) +1.4. The analytic torsion forms. Following [9, § 5.4], for t > 0, we set a rescaling +metric gTX +t += gTX/t and the associated metric gΩ•(X,F ) +t +on Ω•(X, F). +For a ∈ R, let ψa be the automorphism of Λ• (T ∗S) such that, if α ∈ Λk(T ∗S) then +ψaα = akα. For t > 0, put +Ct = ψ−1 +√ +t +√ +tAψ√ +t, +Dt = ψ−1 +√ +t +√ +tBψ√ +t. +(1.15) +For t > 0, let h +� +A′, gΩ•(X,F ) +t +� +be the form as in (1.14) associated with gΩ•(X,F ) +t +. By [9, +(5.28)], we have +h +� +A′, gΩ•(X,F ) +t +� += (2πi)1/2 ϕTrs [h(Dt)] . +(1.16) +By (1.13) and (1.15), we have C2 +t = −D2 +t , together with (1.7) and (1.16), we get +h +� +A′, gΩ•(X,F ) +t +� += (2πi)1/2 ϕTrs +� +Dt exp +� +− C2 +t +�� +. +(1.17) +Let H• (X, F) = ⊕dim X +i=0 +Hi(X, F) be the Z-graded vector bundle over S whose fibre +over b ∈ S is the cohomology H•(Xb, F|Xb) of the sheaf of locally flat sections of F|Xb +on Xb. For the fiberwise Dirac operator DF +X in (1.13), by the Hodge Theorem, we have +Hi (X, F) ∼= ker DF,2 +X +�� +Ωi(X,F ) for any i ∈ N. +(1.18) +By [7, § 2a], H• (X, F) is canonically equipped with a flat connection ∇H•(X,F ) and a +Hermitian metric gH•(X,F ). Let e(TX, ∇TX) ∈ Ω•(M) be the Euler form of TX (see [9, +(1.43)]) and we set h(∇H•(X,F ), gH•(X,F )) = � +j(−1)jh(∇Hj(X,F ), gHj(X,F )). + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +9 +Theorem 1.4 ([7, Theorem 3.16]). The forms h +� +A′, gΩ•(X,F ) +t +� +are real, odd and closed, +and their cohomology class does not depend on t > 0. Moreover, as t → 0, we have +h +� +A′, gΩ•(X,F ) +t +� += +�� +X e(TX, ∇TX)h +� +∇F, gF� ++ O( +√ +t), as t → 0, +h +� +∇H•(X,F ), gH•(X,F )� ++ O(1/ +√ +t), as t → +∞. +(1.19) +Now we review the transgression procedure following [7, § 3.9]. We enlarge M, S to +� +M = M × R∗ ++, �S = S × R∗ ++. Set �π: � +M → �S by �π(x, s) = (π(x), s). Let ρ0 : � +M → M and +ρ1: � +M → R∗ ++ be the projection maps. Let � +X be the fiber of �π, then we have T �X = ρ∗TX. +and we equip T �X with the metric ρ∗gTX/s over M × {s}. We have +∇T � +X = ρ∗∇TX + ds +� ∂ +∂s − 1 +2s +� +, +RT � +X = ρ∗ +0RTX. +(1.20) +On �S, we have the odd form h +� �A′, �gΩ(X,F ) +t +� +analogous to h +� +A′, gΩ(X,F ) +t +� +on S. +Definition 1.5. Let h∧� +A′, gΩ(X,F ) +t +� +be the even form on S satisfies (see [7, (3.114)]) +h +� �A′, �gΩ(X,F ) +t +��� +s=1 = h +� +A′, gΩ(X,F ) +t +� ++ ds ∧ h∧� +A′, gΩ(X,F ) +t +� +. +(1.21) +Set +χ′(X, F) = +m +� +j=0 +(−1)jj dim Hj(X, F), +χ(X, F) = +m +� +j=0 +(−1)j dim Hj(X, F). +(1.22) +By Theorem 1.4 and (1.21) we have: +Theorem 1.6 ([7, Theorems 3.20, 3.21]). The form h∧� +A′, gΩ•(X,F ) +t +� +is even, and +∂ +∂th +� +A′, gΩ•(X,F ) +t +� += 1 +t h∧� +A′, gΩ•(X,F ) +t +� +, +h∧� +A′, gΩ•(X,F ) +t +� += +� +O( +√ +t), as t → 0, +�1 +2χ′(X, F) − m +2 χ(X, F) +� +h′(0) + O(1/ +√ +t), as t → +∞. +(1.23) +Now we give the definition of the analytic torsion form T +� +T HM, gTX, ∇F, gF� +. +Definition 1.7 (see [7, Definition 3.22]). Set +T +� +T HM, gTX, ∇F, gF� += − +� +∞ +0 +� +h∧� +A′, gΩ•(X,F ) +t +� ++ +�m +4 χ(X, F) − 1 +2χ′(X, F) +�� +h′(0) − h′(i +√ +t/2) +��dt +t . +(1.24) +By Theorems 1.4 and 1.6, the above integral is well defined. +Theorem 1.8 ([7, Theorem 3.23]). The form T +� +T HM, gTX, ∇F, gF� +is even and real, +and it verifies the following transgression formula: +dT +� +T HM, gTX, ∇F, gF� += +� +X +e(TX, ∇TX)h +� +∇F, gF� +− h +� +∇H•(X,F ), gH•(X,F )� +. +(1.25) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +10 +1.5. Lichnerowicz type formulas. Let z be an odd Grassmannian variable anticom- +mutes with odd variables we used before. If a ∈ R[z]�⊗π∗Λ• (T ∗S) �⊗Λ• (T ∗X), we set +[a]z = c, +for a = b + zc where b, c ∈ π∗Λ• (T ∗S) �⊗Λ• (T ∗X) . +(1.26) +By (1.17), we get +h +� +A′, gΩ•(X,F ) +t +� += (2πi)1/2 ϕTrs +� +exp +� +− +� +C2 +t − zDt +� ��z +. +(1.27) +From now on, we will always use Latin indices i, j, · · · for vertical variables, and Greek +indices α, β, · · · for horizontal variables. Let e1, · · · , em be a local orthonormal frame of +TX, and let f1, · · · , fr be a basis of TS. The corresponding dual bases are denoted with +upper indices. We set the following Clifford actions on Λ(T ∗X): +c(ej) = ej ∧ −iej, +�c(ej) = ej ∧ +iej, +(1.28) +then for 1 ⩽ i, j ⩽ m we have +[c(ei), c(ej)] = −2δij, +[�c(ei), �c(ej)] = 2δij, +[c(ei), �c(ej)] = 0. +(1.29) +By (1.29), the actions in (1.28) extend to an isomorphism of algebras +c�⊗�c: c(TX)�⊗�c(TX) → End(Λ(T ∗X)). +(1.30) +Put +RF = −1 +4 +� +RTXei, ej +� +�c (ei) �c (ej) − 1 +4ω2� +∇F, gF� +. +(1.31) +Let rX be the scalar curvature of the fibre +� +X, gTX� +. +Definition 1.9. Let ΛF be a section of R[z]�⊗Λ•(T ∗S)�⊗End(Λ(T ∗X)) ⊗ End(F) by +ΛF =1 +4rX + 1 +2c(ei)c(ej)R (ei, ej) + 1 +2fαfβR +� +f H +α , f H +β +� ++ c(ei)fαR +� +ei, f H +α +� ++ 1 +4ω +� +∇F, gF� +(ei)2 + 1 +8�c(ei)�c(ej)ω +� +∇F, gF�2 (ei, ej) +− 1 +2f H +α �c(ei)∇TX⊗Fp,u +fH +α +ω +� +∇F, gF� +(ei) − 1 +2c(ei)�c(ej)∇TX⊗Fp,u +ei +ω +� +∇F, gF� +(ej) +− 1 +2zc(ei)ω +� +∇F, gF� +(ei) − 1 +2zfαω +� +∇F, gF�� +f H +α +� +. +(1.32) +Definition 1.10. Let 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) be the fibrewise connection along X by: +0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) =∇π∗Λ•(T ∗S)�⊗Λ•(T ∗X) + 1 +2 +� +Sei, f H +α +� +c (ei) f α ++ 1 +4 +� +Sf H +α , f H +β +� +f αf β − z +2ei ∧ �c(ei). +(1.33) +By [5, Theorem 4.14], the curvature of 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) is given by +0RR[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)(ei, ej) = 1 +2 +� +RTX(ek, eℓ)ei, ej +�� +c(ek)c(eℓ) − �c(ek)�c(eℓ) +� ++ +� +RTX(fα, eℓ)ei, ej +� +f αc(eℓ) + 1 +2 +� +RTX(fα, fβ)ei, ej +� +f α ∧ f β. +(1.34) +On R[z]�⊗π∗Λ• (T ∗S) �⊗Λ• (T ∗X) ⊗ F, let 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗F,u be the fibrewise +connection induced by 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) and ∇F,u. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +11 +Put LF +t = C2 +t − zDt, which appears in (1.27) and we set LF = LF +4 . Let N1 be the +number operator of the exterior algebra R[z]�⊗π∗Λ•(T ∗S) that acts by multiplication by +the total degree of each term. By (1.15), we have +LF +4t = θ−1 +√ +ttLFθ√ +t, +where θ√ +t = +√ +t +N1. +(1.35) +Theorem 1.11 ([7, Theorem 3.11]). The following Lichnerowicz type formula holds: +LF = − +�0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗F,u +ei +�2 + 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗F,u +∇T X +ei ei ++ ΛF. +(1.36) +2. Toeplitz operators +In this section, we describe the formalism of the Toeplitz operator introduced by +Berezin [2] and Boutet de Monvel-Guillemin [15], and developed by Bordemann-Meinrenken- +Schlichenmaier [13], Schlichenmaier [42] and Ma-Marinescu [25], [26], [27]. +This section is organized as follows. In § 2.1, we introduce the positive line bundle L +on a compact K¨ahler manifold N and the Berezin-Toeplitz quantization, then we give +some uniform estimations that will be used later. In § 2.2, we investigate the exponential +of a Toeplitz operator and give an estimation for each term in the expansion and the +remainder. +2.1. The algebras of Toeplitz operators. Let N be a compact complex manifold of +dimCN = n with the complex structure J. Let gTRN be a J-invariant Riemannian metric +on TRN. Denote the induced Riemannian volume form by dvN. +Let (L, gL) (resp. (ξ, hξ)) be a holomorphic line bundle (resp. holomorphic vector +bundle) on N. Let ∇L (resp. ∇ξ) be the associated Chern connections on L (resp. ξ) +with curvature RL (resp. Rξ). We assume that (L, gL, ∇L) is a positive line bundle. Then +c1(L, gL) = +√−1 +2π RL defines a K¨ahler form on N. We note that gTRN is not necessarily +given by the K¨ahler metric induced by c1(L, gL). +For p ∈ N∗, put Lp = L⊗p, the p-th tensor power of L. We have the L2-inner product +on C ∞(N, Lp ⊗ξ) induced by (gTRN, gL, hξ). We denote the corresponding norm by ∥·∥L2 +and by L2(N, Lp ⊗ ξ) the completion of C ∞(N, Lp ⊗ ξ) with respect to the L2-norm. +We consider the space H(0,0)(N, Lp ⊗ ξ) of holomorphic sections of Lp ⊗ ξ, and let +Pp : L2(N, Lp ⊗ ξ) −→ H(0,0)(N, Lp ⊗ ξ) +(2.1) +be the orthogonal projection with respect to the L2-product. +Definition 2.1 ([25, (7.2.6)]). The Berezin-Toeplitz quantization of a smooth section +H ∈ C ∞(N, End(ξ)) is a sequence of linear operators {TH,p}p∈N∗ given by +TH,p : L2(N, Lp ⊗ ξ) → L2(N, Lp ⊗ ξ), +TH,p = PpHPp. +(2.2) +The operator TH,p has a smooth kernel TH,p(x, x′) with respect to dvN. On the diago- +nal, TH,p(x, x) ∈ End(Lp ⊗ ξ) = End(ξ). For ℓ ∈ N, let |·|C ℓ(N) be the ℓ-th order smooth +norm on C ∞(N, End(ξ)) induced by (∇TRN, ∇E, gTRN, hξ). By the proof of [25, Lemma +7.2.4], we have the following expansion with a uniform estimation for the remainder. +Theorem 2.2. There is a series of differential operators {Ai}+∞ +i=0 of order no more +than 2i such that for any k, ℓ ∈ N, there exists C > 0 such that for any p ∈ N∗ and + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +12 +H ∈ C ∞(N, End(ξ)), +���p−nTH,p(x, x) − +k +� +i=0 +p−iAi(H)(x) +��� +C ℓ(N) ⩽ C |H|C 2k+2(N) p−k−1. +(2.3) +And the operators {Ai}+∞ +i=0 vary smoothly with respect to (J, gTRN, hL, hξ). In particular, +A0(H)dvN = c1(L, gL)n +n! +. +(2.4) +By the Kodaira vanishing Theorem [25, Theorem 1.5.4] and the Riemann-Roch-Hirzebruch +Theorem [25, Theorem 1.4.6], for p ∈ N∗ large enough, dim H(0,0)(N, Lp ⊗ ξ) is a poly- +nomial of p ∈ N∗ with leading term +dim H(0,0)(N, Lp ⊗ ξ) = dimC ξ · +� +N +c1(L, gL)n +n! +pn + O +� +pn−1� +, +(2.5) +and clearly (2.4) is a local version of (2.5). +If A ∈ End (L2 (N, Lp ⊗ ξ)), let ∥A∥ be its operator norm. If A is of trace class, we +denote its trace norm by ∥A∥1. If A = PpAPp, then A is of trace class, by (2.5), there is +C > 0 such that for any p ∈ N∗, +∥A∥1 ⩽ ∥A∥ dimC H(0,0)(N, Lp ⊗ ξ) ⩽ C ∥A∥ pn. +(2.6) +Following Ma-Marinescu [25, Definition 7.2.1], we now define the Toeplitz operator. +Definition 2.3. A Toeplitz operator is a family of operators +� +Tp | Tp ∈ End(L2 (N, Lp ⊗ ξ)) +� +p∈N∗ +bounded with respect to the L2-norm such that Tp = PpTpPp, and that there exists +{Hi | Hi ∈ C ∞(N, End(ξ))}i∈N such that for any k ∈ N, there is ck > 0 such that +���Tp − +k +� +i=0 +p−iTHi,p +��� ⩽ ckp−k−1. +(2.7) +As in [25, (7.2.4), (7.2.5)], we use the notation of formal expansion to denote (2.7) as +Tp = ++∞ +� +i=0 +p−iTHi,p + O +� +p−∞� +, +(2.8) +and we replace �+∞ +i=0 with �k +i=0 if we only refer to the first k terms. +Now we introduce the asymptotic trace symbol for ease of notation. For the Toeplitz +operator {Tp}p∈N∗ in (2.7), for k ∈ N, if we set +Tr[k][Tp] = +� +i+j=k +� +N +Trξ[Ai(Hj)]dvN, +(2.9) +then by (2.3), (2.6) and (2.7), there is Cck,{|Hi|C2i(N)}k+1 +i=1 > 0 which depends on ck and +{|Hi|C 2i(N)}k+1 +i=1 such that +���p−nTrH(0,0)(N,Lp⊗ξ)[Tp] − +k +� +i=0 +p−iTr[i][Tp] +��� ⩽ Cck,{|Hi|C2i(N)}k+1 +i=1 p−k−1. +(2.10) +By the proof of [25, Theorem 7.4.1] and [27, Theorem 0.3], we have the following +product formula of Toeplitz operators with estimation for the remainder. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +13 +Theorem 2.4. The set of Toeplitz operators is an algebra. In particular, for any k ∈ N, +there is C > 0 such that for any H, H′ ∈ C ∞(N, End(ξ)) and p ∈ N∗, we have +��TH,pTH′,p − +k +� +j=0 +p−iTCj(H,H′),p +�� ⩽ Cp−k−1 |H|C 2k+2(N) · |H′|C 2k+2(N) . +(2.11) +where Cj(·, ·) is a smooth bidifferential operator of total degree no more than 2j, and +C0(H, H′) = HH′, +C1(H, H′) − C1(H′, H) = i{H, H′} if H, H′ ∈ C ∞(N, C), +(2.12) +where {H, H′} is the Poisson bracket of 2πc1(L, gL). +2.2. The exponential of Toeplitz operators. To discuss the exponential of Toeplitz +operators, we first study the inverse of a Toeplitz operator. We follow Ma-Zhang [29, +Lemma 4.1] where they have proved that if the principal symbol H0 of a Tp as in (2.7) +is invertible, then T −1 +p +is also a Toeplitz operator, here we give a uniform estimation for +the remainder term in the expansion. +Lemma 2.5. Let Tp be a Toeplitz operator as in (2.7). If H0(x) is invertible for all +x ∈ N, then Tp is invertible for p large, and T −1 +p +is also a Toeplitz operator. Moreover, +if we write T −1 +p += �∞ +i=0 p−iTGi,p + O(p−∞) as in (2.8), then we have G0 = H−1 +0 . +Proof. We recall a basic fact: for invertible operators A and B, we have +A−1 − B = B((1 − (1 − AB))−1 − 1) = B(1 − AB)(1 − (1 − AB))−1. +(2.13) +For k ∈ N and Ci(·, ·) given in Theorem 2.4, we set +G0 = H−1 +0 , +Gk+1 = −H−1 +0 +� +i+j⩽k+1 +(i,j)̸=(0,0) +Ci +� +Hj, Gk+1−i−j +� +. +(2.14) +By (2.7), (2.11) and (2.14), we can prove inductively that +����Tp +� +k +� +i=0 +p−iTGi,p +� +− 1 +���� ⩽ Cp−k−1, +���� +� +k +� +i=0 +p−iTGi,p +� +Tp − 1 +���� ⩽ Cp−k−1. +(2.15) +By (2.15), Tp and �k +i=0 p−iTGi,p have both left and right inverse for p ∈ N∗ large, hence +they are invertible. We take A = Tp, B = �k +i=0 p−iTGi,p in (2.13), by (2.15) and the +obvious inequality ∥(1 − (1 − AB))−1∥ ⩽ (1 − ∥(1 − AB)∥)−1, we get +���T −1 +p +− +k +� +i=0 +p−iTGi,p +��� ⩽ Cp−k−1, +(2.16) +which completes the proof. +□ +Remark 2.6. By the above proof, we can indeed take C in (2.15) and (2.16) as the sum +of terms of the following form: +C|H−1 +0 |α +C 0(N) · +k +� +i=0 +cβi +i |Hi|γi +C 2(k+1)(N) +(2.17) +where C > 0, α ∈ N and β, γ ∈ Nk+1. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +14 +Theorem 2.7. If Tp is a Toeplitz operator with expansion (2.7), exp(−tTp) is also a +Toeplitz operator for any t ⩾ 0 with the following expansion in the sense of (2.8): +exp(−tTp) = ++∞ +� +i=0 +p−iTJi(t),p + O(p−∞), +J0(t) = e−tH0. +(2.18) +Put h = infx∈N Re +� +SpecH0(x) +� +, then for any ε > 0, i, k, ℓ ∈ N, there is C > 0 such that +for any p ∈ N∗, +|Ji(t)|C k(N) ⩽ C exp(−(h − ε)t), +��� exp(−tTp) − +k +� +i=0 +p−iTJi(t),p +��� ⩽ C exp(−(h − ε)t)p−k−1. +(2.19) +Proof. By the product formula (2.11) and (2.12), for k ∈ N, we have T k +p = THk +0,p + +O(p−1). If we ignore the divergence of infinite sums of remainders, we get exp(Tp) = +�+∞ +k=0 +1 +k!THk +0,p + O(p−1) = Texp(H0),p + O(p−1), which is just the first order of (2.18). +Now we give rigorous proof. By Lemma 2.5, if λ /∈ Spec(H0), then λ − Tp is invertible +for p ∈ N∗ large and we have the following expansion in the sense of (2.8): +(λ − Tp)−1 = ++∞ +� +i=0 +p−iTIi(λ),p +(2.20) +where Ii(λ) ∈ C ∞(N, End(ξ)). +As in Theorem 2.4, Ci(·, ·) is a bilinear differential +operator for each i ∈ N, then by (2.14), each Ii(λ) for i ∈ N is the sum of terms of the +following form: +(λ − H0)−n0 f1(λ) (λ − H0)−n1 · · · fj(λ) (λ − H0)−nj +(2.21) +where nj ∈ N and fj(λ) ∈ C ∞(N, End(ξ))[λ], the space of polynomials of λ with coeffi- +cients in C ∞(N, End(ξ)). In particular, the leading term is +I0(λ) = (λ − H0)−1 . +(2.22) +For ε > 0, we choose a bounded curve Γε surrounds Spec(H0) counterclockwise on the +complex plane such that infλ∈Γε Re(λ) ⩾ h−ǫ. By the Cauchy integral formula, we have +exp(−tTp) = +1 +2πi +� +Γε +e−tλ (λ − Tp)−1 dλ. +(2.23) +We denote the C in (2.16) associated with Toeplitz operator ( +� +λ − Tp +� +)−1 by Cλ. By +(2.17), Cλ is uniformly bounded for λ ∈ Γ. By (2.20) and (2.23), we have +���� exp(−tTp) − +k +� +i=0 +p−iT +1 +2πi +� +Γε e−tλIi(λ)dλ,p +���� ⩽ p−k−1 1 +2π +� +Γε +e−tRe(λ)Cλdλ, +(2.24) +from which we get (2.18) and the first inequality of (2.19) with +Ji(t) = +1 +2πi +� +Γε +e−tλIi(λ)dλ. +(2.25) +And by (2.22), we get J0(t) in (2.18). From (2.21) and (2.25), we obtain the second +inequality of (2.19). +□ + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +15 +Remark 2.8. In this study, estimations for each term in (2.19) and the remainder term +are essential. By (2.9) and (2.19), for any t ⩾ 0, we have +���Tr[k] +� +exp(−tTp) +���� ⩽ Ce−(h−ǫ)t. +(2.26) +Moreover, by (2.10), (2.18) and (2.19), we get +���p−nTrH(0,0)(N,Lp⊗ξ)[exp(−tTp)] − +k +� +i=0 +p−iTr[i][exp(−tTp)] +��� ⩽ Cp−k−1e−(h−ǫ)t. +(2.27) +Equations (2.26) and (2.27) ensure the exponential decay of each term in the expansion +and the remainder as t → +∞, and these play important roles in § 4 and § 5. +3. The geometry of bundles {Fp}p∈N∗ +We make the same assumption as in §§ 1.1 and 2.1. The purpose of this section is to +review some properties of bundles {Fp}p∈N∗ over M following [9, § 9]. +This section is organized as follows. In § 3.1, we present the geometry of the fibration +N = PG ×G N → M and the line bundle L → N . In § 3.2, we introduce the bundles +{Fp}p∈N∗ over M and compute the curvature of the unitary connection ∇Fp,u. In § 3.3, for +a smooth family of Toeplitz operators {Tp}p∈N∗, we analyze the asymptotics of TrFp[Tp] +when p → +∞. In § 3.4, we obtain the asymptotics of the odd form h(∇Fp, gFp) for +p → +∞. In § 3.5, we recall the nondegenarated condition of L. +3.1. Geometric settings. Let N be a compact complex manifold of complex dimension +n. Let L and ξ be holomorphic bundles on N with dimC(L) = 1. Let G be a Lie group +acting holomorphically on N, and this action lifts to holomorphic actions on L and ξ. +Let p: PG → M be a principal flat G-bundle. Set +N = PG ×G N. +(3.1) +We denote by q the projection q: N → M with fibre N. We still denote the bundle +PG ×G L (resp. PG ×G ξ) over N by L (resp. ξ). +Let T H +0 N ⊂ TN be the horizontal bundle determined by the flat connection of PG. +Put TRN = ker q∗, the real relative tangent bundle, and let TN be the holomorphic +relative tangent bundle. Let JN be the complex structure on TRN. We clearly have +T H +0 N ∼= q∗TM, then for U ∈ TM, we denote by UH +0 ∈ T H +0 N its horizontal lift. +Let gL (resp. gξ) be a Hermitian metric on L (resp. ξ) over N . Let ∇L (resp. ∇ξ) be +the fibrewise Chern on L (resp. ξ). Using the flat connection on PG, we can extend ∇L +and ∇ξ to connections on L and ξ respectively: for U ∈ TM, put +∇L +UH +0 (or ∇ξ +UH +0 ) = LUH +0 +(3.2) +These connections are in general non-unitary, and we set +ω +� +L, gL� +(U) = +� +gL�−1LUH +0 gL, +ω +� +ξ, gξ� +(U) = +� +gξ�−1LUH +0 gξ. +(3.3) +In what follows, we assume that the first Chern form +c1(L, gL) = +√−1 +2π +� +∇L�2 ∈ Ω2(N ) +(3.4) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +16 +is fibrewisely positive. Let ∂N denote the fibrewise Dolbeault operators. We have the +following identities (see [9, (9.7)]): for U, V ∈ TM and Y ∈ TRN, +c1(L, gL)(UH +0 , V H +0 ) = 0, +c1(L, gL)(UH +0 , Y ) = +√−1 +2π ∂N +Y ω +� +L, gL� +(U). +(3.5) +Let gTRN be a smooth Hermitian metric on TRN, and let dvN be the corresponding +fibrewise volume form. Then we extend dvN to be a form on N that vanishes along the +horizontal direction, and still denote it by dvN ∈ Ω•(N ). Recall that, if U ∈ TM, the +fibrewise Lie derivative operator LUH +0 acts naturally on smooth sections of Λ•(T ∗ +RN ). We +define divN(U) by +LUH +0 dvN = divN(U)dvN. +(3.6) +By (3.5), when gTRN is just the K¨ahler metric gTRN(·, ·) = c1(L, gL)(·, JN·), then for +any U ∈ TM, we can prove the following formula: +divN(U) = 1 +4π∆Nω +� +L, gL� +(U), +(3.7) +where ∆N is the fibrewise (nonnegative) Laplacian operator with respect to the K¨ahler +metric which acts on C ∞(N ). +3.2. The curvature of ∇Fp,u. Recall the definition of Fp in (0.3): +Fp = +� +PG ×G H(0,0)(N, Lp ⊗ ξ) +� +⊗ F, +(3.8) +where the flat bundle (F, ∇F) with metric gF plays the role of a shifting. Let F(F) +be the frame bundle of F, which is a GL(dimC(F))-principal bundle over M. +Since +H(0,0)(N, Lp⊗ξ⊗CdimC(F )) ∼= H(0,0)(N, Lp⊗ξ)⊗CdimC(F ), we could always assume in what +follows that F = C trivial, or we may replace (G, N, L, ξ) by (G×GL(dimC(F)), N, L, ξ⊗ +CdimC(F )). +Put Fp = PG ×G C ∞(N, Lp ⊗ ξ), which is an infinite dimensional bundle on M and +Fp is a subbundle of Fp. The connection ∇Lp⊗ξ naturally induces a flat connection on +∇Fp: if s is a smooth section of Fp and U ∈ TM, set +∇Fp +U s = ∇Lp⊗ξ +UH +0 +s. +(3.9) +This connection preserves Fp, and it induces a flat connection ∇Fp on Fp. We equip Fp +with the L2-metric gFp induced by (gLp, hξ, dvN), which gives a metric gFp on Fp. Let +Pp denote the fibrewise orthogonal projection Fp → Fp. +Let ∇N be a connection on the infinite dimensional bundle C ∞(N ) → M given as +follows: if U ∈ TM and H ∈ C ∞(N ), put ∇N +U = UH +0 (H). Following [9, (9.34)], we set +ϑL = −1 +2ω +� +L, gL� +, +ϑξ = −1 +2ω +� +ξ, gξ� +. +(3.10) +We denote by � +TX a copy of TX. We denote by �ω +� +∇F, gF� +, �ϑL, � +divN the restrictions +of ω +� +∇F, gF� +, ϑL, divN to � +TX. Let {ei}m +i=1 and {�ei}m +i=1 be a local orthonormal frame of +TX and � +TX respectively. We denote the anticommutator by [·, ·]+. +Theorem 3.1 ([9, Theorem 9.27]). We have the following identity: +1 +2pω +� +∇Fp, gFp� += T−ϑL−ϑξ/p+divN/2p,p. +(3.11) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +17 +If H ∈ C ∞(N, End(ξ)) is a smooth section, we have +∇Fp,uTH,p = T∇N H,p + p[TϑL+ϑξ/p−divN/2p,p, TH,p]+ − 2pT(ϑL+ϑξ/p−divN/2p)H,p. +(3.12) +Moreover, combined Theorem 2.4 with (3.11) and (3.12), we see that +1 +4pω(∇Fp, gFp)2, +1 +4p2 �ω(∇Fp, gFp)(ei)2 and ∇Fp,uTH,p are Toeplitz operators. +Remark 3.2. For a general fibration, Ma-Zhang [30, Theorems 1.18, 1.19], [28] showed +that the curvature operator on Fp is a Toeplitz operator. +3.3. The smooth bundle of Toeplitz operators. Analogous to Definition 2.1, we +call {Tp | Tp ∈ C ∞� +M, End(Fp) +� +}p∈N∗ a smooth section of Toeplitz operators if there +exists {Hi | Hi ∈ C ∞(N , End(ξ))}i∈N such that for any k, ℓ ∈ N, there is C > 0 satisfies +���Tp − +k +� +i=0 +p−iTHi,p +��� +C ℓ(M,End(Fp)) ⩽ Cp−k−1, +(3.13) +where the norm |·|C ℓ(M,End(Fp)) is induced by (∇Fp, gFp). +Remark 3.3. As all the expansions in § 2 are smoothly dependent on geometric data +(J, gTRN, hL, hE), all results in § 2 have a similar version for Fp. In particular, by (2.11), +all the operators in Theorem 3.1 are smooth sections of Toeplitz operators as well as their +derivatives. Also, for {Tp}p∈N∗ in (3.13), TrFp[Tp] and Tr[k][Tp] are smooth functions on +M, and we can replace the absolute values |·| on the left hand side of (2.10), (2.26) and +(2.27) by a smooth norm |·|C ℓ(M) +3.4. The asymptotics of h(∇Fp, gFp). Recall the odd form h(∇Fp, gFp) of Fp as in (1.8). +Proposition 3.4. As p → +∞, we have smooth odd forms γi ∈ Ωodd(M), i ∈ N such +that for any k, ℓ ∈ N, there is C > 0 such that +���p−n 1 +√pψ1/√ph +� +∇Fp, gFp� +− +k +� +i=0 +γip−i��� +C ℓ(M) ⩽ Cp−k−1. +(3.14) +Proof. By Theorem 3.1, Remark 3.3 and (1.8), if we take +γi = (2πi)1/2ϕTr[k] +� +exp +� 1 +4pω +� +∇Fp, gFp�2 + 1 +2pzω +� +∇Fp, gFp���z +, +(3.15) +then we get the expansion (3.14). +□ +3.5. Spectral gap. Now we recall the non-degeneracy condition [9, Definition 9.13]. +Definition 3.5. We say that �ϑL is nondegenerated if there is a > 0 such that +m +� +i=1 +�ϑL(ei)2 ⩾ a on N , +(3.16) +or equivalently, �ϑL ∈ C ∞(N , q∗� +T ∗X) is nowhere vanishing on N . +Theorem 3.6 ([9, Theorem 4.4]). If �ϑL is nondegenerate, for any ε > 0 and p ∈ N∗ +large, the fibrewise Dirac operator of Fp (see (1.18)) satisfies +DFp,2 +X +⩾ (a − ε)p2. +(3.17) +In particular, for p ∈ N∗ large enough, DFp,2 +X +is invertible, thus H•(X, Fp) = 0. by the +Hodge Theorem (1.18). + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +18 +4. The asymptotics of the odd superconnection forms +The purpose of this section is to obtain the asymptotics of the odd form h +� +A′, gΩ•(X,Fp) +4t/p2 +� +as p → +∞. The main technique we use is the analytic localization method by Bismut- +Lebeau [6], which is further developed by Dai-Liu-Ma [17] and Ma-Marinescu [25, § 4]. +We treat the two cases 1 ⩽ t < +∞ and 0 ⩽ t ⩽ 1 separately in § 4.1-§ 4.7 and § 4.8 +respectively. The main difficulty is to get an exponential decay term when t → +∞ as +in Theorem 4.1, to do so, we devote a large part of this section to carefully handling +the localization procedure to ensure that at each step, the operator we get is always +“non-negative” as p−2DFp,2 +X +in Theorem 3.6 (see § 4.3-§ 4.4). +Now we give more details on the main points of this section. In § 4.1, we state the main +theorem of this section and express the odd superconnection forms in terms of heat ker- +nels. In § 4.2, by the spectral gap property of LFp, we can use the finite propagation speed +of solutions of hyperbolic equations and localize the original question to a problem on +Rm. In § 4.3, we perform the Bismut-Zhang’s rescaling on the operator θ−1 +1/√pp−2LFpθ1/√p, +then we introduce a smooth family of differential operators � +L Fp +v |0⩽v⩽1/p as in (4.66) which +links the rescaled operator and a limit operator, and we study the Taylor expansion of +� +L Fp +v +with respect to the parameter v. In § 4.4, we introduce graded Sobolev norms with +weights ∥·∥µ,k,p and give some basic elliptic estimations of � +L Fp +v , an important step is to +carefully analyze the structure of � +L Fp +v +and choose a suitable weight µ to preserve the +“positivity” of Spec( � +L Fp +v ). In § 4.5, we study the Sobolev estimations of the resolvent +(λ − � +L Fp +v )−1. In § 4.6, we establish the corresponding convergence of the heat kernel. In +§ 4.7, we analyze the asymptotic trace of the limit kernel when p → +∞ using results +in § 2.2 and properties of Gaussian integral. In § 4.8, we deal with the case 0 < t ⩽ 1. +In § 4.9, we prove the main theorem (see Theorem 4.1) of this section. +4.1. Asymptotic expansion of the odd forms. The constants appearing in the sequel +depend on the compact subset of S we are working on. To simplify the statements in +what follows, we always assume that S is compact. The following theorem is the main +result of this section, and the rest of the section is devoted to its proof. +Recall the odd form h +� +A′, gΩ•(X,Fp) +t +� +and θa in in (1.16) and (1.35) respectively. +Theorem 4.1. If �ϑL is nondegenerate as in (3.16), there exist {ci(t) ∈ Ω•(M) | t > 0}i∈N +such that, for any γ > 0, k, ℓ ∈ N, we have C > 0 that for p ∈ N∗ and t > 0, +����p−n 1 +√pψ1/√ph +� +A′, gΩ•(X,Fp) +4t/p2 +� +− +k +� +i=0 +� +X +ci(t)p−i +���� +C ℓ(S) +⩽ Ce−(a−γ)tp−k−1, +(4.1) +where a ⩾ 0 is given in (3.16). Moreover, there is C > 0 such that for t > 0, we have +���� +� +X +ck(t) +���� +C ℓ(S) +⩽ Ce−(a−γ)t. +(4.2) +For p ∈ N∗ and t > 0, set +MFp = θ√pp−2LFpθ1/√p, +MFp +t += θ√p/ +√ +ttp−2LFpθ√ +t/√p. +(4.3) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +19 +Let exp(−tMFp)(x, x′) and exp(−t′MFp +t )(x, x′) be the smooth heat kernel of MFp and +MFp +t +associated with dvX(x′) respectively. By (1.16) and (4.3), we have +1 +√pψ1/√ph +� +A′, gΩ•(X,Fp) +4t/p2 +� += +√ +2πiϕ +� +θ1/√p +� +X +TrΛ(T ∗X)⊗Fp +s +� +exp(−LFp +4t/p2)(x, x) +� +dvX(x) +�z += +√ +2πiϕ +� +θ−1 +√ +t +� +X +TrΛ(T ∗X)⊗Fp +s +� +exp(−tMFp)(x, x) +� +dvX(x) +�z += +√ +2πiϕ +� � +X +TrΛ(T ∗X)⊗Fp +s +� +exp(−MFp +t )(x, x) +� +dvX(x) +�z +. +(4.4) +Notice that to prove Theorem 4.1, we only need to consider the case a = 0, or we may +replace LFp with LFp − ap2 so that we get extra e−at. +4.2. Localization of the problem. For x0 ∈ M, let X be the fibre containing x0, +we mainly work along this fibre. +For ε > 0, let BX (x0, ε) and BTx0X (0, ε) be the +open balls in X and Tx0X with center x0 and 0 and radius ε respectively. Let expX +x0 +be the exponential map of (X, gTX). For ε small, expX +x0 : BTx0X (0, ε) → BX (x0, ε) is +a diffeomorphism, which gives local coordinates by identifying Tx0X with Rm via an +orthonormal basis {ei}m +i=1 of Tx0X: +Z = (Z1, · · · , Zm) ∈ Rm �−→ Ziei ∈ Tx0X. +(4.5) +We will always identify BX (x0, ε) with BTx0X (0, ε) through the isomorphism in (4.5). +For b ∈ S, let inj(Xb) be the injectivity radius of Xb = π−1(b). For any ε > 0 that +0 < ε < minb∈S inj(Xb)/8, +(4.6) +(we have minb∈Sinj(Xb) > 0 since S is compact), by the compactness of X, we can choose +a finite set {xi}i⩾1 ⊂ X such that +� +BX (xi, ε) +� +i⩾1 is an open covering of X. For now, ε +is not fixed, we will assign it a suitable value after Theorem 4.15. +For Z ∈ BTxiX (0, ε), we identify Fp,Z and +� +R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) +� +Z with Fp,xi +and +� +R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) +� +xi by parallel transport with respect to the connections +∇Fp and 0∇R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) along the curve s → sZ for s ∈ [0, 1]. Let Γ0 and ΓFp,u +be the corresponding connection forms of 0∇R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) and ∇Fp on BTx0X (0, ε) +with respect to this trivialization. +Definition 4.2. Let {fxi}i⩾1 be a partition of unity with respect to {BX (xi, ε)}i⩾1. For +k ∈ N, we define the Sobolev norm ∥·∥Hk,p on C ∞(X, R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) ⊗ Fp): +∥s∥2 +Hk,p = +� +i +� +α∈Nm, 0⩽|α|⩽k +∥∂αfxis∥2 +L2 , +(4.7) +and we denote by Hk,p its completion with respect to the norm ∥·∥2 +Hk,p. +Remark 4.3. Note that ∇Fp,u preserves the metric gFp. Hence the two norms ∥·∥H0,p +and ∥·∥L2(X,Fp) are equivalent uniformly for p and we will not distinguish between them. +Similar to [25, Lemma 1.6.2] and [36, Lemma 3.6], we have the following lemma. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +20 +Lemma 4.4. For k ∈ N, there is C > 0 such that for p ∈ N∗ and s ∈ H2k,p, we have +∥s∥H2k,p ⩽ Cp2k +k +� +j=0 +p−2j��LFp,js +�� +L2. +(4.8) +Proof. By (1.36), on BX (xi, ε) we have +LFp = − +� +ei + ΓFp,u(ei) + Γ0(ei) +�2 + +� +∇TX +ei ei + ΓFp,u(∇TX +ei ei) + Γ0(∇TX +ei ei) +� ++ ΛFp. (4.9) +For any Z ∈ BX (xi, ε), we have a classical relation (see [4, Proposition 1.18]) +ΓFp,u +Z +(∂j) = +� 1 +0 +RFp,u +tZ (R, ∂j)dt, +for R = Zi∂i. +(4.10) +By Theorem 3.1, (1.6), (1.32) and (4.10), we see that p−1ΓFp,u +Z +(∂j) and p−2ΛFp are smooth +family of Toeplitz operators. For a smooth family of Toeplitz operators Tp, by (2.12) +and (3.12), dTp = ∇Fp,uTp − [ΓFp,u, Tp] is also a smooth family of Toeplitz operators. +By (4.9) and the above argument, the operator norms of p−1ΓFp,u and p−2ΛFp are +bounded uniformly for p ∈ N∗ as well as their derivatives, then we get (4.8) exactly as +the proof of [25, Lemma 1.6.2]. +□ +Remark 4.5. Observe that (4.8) is still true if we replace LFp with LFp,∗ or DFp,2 +X +since +they have the same structure as in (4.9). +Choose f ∈ C ∞(R) even, nonincreasing when t ⩾ 0 and +f (t) = +� +1 +if |t| ⩽ 1 +2, +0 +if |t| ⩾ 1. +(4.11) +Definition 4.6. For t, h > 0 and a ∈ C, set +Ft,h (a) = +� +R +e +√ +2iva exp +� +−v2/2 +� +f +�√ +tv/h +� dv +√ +2π, +Gt,h (a) = +� +R +e +√ +2iva exp +� +−v2/2 +� � +1 − f +�√ +tv/h +�� dv +√ +2π +, +Ht,h (a) = +� +R +e +√ +2iva exp +� +−v2/2t +� � +1 − f (v/h) +� dv +√ +2πt. +(4.12) +The functions Ft,h(a), Gt,h(a), Ht,h(a) are even holomorphic functions, thus there exist +holomorphic functions �Ft,h (a), �Gt,h (a) and �Ht,h (a) such that +�Ft,h +� +a2� += Ft,h (a) , +�Gt,h +� +a2� += Gt,h (a) , +�Ht,h +� +a2� += Ht,h (a) . +(4.13) +Moreover, the restriction of �Ft and �Gt to R lies in the Schwartz space S (R), and +Ft,h (a) + Gt,h (a) = exp +� +−a2� +, +Ht,h (a) = Gt,h +�√ +ta +� +. +(4.14) +Now we fix a c > 0, let Vc be the following subset of the complex plane: +Vc = +� +a ∈ C: Re (a) ⩾ 1 +4c2Im (a)2 − c2� +. +(4.15) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +21 +Lemma 4.7. For any h > 0 and k ∈ N, there are C, c1, c2 > 0 such that for t > 0, we +have +sup +a∈Vc +��ak �Ht,h (a) +�� ⩽ C exp +� +c1t − c2 +t +� +. +(4.16) +Proof. By proceeding as in [25, Theorem 4.2.5], when |Im (a) | ⩽ c, as ikakeiva = +∂k +∂vk eiva, +one can integrate by part the expression of akHt,h (a) given in (4.12) to obtain that +sup +|Im(a)|⩽c +|akHt,h(a)| ⩽ C exp +� +c1t − c2 +t +� +. +(4.17) +Note that for c > 0, Vc is just the image of {a ∈ C: |Im(a)| ⩽ c} by the map C → +C: a �→ a2. By (4.13) and (4.17), we get (4.16). +□ +For any δ > 0, let Γ be the contour in C defined by {x ± δi | x ⩾ −δ} ∪ {−δ + xi | +−δ ⩽ x ⩽ δ}. We choose a suitable δ to make Γ ⊂ Vc and mina∈Γ,b∈Vc |a − b| = c2/2 (see +(4.15)), and we denote this contour Γ by Γc. +x +y +Γc +Vc +Figure 2. +Lemma 4.8. For k ∈ N, there are C > 0, ℓ ∈ N∗ such that for any p ∈ N∗, λ ∈ Γc, +��� +λ − LFp�−1s +�� +H2k+2,p ⩽ Cpℓ|λ|ℓ∥s∥H2k,p. +(4.18) +Proof. First, following [36, (3.65)] we prove that for some C > 0, i ∈ N∗, +��� +λ − LFp�−1s +�� +L2 ⩽ Cpi|λ|i∥s∥L2. +(4.19) +By Theorem 1.11, we write LFp = DFp,2 +X ++G Fp where G Fp is a 1-order fibrewise differential +operator with positive degree in π∗Λ (T ∗S), and +� +λ − LFp�−1 = +dim S +� +i=0 +�� +λ − DFp,2 +X +�−1G Fp�i� +λ − DFp,2 +X +�−1. +(4.20) +Note the self adjointness of DFp,2 +X +, for any λ ∈ Γc and x ⩾ 0, we have max{ +1 +|λ−x|, +x +|λ−x|} ⩽ +C +|λ| +|λ−x| + 1 ⩽ C|λ|, so there is C > 0 with +��� +λ − DFp,2 +X +�−1s +�� +L2 ⩽ C∥s∥L2, +��DFp,2 +X +� +λ − DFp,2 +X +�−1s +�� +L2 ⩽ C|λ| · ∥s∥L2. +(4.21) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +22 +By Theorem 3.1, Lemma 4.4, Remark 4.5, (1.32) and (1.36), there is C > 0 such that +for any λ ∈ Γ, +��G Fp� +λ − DFp,2 +X +�−1s +�� +L2 ⩽ Cp2��� +λ − DFp,2 +X +�−1s +�� +H2,p +⩽ Cp2���DFp,2 +X +� +λ − DFp,2 +X +�−1s +�� +L2 + p2∥ +� +λ − DFp,2 +X +�−1s∥L2 +� +⩽ Cp4|λ| · ∥s∥L2. +(4.22) +By (4.20), (4.21) and (4.22), we get (4.19). From (4.20) and an argument similar to +(4.22), for some C > 0, j ∈ N, we have +��LFp(λ − LFp�−1s +�� +L2 ⩽ Cpj|λ|j∥s∥L2 +(4.23) +By (4.23), for any k ∈ N, there are C > 0, ℓ ∈ N∗ such that +��LFp,k+1(λ−LFp�−1s +�� +L2 = +��LFp(λ − LFp�−1LFp,ks +�� +L2 +⩽ Cpj|λ|j∥LFp,ks∥L2 ⩽ Cpℓ|λ|ℓ∥s∥H2k,p. +(4.24) +By (4.8) and (4.24), we get (4.18). +□ +Set the fibre product M ×S M = {(x1, x2) ∈ M × M | π(x1) = π(x2)} with the +projections pri(x1, x2) → xi. For any two bundles V, V ′ over M, Let V ×S V ′ be the +bundle on M ×S M given by pr∗ +1(V ) ⊗ pr∗ +2(V ′). If we have two connections ∇V , ∇V ′ on +V and V ′, let ∇V ×SV ′ be the induced connection on V ×S V ′. We define a bundle +Ep = R[z]�⊗π∗Λ (T ∗S) �⊗ +�� +Λ(T ∗X) ⊗ Fp +� +×S +� +(Λ(T ∗X))∗ ⊗ F ∗ +p +�� +(4.25) +over M ×S M. Let ∇Ep be the connection on Ep induced by 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗Fp,u +and ∇Fp,u (see (1.6) and (1.33)). We notice that �Gt,ε(tLFp) is a smooth section of Ep on +M ×S M. +We also denote the pull back bundle of Ep through the diagonal embedding M → +M ×S M given by x0 �→ (x0, x0), then +Ep,x0 = R[z]�⊗π∗Λ +� +T ∗ +π(x0)S +� �⊗End(Λ(T ∗ +x0X)) ⊗ End(Fp,x0). +(4.26) +Theorem 4.9. For any ε > 0 and ℓ ∈ N, there exist c1, c2 > 0 and k ∈ N such that for +any t > 0 and p ∈ N∗, we have +�� �Gt,ε +� +tLFp��� +C ℓ(M×SM) ⩽ Cpk exp +� +c1t − c2 +t +� +, +(4.27) +where the C ℓ (M ×S M)-norm is induced by ∇Ep. +Proof. By (4.16), for r ∈ N∗, there is a holomorphic function �Ht,h,r(a) defined in a +neighborhood of Vc such that (see also [6, Theorem 13.32]) +1 +(r − 1)! +dr−1 +dar−1 �Ht,h,r(a) = �Ht,h(a), +(4.28) +and for any h > 0, there are C, c1, c2 > 0 such that for any t > 0, we have +sup +a∈Vc +|ak �Ht,h,r (a) | ⩽ C exp +� +c1t − c2 +t +� +. +(4.29) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +23 +Let {Ui}dim S +i=1 +be a set of local vector fields on S. Let {UH +i } denote the set of corresponding +horizontal lift local vector fields. For any multi-index α = (i1, i2, · · ·) in which 1 ⩽ ij ⩽ +dim S, we denote UH +α = ∇Ep +UH +i1 ∇Ep +UH +i2 · · · . By (4.14) and (4.28), we get +UH +α +� �Gt,ε +� +tLFp�� += +1 +2πi +� +Γ +�Ht,ε,r(λ)UH +α (λ − LFp�−rdλ. +(4.30) +Observe that UH +α (λ − LFp�−r is a linear combination of operators in the form of +(λ − LFp�−r0UH +α1 +� +LFp� +(λ − LFp�−r1 · · · UH +αℓ +� +LFp� +(λ − LFp�−rℓ, +(4.31) +where αj = (j1, j2, · · ·) is multi-index, ℓ ∈ N, ri ⩾ 1, � ri = r + ℓ. Note that for any +multi-index α, UH +α +� +LFp� +is a second order fibrewise differential operator satisfies +��UH +α +� +LFp� +s +�� +Hk,p ⩽ Cp2��s +�� +Hk+2,p. +(4.32) +By (4.18) and (4.32), for any k ∈ N, there are i, i′ ∈ N∗ such that +��UH +α +� +LFp�� +λ − LFp�−1s +�� +H2k,p ⩽ pi|λ|i′∥s∥H2k,p. +(4.33) +Let P, Q be differential operators of order 2q and 2q′ with scalar principal symbol +and with compact support in BX (x, ε) and BX (x′, ε) respectively. We take r ⩾ (2q + +2q′ + 2)|α| in (4.30), then there is a rj ⩾ 2(q + q′). By (4.31), we split the operator +PUH +α +� +(λ − LFp)−k� +Q into the product of two parts +P(λ − LFp�−r0UH +α1 +� +LFp� +(λ − LFp�−r1 · · · UH +αj +� +LFp� +(λ − LFp�−1(λ − LFp�−2q, +(λ − LFp�rj−2q′−2q−2(λ − LFp�−2q′−1 · · · UH +αℓ +� +LFp� +(λ − LFp�−rℓQ. +(4.34) +By (4.18) and (4.33), the first part above is a map from L2-space to itself, and the +operator norm is dominated by Cpj0|λ|j′ +0. By Remark 4.5, the adjoint of the second part +has the same structure as the first part, and its operator norm is also dominated by +Cpj1|λ|j′ +1. Therefore, for some j, j′ ∈ N∗, we have +��PUH +α +� +(λ − LFp)−k� +Qs +�� +L2 ⩽ Cpj|λ|j′∥s∥L2. +(4.35) +By (4.29), (4.30), (4.35) and the Sobolev inequality, we find that +��UH +α +� �Gt,ε +� +tLFp�� +(·, ·) +�� +C q(X×X) ⩽ Cpk exp +� +c1t − c2 +t +� +, +(4.36) +which gives (4.27). Here we emphasize that the constant in Sobolev inequality is inde- +pendent on the dimension of bundle Fp. +□ +4.3. Rescaling of the operator LFp. Now we use Bismut-Zhang’s rescaling [12, Chap- +ter 4] for two kinds of Clifford variables, which is analogous to Getzler’s rescaling [20]. +Using the same notation as in the beginning of § 4.2, we fix x0 ∈ M and iden- +tify R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) and Fp to +� +R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) +� +x0 and Fp,x0 on +BTx0X (0, ε). Let Γ0 and ΓFp,u be the corresponding connection forms . +Put X0 = Tx0X ∼= Rm. Let gTX0 be a Riemannian metric on X0 such that +gTX0 = +� +gTX, +on BTx0X (0, inj(X)/2) , +gTx0X, +outside BTx0X (0, inj(X)) , +(4.37) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +24 +and let dvX0 be the associated volume form. Let dvTX be the Riemannian volume form +of +� +Tx0X, gTx0X� +, and κ (x) be the smooth positive function defined by +dvX0 = κ (Z) dvTX, +κ (0) = 1. +(4.38) +Recall the function f in (4.11). For any ε > 0 satisfies (4.6), set fε : Rm → R by +fε(Z) = f(|Z|/ε). +(4.39) +Put +Fp = R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) ⊗ Fp. +(4.40) +On the trivial bundle Fp,x0 over X0, we define +∇Fp,x0 = ∇ + fε(Z) +� +Γ0 +Z + ΓFp,u +Z +� +. +(4.41) +Let ∆Fp,x0 be the Laplacian associated with ∇Fp,x0 and gTX0. Let ∇TX0 be the Levi-Civita +connection of +� +X0, gTX0� +and let (gij) be the inverse of (gij) = +� +gTX0(∂i, ∂j) +� +, then +∆Fp,x0 = −gij� +∇ +Ep,x0 +∂i +∇ +Ep,x0 +∂j +− ∇ +Ep,x0 +∇T X0 +∂i +∂j +� +. +(4.42) +Recall ΛFp in (1.32), locally we view ΛFp as an element in C ∞(BTx0X(0, ε), Ep,x0) (see +(4.26)). Set +Λx0,p = fε(Z)ΛFp ∈ C ∞ +0 (Tx0X, Ep,x0), +L Fp +x0 = ∆Ep,x0 + Λx0,p. +(4.43) +Then L Fp +x0 is a family of differential operators acting on C ∞(Tx0X, Fp,x0) for x0 ∈ M. +Let exp(−tLFp) (x, x′) , (x, x′) be the smooth kernel of exp(−tLFp) with respect to dvX +for ∈ M ×S M, and exp(−tL Fp +x0 ) (Z, Z′) the smooth kernel of exp(−tL Fp +x0 ) with respect +to dvX0 for (Z, Z′) ∈ Tx0X × Tx0X. As in (4.26), we still denote the pull back bundle of +Ep through the projection TX×M TX → M by Ep, then we can view exp(−tL Fp +x0 ) (Z, Z′) +as a smooth section of Ep on TX ×M TX. +Theorem 4.10. For any ε > 0 satisfies (4.6) and ℓ ∈ N, there exist C, c1, c2 > 0 and +k ∈ N such that for any p ∈ N∗, t > 0, we have +��� exp(−tLFp)(x0, x0) − exp(−tL Fp +x0 )(0, 0) +��� +C ℓ(M) ⩽ Cpkec1t− c2 +t , +(4.44) +where |·|C ℓ(M) is the C ℓ norm with respect to the parameter x0 ∈ M. +Proof. By (4.41), (4.42) and (4.43), L Fp +x0 and LFp coincide over BTx0X (0, ε), then from +(4.11), (4.12), (4.13) and the finite propagation speed of wave operator (see [25, Theorem +D.2.1]), we obtain that +�Ft,ε +� +tLFp� +(x0, ·) = �Ft,ε +� +tL Fp +x0 +� +(0, ·) . +(4.45) +By (4.43), L Fp +x0 has the same structure as LFp, especially, Lemmas 4.4 and 4.9 are still +true if we replace LFp with L Fp +x0 . Then (4.44) follows from (4.14). +□ +Put +M Fp +x0 = θ√pp−2L Fp +x0 θ1/√p. +(4.46) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +25 +Corollary 4.11. For any ε > 0 satisfies (4.6), γ > 0 and ℓ ∈ N, there exist C, c1, c2 > 0 +such that for any t > 0 and p ∈ N∗ large enough, we have +���θ−1 +√ +tTrΛ(T ∗X)⊗Fp +s +� +exp(−tMFp)(x0, x0) − exp(−tM Fp +x0 )(0, 0) +���� +C ℓ(M) ⩽ Ceγt− c2 +4t −√γc2p, +(4.47) +where |·|C ℓ(M) is the C ℓ norm with respect to x0 ∈ M, and c1 and c2 are given in (4.16). +Proof. By (2.6), (4.3), (4.46) and (4.44) by replacing t with t/p2, there is C > 0 with +���θ−1 +√ +tTrΛ(T ∗X)⊗Fp +s +� +exp (−tMFp) (x0, x0) − exp (−tM Fp +x0 ) (0, 0) +���� +C ℓ(M) += +���θ√ p +t TrΛ(T ∗X)⊗Fp +s +� +exp (−tp−2LFp) (x0, x0) − exp (−tp−2L Fp +x0 ) (0, 0) +���� +C ℓ(M) +⩽C(1 + t− dim S/2)pdim S/2 exp +� +c1tp−2 − c2p2 +t +� +. +(4.48) +By the mean value inequality, we have +c1tp−2 − c2p2 +t +⩽ γt − c2 +2t − +�γt +2 + c2p2 +2t +� +⩽ γt − c2 +2t − √γc2p. +(4.49) +It is obvious that pdim S/2e−√γc2p and (1+t− dim S)e−c2/4t are uniformly bounded for p ∈ N∗ +and t > 0. Therefore, we get (4.47) from (4.48) and (4.49). +□ +For s ∈ C ∞� +Tx0X, Ep,x0 +� +, Z ∈ Rm, u > 0, we set (Kus) (Z) = s (uZ) and +N Fp +x0 = K1/pM Fp +x0 Kp. +(4.50) +Let exp(−tN Fp +x0 ) (Z, Z′) be the kernel of exp(−tN Fp +x0 ) with respect to dvTX, then +� +exp(−tN Fp +x0 )s +� +(Z) = +� +K1/p exp(−tM Fp +x0 )Kps +� +(Z) += +� +Rm exp(−tM Fp +x0 )(p−1Z, Z′)s(pZ′)κ(Z′)dvTX(Z′) += p−m +� +Rm exp(−tM Fp +x0 )(p−1Z, p−1Z′)κ(p−1Z′)s(Z′)dvTX(Z′). +(4.51) +By (4.38) and (4.51), we get +exp(−tN Fp +x0 )(0, 0) = p−m exp(−tM Fp +x0 )(0, 0). +(4.52) +We introduce another copy � +Tx0X of Tx0X. We will add an extra hat when we refer to +an element in � +Tx0X. For s > 0, put +cs(ej) = 1 +√sej ∧ −√siej +�cs(ej) = 1 +√sej ∧ +√siej. +(4.53) +We denote by � +L Fp +x0 the operator obtained from N Fp +x0 by replacing c(ei), �c(ei) with c1/p(ei) +and �c1/p(�ei) respectively. Set +�Fp = R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) �⊗Λ(� +T ∗X) ⊗ Fp, +(4.54) +then � +L Fp +x0 acts naturally on C ∞(Tx0X, �Fp,x0). +We denote by exp(−t � +L Fp +x0 ) (Z, Z′) the +smooth kernels of exp(−t � +L Fp +x0 ) with respect to dvTX(Z′). Put +�Ep = R[z]�⊗π∗Λ(T ∗S)�⊗End +� +Λ(T ∗X) +��⊗End +� +Λ(� +T ∗X) +� +⊗ End(Fp), +(4.55) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +26 +and denote by π: TX ×M TX → M the natural projection, then exp(−t � +L Fp +x0 ) (Z, Z′) is +a smooth section of π∗�Ep over TX ×M TX. +For any H ∈ End +� +Λ(T ∗X) +��⊗End +� +Λ(� +T ∗X) +� +, it can be expanded uniquely in the fol- +lowing form: +H = +� +1⩽i1<··· 0 and x0 ∈ M, we have +TrΛ(T ∗X)⊗Fp +s +� +exp(−tM Fp +x0 ) (0, 0) +� +dvX = (4π) +m +2 TrFp +� +� +B � +exp(−t � +L Fp +x0 )(0, 0) +�max. (4.61) +Proof. By the definition of � +L Fp +x0 and (4.60), we get +TrΛ(T ∗X)⊗Fp +s +� +exp(−tN Fp +x0 ) (0, 0) +� +dvX(x0) = (4π) +m +2 p−mTrFp +� +� +B � +exp(−t � +L Fp +x0 )(0, 0) +�max, +(4.62) +then (4.61) follows immediately from (4.52) and (4.62). +□ +Let � +N be the number operator of R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) �⊗Λ(� +T ∗X) that acts by +multiplication by the total degree of this exterior algebra. For any a > 0, set +�θa = a +� +N. +(4.63) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +27 +Recall the connection forms Γ0 +Z and ΓFp,u +Z +at the beginning of § 4.3, fε(Z) in (4.39) and +gij as in (4.42). Then fε(Z)Γ0 +Z is a 1-form evaluating in R[z]�⊗π∗Λ(T ∗ +π(x0)S)�⊗End(Λ(T ∗ +x0X)). +By replacing �c(ei) with �c(�ei) in fε(Z)Γ0 +Z, we get a 1-form �Γ0 +Z on Tx0X taking values in +R[z]�⊗π∗Λ(T ∗ +π(x0)S)�⊗End(Λ(T ∗ +x0X))�⊗End(Λ( � +T ∗ +x0X)). Set +�ΓFp,u +Z += fε(Z)ΓFp,u +Z +. +(4.64) +By (4.50) and the construction of � +L Fp +x0 , we have the following identity by evaluating +the tensors on the right-hand side of the equation at Z/p: +� +L Fp +x0,Z = −gij�� +∂i + 1 +p +�θ−1 +1/p�Γ0 (∂i) �θ1/p + p−1�ΓFp,u (∂i) +� +· +� +∂j + 1 +p +�θ−1 +1/p�Γ0 (∂j) �θ1/p + p−1�ΓFp,u (∂j) +� ++ 1 +p +� +∇TX0 +∂i +∂j + 1 +p +�θ−1 +1/p�Γ0� +∇TX0 +∂i +∂j +��θ1/p + p−1�ΓFp,u� +∇TX0 +∂i +∂j +��� ++fε +� rX +4p2 − 1 +8p2 +� +RTX(ei, ej)ek, eℓ +� +c1/p(ei)c1/p(ej)�c1/p (�ek) �c1/p (�eℓ) +− 1 +8p +� +RTX(f H +α , f H +β +� +ek, eℓ +� +f αf β�c1/p (�ek) �c1/p (�eℓ) +− +1 +4p3/2 +� +RTX� +ei, f H +α +� +ek, eℓ +� +c1/p(ei)f α�c1/p (�ek) �c1/p (�eℓ) +− 1 +2pc1/p(ei)c1/p(ej) 1 +4pω +� +∇Fp, gFp�2(ei, ej) +− 1 +2f αf β 1 +4pω +� +∇Fp, gFp�2� +f H +α , f H +β +� +− +1 +p1/2c1/p(ei)f α 1 +4pω +� +∇Fp, gFp�2� +ei, f H +α +� ++ 1 +4p2 �ω +� +∇Fp, gFp� +(ei)2 + 1 +2p�c1/p(�ei)�c1/p(�ej) 1 +4p�ω +� +∇Fp, gFp�2(ei, ej) +− +1 +p1/2f α�c1/p(�ei) 1 +2p∇ +� +TX⊗Fp,u +fH +α +�ω +� +∇Fp, gFp� +(ei) +− 1 +pc1/p(ei)�c1/p(�ej) 1 +2p∇ +� +TX⊗Fp,u +ei +�ω +� +∇Fp, gFp� +(ej) +− +1 +p1/2zc1/p(ei) 1 +2pω +� +∇Fp, gFp� +(ei) − zf α 1 +2pω +� +∇Fp, gFp�� +f H +α +�� +. +(4.65) +Note that dim Fp grows as a polynomial of p ∈ N∗, locally, we cannot reduce Fp to +a fixed bundle, so it is nonsense to consider the limit operator limp→+∞ � +L Fp +x0,Z naively. +In [9, (9.148)], Bismut-Ma-Zhang viewed � +L Fp +x0,Z as a differential operator with Toeplitz +operator (2.8) coefficients and obtained the first order expansion of � +L Fp +x0,Z with respect +to p ∈ N∗ in the sense of Toeplitz operator. +Now we should describe its full “expansion”. We do not expand � +L Fp +x0 directly in the +sense of Toeplitz operator. Instead, we introduce a series of operators { � +L Fp +v }0⩽v⩽1/p such + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +28 +that � +L Fp +1/p = � +L Fp +x0 , and we view the Taylor expansion �ℓ +i=0 +1 +i!pi +∂i � +L +Fp +v +∂vi |v=0 as the ℓ-th order +“expansion” of � +L Fp +x0 . The advantage of this approach is that v is a smooth parameter, +so we could use techniques from Ma-Marinescu [25, § 4.1], and this is also convenient for +the computation of second order expansion in § 6. +Now we give the definition of operators { � +L Fp +v }0⩽v⩽1/p acting on C ∞ +0 (Tx0X, �Fp,x0). By +evaluating the tensors on the right-hand side of the equation at vZ, we take +� +L Fp +v += −gij�� +∂i + v�θ−1 +v �Γ0 +vZ (∂i) �θv + p−1�ΓFp,u +vZ (∂i) +� +· +� +∂j + v�θ−1 +v �Γ0 +vZ (∂j) �θv + p−1�ΓFp,u +vZ (∂j) +� ++ v +� +∇TX0 +∂i +∂j + v�θ−1 +v �Γ0 +vZ(∇TX0 +∂i +∂j)�θv + p−1�ΓFp,u +vZ (∇TX0 +∂i +∂j) +�� ++fε +�v2 +4 rX − v2 +8 +� +RTX(ei, ej)ek, eℓ +� +cv(ei)cv(ej)�cv (�ek) �cv (�eℓ) +− v +8 +� +RTX(f H +α , f H +β +� +ek, eℓ +� +f αf β�cv (�ek) �cv (�eℓ) +− v3/2 +4 +� +RTX� +ei, f H +α +� +ek, eℓ +� +cv(ei)f α�cv (�ek) �cv (�eℓ) +− v +2cv(ei)cv(ej) 1 +4pω +� +∇Fp, gFp�2(ei, ej) +− 1 +2f αf β 1 +4pω +� +∇Fp, gFp�2� +f H +α , f H +β +� ++ 1 +4p2 �ω +� +∇Fp, gFp� +(ei)2 +− v1/2cv(ei)f α 1 +4pω +� +∇Fp, gFp�2� +ei, f H +α +� ++ v +2�cv(�ei)�cv(�ej) 1 +4p�ω +� +∇Fp, gFp�2(ei, ej) +− v1/2f α�cv(�ei) 1 +2p∇ +� +TX⊗Fp,u +fH +α +�ω +� +∇Fp, gFp� +(ei) +− vcv(ei)�cv(�ej) 1 +2p∇ +� +TX⊗Fp,u +ei +�ω +� +∇Fp, gFp� +(ej) +− v1/2zcv(ei) 1 +2pω +� +∇Fp, gFp� +(ei) − zf α 1 +2pω +� +∇Fp, gFp�� +f H +α +�� +. +(4.66) +Combing (4.65) and (4.66), we clearly have � +L Fp +1/p = � +L Fp +x0 . Now we discuss the Taylor +expansion of � +L Fp +v +with respect to v. Set +R +� +TX = +� +RTXei, ej +� +∧ �ej ∧ i�ei, +�RTX = 1 +2 +� +RTX(ek, eℓ)ei, ej +� +�ek ∧ �eℓ ∧ ej ∧ iei. +(4.67) +Theorem 4.13. For r ∈ N, p ∈ N∗, 0 ⩽ v ⩽ 1/p, we can write +∂r � +L Fp +v +∂vr += X Fp,(r) +v,ij +(Z)∂i∂j+Y Fp,(r) +v,i +(Z)∂i+Z Fp,(r) +v +(Z), for X Fp,(r) +v,ij +(Z) = −∂rgij +vZ +∂vr +(4.68) +and Y Fp,(r) +v,i +(Z), Z Fp,(r) +v +(Z) ∈ C ∞(Tx0X, �Ep,x0) with the following properties: + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +29 +(1) The function X Fp,(r) +v,ij +is a constant outside a compact set of Tx0X. Also, if v ̸= 0, +Y Fp,(r) +v,i +(Z) and Z Fp,(r) +v +(Z) have compact supports. +(2) There is C > 0 such that for p ∈ N∗ and 0 ⩽ v ⩽ 1/p, the operator norm ∥·∥�Ep,x0 +of X Fp,(r) +v,ij +(Z), Y Fp,(r) +v,i +(Z) and Z Fp,(r) +v +(Z) are dominated by C(1 + |Z|r). +(3) The operator X Fp,(r) +0,ij +(Z) (resp. Y Fp,(r) +0,i +(Z), Z Fp,(r) +0 +(Z)) is polynomial in Z (resp. +with coefficients in �Ep,x0). Moreover, X Fp,(r) +0,ij +(Z) is a homogeneous polynomial of +Z of degree r, the degree in Z of Y Fp,(r) +0,i +(Z) ( Z Fp,(r) +0 +(Z)) is no more than r. +(4) In particular, +� +L Fp +0 += − ∆Tx0X − 1 +4 +� +RTXei, ej +� +x0�ei ∧ �ej − 1 +4pω +� +∇Fp, gFp�2 +x0 ++ 1 +4p2 �ω +� +∇Fp, gFp� +x0(ei)2 + 1 +4p�ω +� +∇Fp, gFp�2 +x0 +− 1 +2p∇ +� +TX⊗Fp,u�ω +� +∇Fp, gFp� +x0 − 1 +2pzω +� +∇Fp, gFp� +x0, +∂ � +L Fp +v +∂v +��� +v=0 = − Zj +4 +�� +RTX +x0 ∂i, ∂j +� +− +� �RTX +x0 ∂i, ∂j +� +− 1 +2pω +� +∇Fp, gFp�2 +x0 +� +∂i, ∂j +�� +∂i ++ QFp(Z) + QFp′, +(4.69) +where QFp(Z) (resp. QFp′) is a homogeneous polynomials in Z of degree 1 (resp. +0) with coefficients in �Ep,x0, and +QFp′ =1 +2R +� +TX − 1 +2 +�RTX − 1 +4 +� +RTX(f H +α , ei)ek, el +� +�ek ∧ �el ∧ f α ∧ iei ++ ei ∧ iej +1 +4pω2 +p(ei, ej) + f α ∧ iei +1 +4pω2 +p +� +f H +α , ei +� ++ �ei ∧ i�ej +1 +4p�ω2 +p(�ei, �ej) +− f α ∧ i�ei +1 +2p∇ +� +TX⊗Fp,u +fH +α +�ωp(ei) − ei ∧ i�ej +1 +2p∇ +� +TX⊗Fp,u +ei +�ωp(ej) +− �ej ∧ iei +1 +2p∇ +� +TX⊗Fp,u +ei +�ωp(ej) + ziei +1 +2pωp(ei). +(4.70) +Proof. As in the proof of Lemma 4.4, by Theorem 3.1, (1.6) and (4.10), we get +p−1�ΓFp,u +vZ +� +∂i +� +∼ vZj +2 +· 1 +4pω +� +∇Fp, gFp�2 +x0 +� +∂i, ∂j +� ++ O(v2). +(4.71) +By (4.53), we have +v1/2cv(ei) = ei ∧ −viei, +v1/2�cv(�ei) = �ei ∧ +vi�ei. +(4.72) +By (4.72), similar to (4.71),we have +�θ−1 +v (�Γ0 +vZ)�θv +� +∂i +� +∼ −1 +4Zj +�� +RTX +x0 (ek, eℓ)∂i, ∂j +� +ek ∧ eℓ − ⟨RTX +x0 (ek, eℓ)∂i, ∂j +� +�ek ∧ �eℓ ++ +� +RTX +x0 (fα, eℓ)∂i, ∂j +� +f α ∧ eℓ + 1 +2 +� +RTX +x0 (fα, fβ)∂i, ∂j +� +f α ∧ f β� ++ O(v). +(4.73) +By (4.66), (4.71), (4.72) and (4.73), we can write +� +L Fp +v += X Fp +v,ij∂i∂j + Y Fp +v,i ∂i + Z Fp +v , +where X Fp +v,ij(Z) = −gij +vZ, +(4.74) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +30 +and each of {Y Fp +v,i , Z Fp +v } is the sum of terms in the form of vkΓvZ, where k ∈ N, Γ ∈ +C ∞ +0 (Tx0X, �Ep,x0) is a smooth family of Toeplitz operators (see § 3.3) and has support in +BTx0X(0, ε), so its norm |·|C (Tx0X,�Ep,x0) is bounded uniformly for p ∈ N∗ as well as all its +derivatives. For r ∈ N, we have +dr +dvr vkΓvZ = +� +r1+r2=r +r1⩽k +� +|α|=r2 +CαZαvk−r1� ∂αΓ +∂Zα +� +vZ, +(4.75) +where α is a multi-index. By (4.75), ∂r � +L +Fp +v +∂vr +is the sum of terms in the form of +� +|α|⩽r +CαvkZαΓα,vZ∂i∂j + C′ +αvk′ZαΓ′ +α,vZ∂i + C′′ +αvk′′ZαΓ′′ +α,vZ, +(4.76) +where Γα, Γ′ +α, Γ′′ +α ∈ C ∞ +0 (Tx0X, �Ep,x0) verify similar property as Γ. By (4.76), we get the +first three statements. +By (4.66), (4.71) and (4.73), we get the first identity in (4.69), also, together with the +property of geodesic coordinates (∂igjk� +(0) = 0, we obtain X Fp +ij,1(0) and Y Fp +i,1 (0) in the +second identity of (4.69). By (4.72), the contribution of the first four terms in (4.66) to +QFp′ is given by +− 1 +4 +� +RTX(ei, ej)ek, eℓ +� +ei ∧ ej ∧ �ek ∧ i�eℓ + 1 +4 +� +RTX(ei, ej)ek, eℓ +� +�ek ∧ �eℓ ∧ ei ∧ iej +− 1 +4 +� +RTX(f H +α , f H +β +� +ek, eℓ +� +f αf β�ek ∧ i�eℓ +− 1 +4 +� +RTX� +ei, f H +α +� +ek, eℓ +� +ei ∧ f α ∧ �ek ∧ i�eℓ + 1 +4 +� +RTX� +ei, f H +α +� +ek, eℓ +� +�ek ∧ f αiei, +(4.77) +which yields the first three terms in (4.70). The other terms in (4.70) follows from (4.66) +and (4.72). We finish the proof. +□ +4.4. Sobolev spaces with weights and estimates on the resolvents. In this sub- +section, we follow the strategy of Bismut-Lebeau [6, § 11 k)-l)] with some modifications. +In particular, we should choose the weight in Theorem 4.15 and the path of integral in +Theorem 4.18 carefully to ensure the “positivity” of � +L Fp +v : by (4.69), we see that � +L Fp +v += +� +L Fp +0 ++O(v), and � +L Fp +0 +is the sum of a non-negative part −∆Tx0X + 1 +4p|�ω(∇Fp, gFp)|2 +x0 and +a nilpotent part. So, we want to prove that � +L Fp +v +is “nearly” a non-negative operator +when v is small: for any γ > 0, there is C > 0 such that for t > 0 large enough, we have +∥ exp(−t � +L Fp +v )(0, 0)∥ ⩽ C exp(γt). To eliminate the effect of the nilpotent part in � +L Fp +v , +we introduce Sobolev norms with weights. +From now on, we will fix a γ > 0. Note that �Fp and �θµ are defined in (4.54) and (4.63). +Definition 4.14. For 0 < µ < 1, k ∈ N and s, s′ ∈ C ∞ +0 (Tx0X, �Fp,x0), set +⟨s, s′⟩0,p,µ = +� +Rn +��θµs, �θµs′� +dvTX, +⟨s, s′⟩k,p,µ = +� +|α|⩽k +� +∂αs, ∂αs′� +0,p,µ, +(4.78) +and let ∥·∥0,p,µ and ∥·∥k,p,µ be the corresponding norms. Let Hk,p,µ be the Sobolev space +which is the completion of C ∞ +0 (Tx0X, �Fp,x0) with respect to ∥·∥k,p,µ. Let H−k,p,µ be the + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +31 +Sobolev space of negative order with the norm: for s ∈ C ∞ +0 (Tx0X, �Fp,x0), +∥s∥−k,p,,µ = +sup +0̸=s′∈Hk,p,µ +�� ⟨s, s′⟩0,p,µ +�� +∥s′∥k,p,µ +. +(4.79) +For k1, k2 ∈ Z, on the space of bounded linear map from Hk1,p,µ to Hk2,p,µ, we denote by +∥·∥k1,k2 +p,µ +the operator norm. +By (4.78), it is clear that +��ei∧ +��0,0 +p,µ ⩽ µ, +∥iei∥0,0 +p,µ ⩽ 1 +µ, +(4.80) +and (4.80) is tailor-made for the next theorem. +Theorem 4.15 (Elliptic estimations). There are µ, ε > 0 such that we have Ci > 0 for +1 ⩽ i ⩽ 4 in which C2 < γ/2 and p0 ∈ N∗ such that for any p ⩾ p0, 0 ⩽ v ⩽ 1/p and +s, s′ ∈ C ∞ +0 (Tx0X, �Fp,x0), we have +Re +� � +L Fp +v s, s +� +0,p,µ ⩾ C1 ∥∇s∥2 +0,p,µ − C2∥s∥2 +0,p,µ, +���Im +� � +L Fp +v s, s +� +0,p,µ +��� ⩽ C3∥s∥1,p,µ∥s∥0,p,µ, +��� +� � +L Fp +v s, s′� +0,p,µ +��� ⩽ C4∥s∥1,p,µ · ∥s′∥1,p,µ. +(4.81) +Proof. We now carefully discuss the structure of � +L Fp +v . First, we focus on a single term +� +∂i + v�θ−1 +v �Γ0 +vZ (∂i) �θv + p−1�ΓFp,u +vZ (∂i) +� +(4.82) +as in (4.66). Let �R0 be the operator obtained by replacing �c(ej) with �c(�ej) on the left +hand side of (1.34). By (1.34) and (4.10), we get +v�θ−1 +v �Γ0 +vZ (∂i) �θv = −v +2 +� 1 +0 +fε(vZ)vZj +��θ−1 +v +�R0 +svZ�θv +� +(∂i, ∂j)ds, +p−1�ΓFp,u +vZ (∂i) = − +� 1 +0 +fε(vZ)vZj +1 +4pω +� +∇Fp, gFp�2 +svZ(∂i, ∂j)ds. +(4.83) +Here we use the integral form rather than the Taylor expansion to get an estimation +uniformly for p ∈ N∗. By Theorem 3.1, the operator norm of +1 +4pω(∇Fp, gFp)2 is uniformly +bounded for p ∈ N∗. Since the support of fε(Z) is contained in BTx0X(0, 2ε), by Theorem +3.1 and (4.83), there is C > 0 such that for any ε satisfies (4.6), p ∈ N∗, 0 ⩽ v ⩽ 1/p and +Z ∈ Tx0X, we have +v∥�θ−1 +v �Γ0 +vZ (∂i) �θv∥�Ep,x0 ⩽ Cv, +��p−1�ΓFp,u +vZ (∂i) +���Ep,x0 ⩽ Cε. +(4.84) +Consider the expansion in (4.74) +� +L Fp +v += X Fp +v,ij(Z)∂i∂j + Y Fp +v,i (Z)∂i + Z Fp +v (Z) += ∂iX Fp +v,ij(Z)∂j + +� +Y Fp +v,i (Z) − ∂j(X Fp +v,ji)(Z) +� +∂i + Z Fp +v (Z). +(4.85) +By (4.66), (4.72) (4.83) and (4.84), we see that Z Fp +v (Z) can be separated into three parts +Z Fp +v (Z) = Z Fp +v +′(Z) + Z Fp +v +′′(Z) + Z Fp +v +′′′(Z), where Z Fp +v +′(Z) is a non-negative operator, + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +32 +Z Fp +v +′′(Z) adds the degree of exterior algebra in �Fp,x0 and there is C > 0 such that +∥Z Fp +v +′′(Z)∥�Ep,x0 ⩽ C, +∥Z Fp +v +′′′(Z)∥�Ep,x0 ⩽ Cv. +(4.86) +By (4.37) and the positivity of the matrix {gij +vZ}, there is C > 0 such that +�� +∂iX Fp +v,ij(Z)∂j +� +s, s +� +0,p,µ ⩾ C ∥∇s∥2 +0,p,µ . +(4.87) +For some C > 0, we denote Cε,µ,v = C(ε + µ + v +ε + v +µ). Note that if we let ∂i act on the +left hand side of (4.83), we get an extra v +ε. This together with (4.80) and (4.84) implies +���� +Y Fp +v,i − ∂j(X Fp +v,ji) +� +∂is, s +� +0,p,µ +�� ⩽ Cε,µ,v∥s∥1,p,µ∥s∥0,p,µ. +(4.88) +Moreover, by (4.80), (4.84) and (4.86), we have +��� +�� +Z Fp +v +′′ + Z Fp +v +′′′� +s, s +� +0,p,µ +��� ⩽ Cε,µ,v∥s∥2 +0,p,µ. +(4.89) +By (4.87), (4.88) and (4.89), we have +Re +� � +L Fp +v s, s +� +0,p,µ ⩾ C ∥∇s∥2 +0,p,µ − Cε,µ,v +� +∥∇s∥0,p,µ∥s∥0,p,µ + ∥s∥2 +0,p,µ +� +⩾ +� +C − Cε,µ,v +� +∥∇s∥2 +0,p,µ − Cε,µ,v ∥s∥2 +0,p,µ , +���Im +� � +L Fp +v s, s +� +0,p,µ +��� ⩽ Cε,µ,v +� +∥∇s∥0,p,µ∥s∥0,p,µ + ∥s∥2 +0,p,µ +� +. +(4.90) +By (4.90), we get the first two inequalities in (4.81), and the third one is obvious. +□ +Remark 4.16. By (4.83), there is C > 0 such that for any h satisfies (4.6), p ∈ N∗, 0 ⩽ +v ⩽ 1/p and Z ∈ Tx0X, we have p−1���ΓFp,u +vZ (∂i) +�� +End(Fp,x0) ⩽ Cv|Z|. Even if we get an +O(v), it is not uniformly bounded for Z ∈ Tx0X. This explains the necessity of (4.84). +In the rest of this section, we will always fix a couple of (µ, ε) satisfying Theorem 4.15. +Proposition 4.17. For k ∈ N, there exist C > 0 and p0 ∈ N∗ such that for p ⩾ p0, 0 ⩽ +v ⩽ 1/p and Q1, · · ·Qk ∈ {∂i, Zi}m +i=1, we have +��� +�� +Q1, [Q2, · · ·[Qk, � +L Fp +v ] · · · ] +� +s, s′� +0,p,µ +��� ⩽ C∥s∥1,p,µ∥s′∥1,p,µ. +(4.91) +Proof. By (4.74) and (4.75), for i, j ∈ N, we have +[Zj, ∂i] = δij, +∂i +� +ΓvZ +� += v(∂iΓ)vZ, +(4.92) +where ΓZ ∈ C ∞ +0 (Tx0X, �Ep,x0). By (4.74) and (4.92), +� +Q1, [Q2, · · ·[Qk, � +L Fp +v ] · · · ] +� +has the +same structure as � +L Fp +v +in (4.74). Hence we easily get (4.92) by (4.81). +□ +For d1, d2 > 0, set +Vd1,d2 = +� +λ ∈ C | Re(λ) ⩽ d1Im(λ)2 − d2 +� +, +(4.93) +and let ∂Vd1,d2 be its boundary with counterclockwise orientation. +Recall the norm +∥·∥k1,k2 +p,µ +given in Definition 4.14. +Theorem 4.18 (Elliptic regularity). There exist d, C > 0 and p0 ∈ N∗ such that for any +p ⩾ p0, 0 ⩽ v ⩽ p−1 and λ ∈ Vd,γ/2, the resolvent +� +λ − � +L Fp +v +�−1 exists, and +��� +λ − � +L Fp +v +�−1��0,0 +p,µ ⩽ C, +��� +λ − � +L Fp +v +�−1��−1,1 +p,µ ⩽ C +� +1 + |λ|2 � +. +(4.94) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +33 +Proof. Since γ could be arbitrarily small, (4.94) almost indeed ensures the “non- +negativity” of Spec( � +L Fp +v ). +For a, b ∈ R and λ = a + ib ∈ C, we have +���� +λ − � +L Fp +v +� +s, s +� +0,p,µ +�� ⩾ sup +� +Re +� � +L Fp +v s, s +� +0,p,µ − a ∥s∥2 +0,p,µ , +���Im +� � +L Fp +v s, s +� +0,p,µ − b ∥s∥2 +0,p,µ +��� +� +⩾ sup +� +C1 ∥s∥2 +1,p,µ − (a + C1 + C2) ∥s∥2 +0,p,µ , |b| ∥s∥2 +0,p,µ − C3 ∥s∥0,p,µ ∥s∥1,p,µ +� +. +(4.95) +It is obvious that ∥s∥0,p,µ ⩽ ∥s∥1,p,µ, then we deduce from (4.95) that +���� +λ − � +L Fp +v +� +s, s +� +0,p,µ +�� ⩾ C(λ) ∥s∥2 +0,p,µ , +(4.96) +where C(λ) is given by +C(λ) = inf +w⩾1 sup +� +C1w2 − (a + C1 + C2), |b| − C3w +� +. +(4.97) +For any δ > 0, by (4.97), C(λ) ⩾ δ if and only if {w ⩾ 1} is contained in +{w ∈ R | C1w2 − (a + C1 + C2) ⩾ δ} ∪ {v ∈ R | |b| − C3w ⩾ δ}, +(4.98) +which means one of the following conditions holds +� +C1 − (a + C1 + C2) ⩾ δ, +� +C−1 +1 (a + C1 + C2 + δ) ⩽ C−1 +3 (|b| − δ). +(4.99) +By (4.99), C(λ) ⩾ δ if and only if one of the following inequalities holds +a ⩽ +� +−C2 − δ, +C1C−2 +3 +(|b| − δ)2 − C1 − C2 − δ, +(4.100) +which is the red part of the figure below. By Theorem 4.15, let δ small enough that +C1C−2 +3 δ2 − C1 − C2 − δ < γ/2 < −C2 − δ, +(4.101) +then by the figure below, we can choose a d that satisfies 0 < d < C1C−2 +3 +to make sure +that Vd,γ/2 is a subset of the red part, in other words, for any λ = a + bi ∈ Vd,γ/2, it +satisfies one of the two inequalities (4.100). Therefore, we have infλ∈Vd,γ/2 C(λ) ⩾ δ > 0. +Then by (4.96), if λ ∈ Vd,γ/2, the the resolvent +� +λ − � +L Fp +v +�−1 exists and +��� +λ − � +L Fp +v +�−1��0,0 +p,µ ⩽ C. +(4.102) +By (4.81), it is clear that � +L Fp +v +can be extended to a continuous linear map from H1,p,µ +into H−1,p,µ. Using (4.95), if we fix λ0 = −1 − C1 − C2, then +��� +λ0 − � +L Fp +v +�−1��−1,1 +p,µ ⩽ 1. +(4.103) +If λ ∈ Vd,γ/2, then +� +λ − � +L Fp +v +�−1 = +� +λ0 − � +L Fp +v +�−1 + (λ0 − λ) +� +λ − � +L Fp +v +�−1� +λ0 − � +L Fp +v +�−1. +(4.104) +By (4.102), (4.103) and (4.104), we see that +��� +λ − � +L Fp +v +�−1��−1,0 +p,µ ⩽ 1 + C |λ − λ0| , +(4.105) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +34 +a = −C2 − δ +a = db2 − γ/2 +a = C1C−2 +3 +(|b| − δ)2 − C1 − C2 − δ +a +−γ/2 +b +Figure 3. +on the other hand, we also have +� +λ − � +L Fp +v +�−1 = +� +λ0 − � +L Fp +v +�−1 + (λ0 − λ) +� +λ0 − � +L Fp +v +�−1� +λ − � +L Fp +v +�−1. +(4.106) +From (4.105) and (4.106), we obtain +��� +λ − � +L Fp +v +�−1��−1,1 +p,µ ⩽ 1 + C |λ0 − λ| (1 + C |λ − λ0|) ⩽ C +� +1 + |λ|2 � +. +(4.107) +This completes the proof. +□ +In § 4.5-§ 4.7, we will always work under the assumption that p ⩾ p0, 0 ⩽ v ⩽ p−1 +and λ ∈ Vd,γ/2 as in Theorem 4.18. +4.5. Regularizing properties of the resolvents. In this subsection, we follow closely +Ma-Marinescu [25, Theorems 4.1.12-4.1.14] with some necessary modifications. +Theorem 4.19 (Higher regularity). For any k ∈ N, the resolvent +� +λ− � +L Fp +v +�−1 in (4.94) +maps Hk,p,µ to Hk+1,p,µ. For any multi-index α ∈ Nm, there is C > 0 such that +��Zα� +λ − � +L Fp +v +�−1s +�� +k+1,p,µ ⩽ C +� +1 + |λ|2 �k+|α|+1 � +α′⩽α +��Zα′s +�� +k,p,µ +(4.108) +Proof. Using Proposition 4.17 and Theorem 4.18 in the same way as [25, Theorem +1.6.10] follows from [25, Theorem 1.6.8, Proposition 1.6.9], we get Theorem 4.19. +□ +For k ∈ N∗, r ∈ N, set +Ik,r = +� +(k, r) = (ki, ri) ∈ (N∗)j+1 × Nj ��� +j +� +i=0 +ki = k + j, +j +� +i=1 +ri = r +� +. +(4.109) +For (k, r) ∈ Ik,r, we set +Ak +r (λ, v, p) = +� +λ − � +L Fp +v +�−k0 ∂r1 � +L Fp +v +∂vr1 +� +λ − � +L Fp +v +�−k1 · · · ∂rj � +L Fp +v +∂vrj +� +λ − � +L Fp +v +�−kj. +(4.110) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +35 +Then there exist ak +r ∈ R such that +∂r +∂vr +� +λ − � +L Fp +v +�−k = +� +(k,r)∈Ik,r +ak +rAk +r (λ, v, p) . +(4.111) +Theorem 4.20. For any ℓ ∈ N, k > 2(ℓ + r + 1) and (k, r) ∈ Ik,r, there are C > 0 and +j ∈ N such that for any λ ∈ Vd,γ/2 and α, α′ ∈ Nm that |α| , |α′| ⩽ ℓ, we have +��∂αAk +r (λ, v, p) ∂α′s +�� +0,p,µ ⩽ C(1 + |λ|)j � +|α|⩽r +∥Zαs∥0,p,µ . +(4.112) +Proof. By Theorem 4.19, if |α| ⩽ ℓ, there is C > 0 such that for any λ ∈ Vd,γ/2 +��∂α� +λ − � +L Fp +v +�−ℓ��0,0 +p,µ ⩽ C +� +1 + |λ|2 �ℓ+1. +(4.113) +Note that if µ = 1 in (4.78), the product ⟨·, ·⟩0,p,1 is just the usual L2 inner prod- +uct. Let � +L Fp,∗ +v +be the formal adjoint operator of � +L Fp +v +with respect to ⟨·, ·⟩0,p,1. Then +� +L Fp,∗ +v +has essentially the same structure as the operator � +L Fp, except that the operators +ei∧, �ei∧, iei, i�ei are changed into iei, i�ei, ei∧, �ei∧ respectively. We have +� +s, ∂α� +λ − � +L Fp,∗ +v +�−ℓs′� +0,p,1 = (−1)|α|�� +λ − � +L Fp +v +�−ℓ∂αs, s′� +0,p,1. +(4.114) +Note that +⟨s, s′⟩0,p,1 = ⟨�θµs, �θµ−1s′⟩0,p,1, +∥s∥0,p,µ = +���θµs +�� +0,p,1, +∥s′∥0,p,µ−1 = +���θµ−1s′�� +0,p,1. (4.115) +Now if we consider the weighted product ∥·∥0,p,µ−1 for � +L Fp,∗ +v +, an obvious analogue of +(4.113) still holds: ∥(λ− � +L Fp,∗ +v +)−ℓ∥0,0 +p,µ−1 ⩽ C(1+|λ|2)ℓ+1. This together with (4.114) and +(4.115) yields +��� +s, ∂α� +λ − � +L Fp,∗ +v +�−ℓs′� +0,p,1 +�� ⩽ ∥s∥0,p,µ +��� +λ − � +L Fp,∗ +v +�−ℓs′�� +0,p,µ−1 +⩽ C +� +1 + |λ|2 �ℓ+1∥s∥0,p,µ∥s′∥0,p,µ−1. +(4.116) +By (4.114), (4.115) and (4.116), we see that +��� +λ − � +L Fp +v +�−ℓ∂αs +�� +0,p,µ = +sup +∥s′∥0,p,µ−1⩽1 +��� +�� +λ − � +L Fp +v +�−ℓ∂αs, s′� +0,p,1 +��� +⩽ C +� +1 + |λ|2 �ℓ+1��s +�� +0,p,µ. +(4.117) +By (4.113) and (4.117), we prove (4.112) for r = 0. +For r > 0, by (4.76) and (4.92), we can rewrite +∂r +∂vr � +L Fp +v +in (4.110) as +� +|β|⩽r +CβΓβ,vZ∂i∂jZβ + C′′ +βΓ′ +β,vZ∂iZβ + C′′ +βΓ′′ +β,vZZβ, +(4.118) +where Γβ,Z, Γβ′,Z, Γβ′′,Z ∈ C ∞ +0 (Tx0X, �Ep,x0) are smooth families of Toeplitz operators (see +§ 3.3). Let R′ +v,p be the family of operators of the form +R′ +v,p = +�� +Γ1Q1, +� +Γ2Q2, · · · +� +ΓkQk, � +L Fp +v +� +· · · +��� +(4.119) +where Γi ∈ C ∞ +0 (Tx0X, �Ep,x0) are bounded with respect to |·|C 0(Tx0X,�Ep,x0) uniformly for +p ∈ N∗ with all derivatives and Q1, · · ·Qk ∈ {∂i, Zi}m +i=1. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +36 +We handle now the operator Ak +r (λ, v, p) ∂α′. For each ∂ri � +L +Fp +v +∂vri , we write it in the form +of (4.118) and move all the terms to the right-hand side as in [25, Theorem 4.1.13]. +Then ∂αAk +r (λ, v, p) ∂α′ is as the form � +|β|⩽r Lv,βZβ where Lv,β is a linear combination +of operators that can be split into the product of two parts: +∂α� +λ − � +L Fp +v +�−k′ +0R1 +� +λ − � +L Fp +v +�−k′ +1 · · · Ri +� +λ − � +L Fp +v +�−k′′ +i +� +λ − � +L Fp +v +�−(k′ +i−k′′ +i ) · Ri+1 · · · Rj′� +λ − � +L Fp +v +�−k′ +j′∂β′, +(4.120) +where Ri ∈ R′ +v,p, �i−1 +j=0(k′ +j − 1) + k′′ +i − ℓ > 0 and �j′ +j=i(k′ +j − 1) − k′′ +i > 2r + ℓ + 1, then +the ∥·∥0,0 +p,µ norm of each part is bounded by C(1 + |λ|)N. This finishes the proof. +□ +Theorem 4.21. For any r ⩾ 0 and k > 0, there exist C > 0 and ℓ ∈ N∗ such that +���� +�∂r � +L Fp +v +∂vr +− ∂r � +L Fp +v +∂vr +��� +v=0 +� +s +���� +−1,p,µ +⩽ Cv +� +|α|⩽r+1 +∥Zαs∥1,p,µ , +���� +� ∂r +∂vr +� +λ − � +L Fp +v +�−k − +� +(k,r)∈Ik,r +ak +rAk +r (λ, 0, p) +� +s +���� +0,p,µ +⩽ Cv (1 + |λ|)ℓ +� +|α|⩽4r+1 +∥Zαs∥0,p,µ . +(4.121) +Proof. An application of Taylor expansion for (4.74) and (4.76) implies +��∂r � +L Fp +v +∂vr +− ∂r � +L Fp +v +∂vr +��� +v=0 +� +s, s′� +0,p,µ ⩽ Cv ∥s′∥1,p,µ +� +|α|⩽r+1 +∥Zαs∥1,p,µ , +(4.122) +from which we get the first inequality of (4.121). +The second inequality of (4.121) +follows from Theorems 4.18, 4.19 and the first inequality of (4.121) as [25, Theorem +4.1.14] follows from [25, Theorems 4.1.10, 4.1.12]. +□ +4.6. Uniform estimation on the heat kernel. This section is analogous to [25, +§ 4.1.5], with the necessary changes made. +Theorem 4.22. For any k, ℓ, r ∈ N, there is C > 0 such that for t ⩾ 1, Z, Z′ ∈ Tx0X +with |Z| , |Z′| ⩽ 1, we have +sup +|α|,|α′|⩽k +���� +∂|α|+|α′| +∂Zα∂Z′α′ +∂r +∂vr exp(−t � +L Fp +v ) (Z, Z′) +���� +C ℓ(M) +⩽ Ceγt/2. +(4.123) +Proof. Recall Vd,γ/2 given in (4.93), by the Cauchy integral formula, for k ∈ N∗, we have +exp(−t � +L Fp +v ) = +(k − 1)! +2πi (−t)k−1 +� +∂Vd,γ/2 +e−tλ� +λ − � +L Fp +v +�−kdλ. +(4.124) +From Theorem 4.20 and (4.124), we obtain that for |αi| ⩽ ℓ and t ⩾ 1, we have +��∂α1 exp(−t � +L Fp +v )∂α′��0,0 +p,µ ⩽ Ceγt/2. +(4.125) +Now by (4.125) and the Sobolev inequality, for Z, Z′ ∈ Tx0X that |Z| , |Z′| ⩽ 1, we have +sup +|α|,|α′|⩽l +���� +∂|α|+|α′| +∂Zα∂Z′α′ exp(−t � +L Fp +v ) (Z, Z′) +���� ⩽ Ceγt/2. +(4.126) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +37 +This implies (4.123) for r = ℓ = 0. To obtain (4.123) for r ⩾ 1, we see from (4.124) that +for k ∈ N∗, +∂r +∂vr exp(−t � +L Fp +v ) = +(k − 1)! +2πi (−t)k−1 +� +∂Vd,γ/2 +e−tλ ∂r +∂vr +� +λ − � +L Fp +v +�−kdλ. +(4.127) +By (4.111), (4.113) and (4.127), we get +��� ∂r +∂vr ∂α1 exp(−t � +L Fp +v )∂α′s +��� +0,p,µ ⩽ Ceγt/2 � +|α|⩽r +∥Zαs∥0,p,µ. +(4.128) +Taking supp(s) ∈ BTx0X(0, 2), by Sobolev inequality again, we get (4.123) for ℓ = 0. +Finally, for any vector U on M, +∇π∗�Ep +U +exp(−t � +L Fp +v ) = +(k − 1)! +2πi (−t)k−1 +� +∂Vd,γ/2 +e−tλ∇π∗�Ep +U +� +λ − � +L Fp +v +�−kdλ. +(4.129) +Now we use a similar formula as (4.111) for ∇π∗�Ep +U +� +λ − � +L Fp +v +�−k by replacing +∂ri +∂vri � +L Fp +v +with ∇π∗�Ep +U +� +L Fp +v . Remark that ∇π∗�Ep +U +� +L Fp +v +is a differential operator on Tx0X with the +same structure as � +L Fp +v , it has the same type as (4.74). Then using the above argument, +we conclude that (4.111) also holds for ℓ ⩾ 1. We complete the proof. +□ +For k ∈ N∗ large enough, set +S (r) +t,p = +(k − 1)! +2πi(−t)k−1r! +� +∂Vd,γ/2 +e−tλ +� +(k,r)∈Ik,r +ak +rAk +r (λ, 0, p) dλ, +S (r) +v,t,p = 1 +r! +∂r +∂vr exp(−t � +L Fp +v ) − S (r) +t,p (t), +(4.130) +then S (r) +t,p and S (r) +v,t,p do not depend on the choice of k. We denote by S (r) +t,p (Z, Z′) (resp. +S (r) +v,t,p(Z, Z′)) the smooth kernel of S (r) +p (t) (resp. S (r) +v,t,p) with respect to dvTX. +Theorem 4.23. For any r ∈ N, there is C > 0 such that for t ⩾ 1, Z, Z′ ∈ Tx0X with +|Z| , |Z′| ⩽ 1, we have +���S (r) +v,t,p(Z, Z′) +��� ⩽ Cv1/(m+1)eγt/2. +(4.131) +Proof. By (4.130) and the Cauchy integral formula, we get +S (r) +v,t,p = +(k − 1)! +2πi (−t)k−1 r! +� +∂Vd,γ/2 +e−tλ · +� ∂r +∂vr +� +λ − � +L Fp +v +�−k +− +� +(k,r)∈Ik,r +ak +rAk +r (λ, 0, p) +� +dλ. +(4.132) +By (4.121) and (4.132), when t ⩾ 1, we have +��S (r) +v,t,p(t)s +�� +0,p,µ ⩽ Cveγt/2 +� +|α|⩽4r+1 +∥Zαs∥0,p,µ . +(4.133) +By Theorem 4.22 and (4.133), using a similar argument as [25, Theorem 4.17] follows +from [25, Theorem 4.16, (4.1.67)], we get (4.131). +□ + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +38 +Theorem 4.24. For any r, ℓ ∈ N, there is C > 0 such that for t ⩾ 1, we have +��� exp(−t � +L Fp +v )(0, 0) − +r +� +i=0 +S (i) +t,p (0, 0)vi��� +C ℓ(M) ⩽ Cvr+1eγt/2. +(4.134) +Proof. By Theorem 4.23 and (4.130), we have +1 +r! +∂r +∂vr exp(−t � +L Fp +v ) +��� +v=0 = S (r) +t,p . +(4.135) +Now by Theorem 4.22 and (4.130), S (r) +t,p has the same estimation as +∂r +∂vr exp(−t � +L Fp +v ) in +(4.123). From (4.123) and the Taylor expansion +f(v) − +k +� +r=0 +1 +r! +∂rf +∂vr (0)vr = 1 +k! +� v +0 +(v − y)k∂k+1f +∂vk+1 (y)dy, +(4.136) +we get (4.134). +□ +4.7. The asymptotics of S (i) +t,p (0, 0). Let PFp +t (Z, Z′) be the kernel of exp(−t � +L Fp +0 ) with +respect to dvTX. Put +σFp = − 1 +4 +� +RTXei, ej +� +x0�ei ∧ �ej − 1 +4pωFp,2 +x0 ++ 1 +4p2 +���ωFp��2 +x0 + 1 +4p�ωFp,2 +x0 +− 1 +2p +� +∇ +� +TX⊗Fp,u�ωFp� +x0 − 1 +2pzωFp +x0 . +(4.137) +By (4.69), we have +PFp +t (Z, Z′) = +1 +(4πt) +m +2 e− |Z−Z′|2 +4t +−tσFp. +(4.138) +For k ∈ N, the k-simplex △k is given by {(t1, · · · , tk) | 0 ⩽ t1 ⩽ t2 · · · ⩽ tk ⩽ 1}. For +t > 0, we will write t△k for the rescaled simplex {(t1, · · · , tk) | 0 ⩽ t1 ⩽ t2 · · · ⩽ tk ⩽ t}. +For multi-index r = (r1, · · · , rk)k∈N where r1, · · · , rk ∈ N∗, set +St,r,p = +� +t△k +� +PFp +t−tk +∂rk � +L Fp +v +∂vrk +��� +v=0PFp +tk−tk−1 · · · ∂r1 � +L Fp +v +∂vr1 +��� +v=0PFp +t1 +� +(0, 0) +k +� +j=1 +dtj += +� +t△k +� +(TX)k PFp +t−tk(0, Z(k)) +�∂rk � +L Fp +v,Z(k) +∂vrk +��� +v=0PFp +tk−tk−1 +� +(Z(k), Z(k−1)) +· · · +�∂r1 � +L Fp +v,Z(1) +∂vr1 +��� +v=0PFp +t1 +� +(Z(1), 0) +k +� +j=1 +dvTX +� +Z(j)� +k +� +j=1 +dtj +(4.139) +where (TX)k means integral with respect to k-copies of Tx0X: +� +Z(i) = +� +Z(i) +1 , · · · , Z(i) +m +� +| +1 ⩽ i ⩽ k +� +, and the subscript Z(j) in +∂rj � +L +Fp +v,Z(j) +∂vrj +means acting on the coordinate Z(j). By +the Duhamel’s principle [4, § 2.7], Theorem 4.13, (4.110), (4.130) and (4.135), we have +S (i) +t,p (0, 0) = +� +r1+···+rk=i +(−1)k +�k +j=1 rj! +St,r,p. +(4.140) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +39 +In particular, +S (0) +t,p (0, 0) = e−tσFp +(4πt) +m +2 , S (1) +t,p (0, 0) = − +� t +0 +� +Tx0X +PFp +t−t1(0, Z) +�∂ � +L Fp +v,Z +∂v +��� +v=0PFp +t1 +� +(Z, 0)dZdt1. +(4.141) +By (4.138), we see that there are singularities inside the integral of (4.139), and we will +prove that there are no singularities after taking the integral. Let us give an auxiliary +result first. Put Pt(x, y) the classical heat kernel on R: Pt(x, y) = (4πt)−1/2e−(x−y)2/4t. +For t > 0, (t1, · · · , tk) ∈ t△k and {α(j) +i , β(j) +i +∈ N}1⩽i⩽m,1⩽j⩽k, set +ft(tk, · · · , t1) = +m +� +i=1 +� +Pt−tk(0, Z(k) +i +) · Z +(k),α(k) +i +i +∂β(k) +i +∂Z +(k),β(k) +i +i +Ptk−tk−1(Z(k) +i +, Z(k−1) +i +) +· · · Z +(2),α(2) +i +i +∂β(2) +i +∂Z +(2),β(2) +i +i +Pt2−t1(Z(2) +i , Z(1) +i ) · Z(1),αi +i +∂βi +∂Z(1),βi +i +Pt1(Z(1) +i , 0) +k +� +j=1 +dZ(j) +i +� +. +(4.142) +Lemma 4.25. For k ∈ N∗, there are ℓ ∈ N and C > 0 such that for t ⩾ 1 and +(t1, · · · , tk) ∈ t△k, we have +|ft(tk, · · · , t1)| ⩽ C +� +1 + tℓ� +. +(4.143) +Proof. By (4.142), we only need to prove for the special case m = 1. We prove by +induction on k ∈ N∗. Let us consider the integral of the following form: +� ∞ +−∞ +Pt1(Z2, Z1)Zℓ′ +1 +∂ℓ +∂Zℓ +1 +Pt0(Z1, Z0)dZ1, +for Z0, Z1, Z2 ∈ R, t0, t1 > 0. +(4.144) +We introduce the generating function of the integral in (4.144). Put +f1(Z2, Z0) = +� +ℓ,ℓ′∈N +� ∞ +−∞ +Pt1(Z2, Z1)(µ1Z1)ℓ′ +(ℓ′)! +λℓ +1 +ℓ! +∂ℓ +∂Zℓ +1 +Pt0(Z1, Z0)dZ1, +(4.145) +For any analytic function g(y), we have � +ℓ∈N +λℓ +ℓ! +∂ℓ +∂yℓg(y) = g(y + λ), this gives +f1(Z2, Z0) = +� ∞ +−∞ +Pt1(Z2, Z1) exp(µ1Z1)Pt0(Z1 + λ1, Z0)dZ1 += +� ∞ +−∞ +1 +4π(t0t1) +1 +2 exp +� +−t1 + t0 +4t1t0 +� +Z1 − t0Z2 +t1 + t0 +− t1(Z0 − λ1) +t1 + t0 +− 2t1t0µ1 +t1 + t0 +�2� +dZ1 +· exp +� +−(Z2 − Z0 + λ1)2 +4(t0 + t1) ++ µ1 +� t0Z2 +t1 + t0 ++ t1(Z0 − λ1) +t1 + t0 +� ++ +µ2 +1t0t1 +(t1 + t0) +� += Pt1+t0(Z2, Z0 − λ1) exp +� +µ1 +� t0Z2 +t1 + t0 ++ t1(Z0 − λ1) +t1 + t0 +� ++ +µ2 +1t0t1 +(t1 + t0) +� +. +(4.146) +Similar to (4.145) and (4.146), we define +fj(Zj+1, Z0) = +� ∞ +−∞ +Ptj(Zj+1, Zj) exp(µjZj)fj−1(Zj + λj, Z0)dZj, +w0 = 0, +wj = +� +j +� +i=0 +ti +�−1 � +0⩽i<ℓ⩽j +µℓti for j ∈ N∗. +(4.147) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +40 +Then we can prove inductively that +fj(Zj+1, Z0) = P�j +i=0 ti +� +Zj+1, Z0 − +j +� +i=1 +λj +� +exp +� +wjZj+1 + +j +� +i=1 +(µj + wj−1)witi ++ +j +� +i=1 +(µi + wi−1 − wi)(Z0 − +i +� +ℓ=1 +λℓ) +� +. +(4.148) +Plugging Z0 = Zj+1 = 0 in (4.148), we obtain +fj(0, 0) = +� +4π +j +� +i=0 +ti +�− 1 +2 exp +� +− +� +4 +j +� +i=0 +λi +�−1� +j +� +i=1 +λi +�2 ++ +j +� +i=1 +(µj + wj−1)witi +− +j +� +i=1 +(µi + wi−1 − wi) +� +i +� +ℓ=1 +λℓ +�� +. +(4.149) +By (??) and (4.149), for k ∈ N, there is C > 0 such that for any multi-index α, β ∈ Nj +with |α| , |β| ⩽ k and �j +i=0 tj = t ⩾ 1, the coefficient of λαµβ in fj(0, 0) is dominated by +C(1 + tk), which implies (4.143). +□ +Recall the asymptotic trace symbol Tr[i][Tp] in (2.10). +Theorem 4.26. For i, j, ℓ ∈ N and δ > 0, there exists C > 0 such that for t ⩾ 1, +���Tr[j] +� +S (i) +t,p (0, 0) +���� +C ℓ(M) ⩽ Ceδt, +����p−nTrFp[S (i) +t,p (0, 0)] − +j +� +k=0 +p−kTr[k][S (i) +t,p (0, 0)] +���� +C ℓ(M) +⩽ Ceδtp−j−1. +(4.150) +Proof. By Theoreom 4.13, (4.138), (4.139) (4.143), Sr,p is a sum of integrals of the +form +� +t△k +ft(tk, · · · , t1)e−(t−tk)σFpτke−(tk−tk−1)σFp · · · τ1e−(t1−t0)σFp, +(4.151) +where τj is a smooth family of Toeplitz operators (see § 3.3) with respect to the parameter +x0 ∈ M for 0 ⩽ j ⩽ k, and ft(tk, · · · , t1) is the sum of the product of functions in the +form of (4.142). By (4.143), we get +��ft(tk, · · · , t1) +�� ⩽ C(1 + tℓ) +(4.152) +for some C > 0, ℓ ∈ N. By (4.137), it is clear that +Spec +� +σFp� += Spec +� 1 +4p2 +���ωFp��2 +x0 +� +. +(4.153) +By Theorem 2.7, (4.140), (4.151), (4.152), (4.153), S (i) +t,p (0, 0) verifies estimations the +same as (2.26) and (2.27), from which we get (4.150) for ℓ = 0. Since S (i) +t,p (0, 0) is a +smooth family of Toeplitz operators with respect to x0 ∈ M, by Remark 3.3, we get +(4.150) for the general norm |·|C ℓ(M). +□ + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +41 +4.8. The convergence when 0 < t ⩽ 1. As t → 0, there is singularity in (4.134), we +consider a different rescaling following the proof of [9, Theorem 9.31]. In this subsection, +we always assume that 0 < t ⩽ 1. Set +N Fp +t,x0 = t +p2θ√ p +t K √ +t +p L Fp +x0 K p +√ +tθ� +t +p. +(4.154) +We denote by � +L Fp +t,x0 the operator obtained from N Fp +t,x0 by replacing c(ei), �c(ei) with c t +p(ei) +and �c 1 +p(�ei) respectively. Like (4.61), we have +θ 1 +√ +tTrΛ(T ∗X)⊗Fp +s +� +exp(−tM Fp +x0 ) (0, 0) +� +dvX = (4π) +m +2 TrFp +� +� +B � +exp(− � +L Fp +t,x0)(0, 0) +�max +. +(4.155) +Let N2 and N3 be the number operators of the exterior algebras R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) +and Λ(� +T ∗X) respectively acting by multiplication of the degree. For any a > 0, set +θa = aN2, +�θa = aN3. +(4.156) +By evaluating the tensors on the right hand side at v +√ +tZ, we take +� +L Fp +v,t = −gij�� +∂i + v +√ +tθ +−1 +√ +vt�θ−1 +√v�Γ0 (∂i) �θ√vθ√ +vt + +√ +t +p +�ΓFp,u (∂i) +� +· +� +∂j + v +√ +tθ +−1 +√ +vt�θ−1 +√v�Γ0 (∂i) �θ√vθ√ +vt + +√ +t +p +�ΓFp,u (∂j) +� ++v +√ +t +� +∇TX0 +∂i +∂j + v +√ +tθ +−1 +v +√ +t�θ−1 +v �Γ0(∇TX0 +∂i +∂j)�θ√vθ√ +vt + +√ +t +p +�ΓFp,u(∇TX0 +∂i +∂j) +�� ++fε +�v2t +4 rX − v2t +8 +� +RTX(ei, ej)ek, eℓ +� +cvt(ei)cvt(ej)�cv (�ek) �cv (�eℓ) +− v +√ +t +8 +� +RTX(f H +α , f H +β +� +ek, eℓ +� +f αf β�cv (�ek) �cv (�eℓ) +− v +√ +vt +4 +� +RTX� +ei, f H +α +� +ek, eℓ +� +cvt(ei)f α�cv (�ek) �cv (�eℓ) +− vt +2 cvt(ei)cvt(ej) 1 +4pωFp,2(ei, ej) − 1 +2f αf β 1 +4pωFp,2� +f H +α , f H +β +� +− +√ +vtcvt(ei)f α 1 +4pωFp,2� +ei, f H +α +� ++ +t +4p2 +���ωFp��2 ++ vt +2 �cv(�ei)�cv(�ej) 1 +4p�ωFp,2(ei, ej) − +√ +vtf α�cv(�ei) 1 +2p∇ +� +TX⊗Fp,u +fH +α +�ωFp(ei) +− vtcvt(ei)�cv(�ej) 1 +2p∇ +� +TX⊗Fp,u +ei +�ωFp(ej) +− +√ +vtzcvt(ei) 1 +2pωFp(ei) − zf α 1 +2pωFp� +f H +α +�� +, +(4.157) +then we get � +L Fp +1/p,t = � +L Fp +t,x0. Similar to (4.71), we have +v +√ +tθ +−1 +√ +vt�θ−1 +√v�Γ0 +v +√ +tZ (∂i) �θ√vθ√ +vt = vZjΓ′ +ij,v +√ +tZ − vtZjΓ′′ +ij,v +√ +tZ, +(4.158) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +42 +where the support of Γ′ +i,Z, Γ′′ +i,Z are contained in BTx0X(0, 2ε), and +Γ′ +ij,0 =1 +4 +� � +k<ℓ +� +RTX(ek, eℓ)∂i, ∂j +� +ek ∧ eℓ + +� � +RTX(fα, ek)∂i, ∂j +� +f α ∧ ek ++ +� +α<β +� +RTX(fα, fβ)∂i, ∂j +� +f α ∧ f β� +, +Γ′′ +ij,0 = − 1 +4 +� +k<ℓ +� +RTX(ek, eℓ)∂i, ∂j +� +ek ∧ eℓ. +(4.159) +Since the first term on the right-hand side of (4.158) is not bounded for Z ∈ Tx0X, +we set a new norm to make the following operators uniformly bounded for 0 ⩽ v, t ⩽ 1: +1v +√ +t|Z|⩽ε · vZj(ei ∧ −vtiei), +1v +√ +t|Z|⩽ε · (ei ∧ −vtiei), +(4.160) +where +1A for A ⊆ Tx0X represents the characteristic function of A. +Definition 4.27. For k, ℓ ∈ N, if N1s = ℓs (see (4.156)), we set +∥s∥2 +0,p = +� +Rn +��s(Z) +��2(1 + |Z|2)m+1−ℓdZ, +∥s∥2 +k,p = +� +α∈Nm,|α|⩽k +∥∂αs∥2 +0,p. +(4.161) +Let Hk,p be the completion Sobolev space with respect to ∥·∥k,p. Let H−k,p be the Sobolev +space of negative order with the norm given by +∥s∥−k,p = +sup +0̸=s′∈Hk,p +�� ⟨s, s′⟩0,p +�� +∥s′∥k,p +(4.162) +Theorem 4.28. For r ∈ N, 0 ⩽ v, t ⩽ 1, p ∈ N∗, we have +∂r +∂vr � +L Fp +v,t = X (r) +v,t,ij(Z)∂i∂j + Y (r) +v,t,i(Z)∂i + Z (r) +v,t (Z), +for X (r) +v,t,ij(Z) = − +∂rgij +v +√ +tZ +∂vr +, (4.163) +where Y Fp,(r) +v,t,i +, Z Fp,(r) +v,t +are in C ∞(Tx0X, �Ep,x0) with the following properties: +(1) There is C > 0 such that the ∥·∥0,p norm of X Fp,(r) +v,t,ij +Y Fp,(r) +v,t,i +, Z Fp,(r) +v,t +is dominated +by C +��(1 + |Z|r)s +�� +0,p uniformly for 0 ⩽ v, t ⩽ 1, p ∈ N∗. +(2) Operators X Fp,(r) +0,t,ij +Y Fp,(r) +0,t,i +, Z Fp,(r) +0,t +are polynomials in Z and +√ +t. Moreover, X Fp,(r) +0,t,ij (Z) +is a homogeneous polynomial in Z of degree r, the degrees in Z of Y Fp,(r) +0,t,i +(Z) and +Z Fp,(r) +0,t +(Z) are no more that r. +(3) In particular, +� +L Fp +0,t = − ∆Tx0X − 1 +4 +� +RTXei, ej +� +x0�ei ∧ �ej − 1 +4pωFp,2 +x0 ++ +t +4p2 +���ωFp��2 +x0 ++ t +4p�ωFp,2 +x0 +− +√ +t +2p ∇ +� +TX⊗Fp,u�ωFp +x0 − 1 +2pzωFp +x0 , +∂ +∂v +� +L Fp +v,t +��� +v=0 =Y Fp,(1) +0,t,i +(Z)∂i + Z Fp,(1) +0,t +(Z) = −Zj +4 +�� +RTX +x0 ∂i, ∂j +� +− t +� �RTX +x0 ∂i, ∂j +� +− t +2pωFp,2 +x0 +� +∂i, ∂j +�� +∂i + Qt(Z) + Q′ +t, +(4.164) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +43 +where QFp +t (Z) and QFp +t +′ are homogeneous polynomials in Z of degree 1 and 0 +respectively with coefficients in �Ep,x0, and +QFp +t +′ = t +2R +� +TX − 1 +2 +�RTX − t +4 +� +RTX(f H +α , ei)ek, el +� +�ek ∧ �el ∧ f α ∧ iei ++ ei ∧ iej +t +4pωFp,2(ei, ej) + f α ∧ iei +t +4pωFp,2� +f H +α , ei +� ++ �ei ∧ i�ej +t +4p�ωFp,2(�ei, �ej) +− f α ∧ i�ei +√ +t +2p ∇ +� +TX⊗Fp,u +fH +α +�ωFp(ei) − ei ∧ i�ej +√ +t +2p ∇ +� +TX⊗Fp,u +ei +�ωFp(ej) +− �ej ∧ iei +t3/2 +2p ∇ +� +TX⊗Fp,u +ei +�ωFp(ej) + ziei +t +2pωFp(ei). +(4.165) +Proof. The first statement follows from (4.157), (4.158) and the weight in the norm +(4.161). The rest part follows from (4.157) as we obtain Theorem 4.13 from (4.66). +□ +By Theorem 4.28, using the same way as we get Theorems 4.15-4.26, we could prove +the following Theorems 4.29-4.38 that all hold uniformly for 0 ⩽ v, t ⩽ 1, p ∈ N∗. +Theorem 4.29. There exists C > 0 such that +Re +� � +L Fp +v,t s, s +� +0,p ⩾ C−1 ∥∇s∥2 +0,p − C∥s∥2 +0,p, +���Im +� � +L Fp +v,t s, s +� +0,p +��� ⩽ C∥s∥1,p∥s∥0,p, +��� +� � +L Fp +v,t s, s′� +0,p +��� ⩽ C∥s∥1,p · ∥s′∥1,p. +(4.166) +Proposition 4.30. For k ∈ N, there is C > 0 such that for any Q1, · · · Qk ∈ {∂i, Zi}m +i=1, +��� +�� +Q1, [Q2, · · ·[Qk, � +L Fp +v,t ] · · · ] +� +s, s′� +0,p +��� ⩽ C∥s∥1,p∥s′∥1,p. +(4.167) +Theorem 4.31. There exist d1, d2 > 0 such that if λ ∈ Vd1,d2 where Vd1,d2 is given in +(4.93), the resolvent +� +λ − � +L Fp +v,t +�−1 exists. Moreover, there is C > 0 such that +��� +λ − � +L Fp +v,t +�−1��0,0 +p +⩽ C, +��� +λ − � +L Fp +v,t +�−1��−1,1 +p +⩽ C +� +1 + |λ|2 � +. +(4.168) +Proposition 4.32. For any λ ∈ Vd1,d2, k ∈ N, the resolvent +� +λ − � +L Fp +v,t +�−1 maps Hk,p to +Hk+1,p. Moreover, for any multi-index α ∈ Nm, there exists C > 0 such that +��Zα� +λ − � +L Fp +v,t +�−1s +�� +k+1,p ⩽ C +� +1 + |λ|2 �k+|α|+1 � +α′⩽α +��Zα′s +�� +k,p. +(4.169) +For (k, r) ∈ Ik,r, λ ∈ Vd1,d2, we set +Bk +r (v, λ, p) = +� +λ − � +L Fp +v,t +�−k0 ∂r1 � +L Fp +v,t +∂vr1 +� +λ − � +L Fp +v,t +�−k1 · · · ∂rj � +L Fp +v,t +∂vrj +� +λ − � +L Fp +v,t +�−kj. +(4.170) +Then there exist ak +r ∈ R such that +∂r +∂vr +� +λ − � +L Fp +v,t +�−k = +� +(k,r)∈Ik,r +ak +rBk +r (v, λ, p) +(4.171) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +44 +Theorem 4.33. Let ℓ ∈ N, for any multi-index α, α′ ∈ Nm with |αi| ⩽ ℓ and k > +2(ℓ + r + 1), if (k, r) ∈ Ik,r, there are C > 0 and j ∈ N such that for any λ ∈ Vd1,d2, +��∂αBk +r (v, λ, p) ∂α′s +�� +0,p ⩽ C(1 + |λ|)j � +|α|⩽r +∥Zαs∥0,p . +(4.172) +Theorem 4.34. For any r ∈ N and k ∈ N∗, there exist C > 0 and ℓ ∈ N such that for +λ ∈ Vd1,d2, we have +���� +�∂r � +L Fp +v,t +∂vr +− ∂r � +L Fp +v,t +∂vr +��� +v=0 +� +s +���� +−1,p +⩽ Cv +� +|α|⩽r+1 +∥Zαs∥1,p , +���� +� ∂r +∂vr +� +λ − � +L Fp +v,t +�−k − +� +(k,r)∈Ik,r +ak +rBk +r (0, λ, p) +� +s +���� +0,p +⩽ Cv (1 + |λ|)ℓ +� +|α|⩽4r+1 +∥Zαs∥0,p . +(4.173) +Theorem 4.35. For k, ℓ, r ∈ N, there is C > 0 such that for Z, Z′ ∈ Tx0X with +|Z| , |Z′| ⩽ 1, we have +sup +|α|,|α′|⩽k +���� +∂|α|+|α′| +∂Zα∂Z′α′ +∂r +∂vr exp(− � +L Fp +v,t ) (Z, Z′) +���� +C ℓ(M) +⩽ C. +(4.174) +For k large enough, set +T (r) +t,p = +(k − 1)! +2πi(−1)k−1r! +� +∂Vd1,d2 +e−λ +� +(k,r)∈Ik,r +ak +rBk +r (0, λ, p) dλ, +T (r) +v,t,p = 1 +r! +∂r +∂vr exp(−t � +L Fp +v,t ) − T (r) +t,p . +(4.175) +Then T (r) +t,p and T (r) +v,t,p do not depend on the choice on k. We denote by T (r) +t,p (Z, Z′) (resp. +T (r) +v,t,p(Z, Z′)) the smooth kernel of T (r) +t,p (resp. T (r) +v,t,p) with respect to dvTX. +Theorem 4.36. For r ∈ N, there is C > 0 such that for Z, Z′ ∈ Tx0X with |Z| , |Z′| ⩽ 1, +���T (r) +v,t,p(Z, Z′) +��� ⩽ Cv1/(m+1). +(4.176) +Theorem 4.37. For any r, ℓ ∈ N, there is C > 0 such that +��� exp(− � +L Fp +v,t )(0, 0) − +r +� +i=0 +T (i) +t,p (0, 0)vi��� +C ℓ(M) ⩽ Cvr+1. +(4.177) +For s > 0, let PFp +s (x, y) be the kernel of exp(−s � +L Fp +0,t ) with respect to dvTX. +For +multi-index r = (r1, · · · , rk)k∈N where r1, · · · rk ∈ N∗, like (4.139), we set +Tr,t,p = +� +△k +� +PFp +1−tk +∂rk � +L Fp +v,t +∂vrk +��� +v=0Ptk−tk−1 · · · ∂r1 � +L Fp +v,t +∂vr1 +��� +v=0PFp +t1 +� +(0, 0) +k +� +j=1 +dtj, +(4.178) +then by the Duhamel’s principle [4, § 2.7], we have +T (i) +t,p (0, 0) = +� +r1+···+rk=i +(−1)k +�k +j=1 rj! +Tr,t,p, +T (0) +t,p (0, 0) = e−σ +Fp +t +(4π) +m +2 . +(4.179) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +45 +Theorem 4.38. For i, j, ℓ ∈ N, there exists C > 0 such that +���Tr[j] +� +T (i) +t,p (0, 0) +���� +C ℓ(M) ⩽ C, +����p−nTrFp[T (i) +t,p (0, 0)] − +j +� +k=0 +p−kTr[k][T (i) +t,p (0, 0)] +���� +C ℓ(M) +⩽ Cp−j−1. +(4.180) +Note that there is singularity inside the integral of (4.178), so similar to the proof of +Theorem 4.26, here we use (4.143) when t = 1 to get Theorem 4.38. +4.9. The proof of Theorem 4.1. Now we begin to prove the main theorem of this +section. First, we deal with the case of t ⩾ 1. Recall the symbols [·]z in (1.26), [·]max in +(4.57) and +� � +B in (4.58), then we set +Ix0 = (4π) +m +2 p−nTrFp� +θ−1 +√ +t +� +� +B � +exp(−t � +L Fp +x0 )(0, 0) +�max�z +. +(4.181) +By (4.4), (4.47) and (4.61), we have +����p−n 1 +√pψ1/√ph +� +A′, gΩ•(X,Fp) +4t/p2 +� +− +� +X +I +���� +C ℓ(S) +⩽ Ceγt−√c1c2cεp. +(4.182) +Taking v = 1/p in (4.134), we get +��� exp(−t � +L Fp +x0 )(0, 0) − +r +� +i=0 +p−iS (i) +t,p (0, 0) +��� +C ℓ(M) ⩽ Cp−r−1eγt. +(4.183) +Put +ci(t) = +√ +2πiϕ +� +(4π) +m +2 θ−1 +√ +t +� +� +B � +0⩽j⩽i +Tr[i−j] +� +S (j) +t,p (0, 0) +�max +�z +. +(4.184) +By (4.150), (4.181), (4.182) and (4.183), we get (4.1) and (4.2) for t ⩾ 1. +Likewise, if we take +ci(t) = +√ +2πiϕ +� +(4π) +m +2 +� +� +B � +0⩽j⩽i +Tr[i−j] +� +T (j) +t,p (0, 0) +�max +�z +, +(4.185) +then by (4.4), (4.47), (4.155), (4.177) and (4.180), we see that (4.1) and (4.2) hold for +0 < t ⩽ 1. Note that ci(t) ̸= ci(t) in general, while when separating them with respect +to (1.2), they contain the same component in top degree of Λ(T ∗X), this gives +� +X +ci(t) = +� +X +ci(t) +(4.186) +for i ∈ N and t > 0. Hence for t > 0, we can always define ci(t) by (4.185). This +completes the proof. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +46 +5. The full asymptotics of the analytic torsion forms +In this section, when �ϑL is nondegenerate, we obtain the full asymptotics of the analytic +torsion forms T +� +T HM, gTX, ∇Fp, gFp� +as p → +∞. +This section is organized as follows. In § 5.1, we give the existence of the full asymptotic +expansion of torsion forms. The proof is divided into two key steps, involving large and +small values of the parameter t > 0. In § 5.2, we prove that the Γ-torsion forms share +the same asymptotic behavior as the torsion forms. In § 5.3, we prove that T1,t,p(0, 0) +does not contribute to the first two terms in the expansion of torsion forms. +5.1. The main result: a precise version of Theorem 0.1. For the enlarged fibration +� +M → �S as in § 1.4, by (1.21) we have +h +� �A′, gΩ•(X,Fp) +4t/p2 +��� +s=1 = h +� +A′, gΩ•(X,Fp) +4t/p2 +� ++ ds ∧ h∧� +A′, gΩ•(X,Fp) +4t/p2 +� +. +(5.1) +By Theorem 4.1, there exist smooth forms {�ci(t)}i∈N as in (4.1) for h +� �A′, gΩ•(X,Fp) +4t/p2 +� +, then +for some smooth forms {di(t)}i∈N, for t > 0 we have +�ci(t) = ci(t) + ds ∧ di(t). +(5.2) +According to (4.1), (5.1) and (5.2), for k, ℓ ∈ N, there is C > 0 such that for any t > 0, +����p−n−1ψ1/√ph∧� +A′, gΩ•(X,Fp) +4t/p2 +� +− +k +� +i=0 +� +X +di(t)p−i +���� +C ℓ(S) +⩽ Ce−(a−γ)tp−k−1, +(5.3) +moreover, +���� +� +X +dk(t) +���� +C ℓ(S) +⩽ Ce−(a−γ)t. +(5.4) +By (1.20), the restriction on M ×{1} of the operator � +L Fp +v,t on � +M is exactly the operator +in (4.157) by counting also the direction +∂ +∂s in the term − +√ +vtf α�cv(�ei) 1 +2p∇ +� +TX⊗Fp,u +fH +α +�ωFp(ei), +so we only need to add the following extra term in (4.157): +1 +2 +√ +vtds ∧ �cv(�ei) 1 +2p�ωFp(ei). +(5.5) +By (5.5), the degrees of +√ +t in all the terms of (4.157) that contain ds are no less than +1. Therefore, when 0 < t ⩽ 1, (5.3) and (5.4) can be replaced by +����p−n−1ψ1/√ph∧� +A′, gΩ•(X,Fp) +4t/p2 +� +− +k +� +i=0 +� +X +di(t)p−i +���� +C ℓ(S) +⩽ C +√ +tp−k−1, +���� +� +X +dk(t) +���� +C ℓ(S) +⩽ C +√ +t. +(5.6) +Now we state and prove the main result of this article. +Theorem 5.1. If �ϑL is nondegenerate, then for any i ∈ N, the integral +W L,ξ +i += − +√ +2πiϕ +� +∞ +0 +di(t)dt +t ∈ Ω•(M, o(TX)) +(5.7) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +47 +is well defined. Moreover, for k, ℓ ∈ N, there is C > 0 such that as p → +∞, we have +����p−n−1ψ1/√pT +� +T HM, gTX, ∇Fp, gFp� +− +k +� +i=0 +� +X +W L,ξ +i +p−i +���� +C ℓ(S) +⩽ Cp−k−1. +(5.8) +Also, for differential forms γi defined in (3.14), we have +dW L,ξ +i += +� +X +� +e +� +TX, ∇TX� +γi +� +. +(5.9) +Proof. By Theorem 3.6, we have H•(X, Fp) = 0 for p ∈ N∗ large enough, then by (1.24), +T +� +T HM, gTX, ∇Fp, gFp� += − +� +∞ +0 +h∧� +A′, gΩ•(X,Fp) +t +�dt +t . +(5.10) +Using the change of variable t → 4t +p2 on the right hand side of (5.10), the integral becomes +T +� +T HM, gTX, ∇Fp, gFp� += − +� +∞ +0 +h∧� +A′, gΩ•(X,Fp) +4t/p2 +�dt +t . +(5.11) +By (5.4) and (5.6), the form W L,ξ +i +in (5.7) is well-defined, and we have +pk+1 +����p−n−1ψ1/√pT +� +T HM, gTX, ∇Fp, gFp� +− +k +� +i=0 +� +X +W L,ξ +i +p−i +���� +C ℓ(S) +⩽ +� 1 +0 +C +√ +tdt +t + +� +∞ +1 +Ce−atdt +t ⩽ C, +(5.12) +from which we get (5.8). And (5.9) follows directly from (1.25), (3.14) and (5.8). +□ +5.2. The asymptotics of the Γ-torsion forms. Let � +M → S be a smooth fibre bundle +with fibre � +X, and let Γ be a discrete group acting fibrewise freely and properly discon- +tinuously on � +M such that the Γ\� +M = M. Let �π: � +M → M be the obvious projection. +Let F be a Hermitian vector bundle on M. +If Q(�x, �x′) be a continuous fibrewise +kernel acting on Ω•(� +M, �π∗F) that commute with Γ, we can define its Von Neumann +Γ-supertrace. Note that Trs[Q(x, x)] is Γ-invariant that it descends to M, we define +TrΓ +s [Q] as the fibrewise integral of Trs[Q(x, x)] on M: for each b ∈ S, let � +Xb and Xb be +the corresponding fibres in � +M and M, then we have an isomorphism Γ\ � +Xb ∼= Xb, let +Xb ⊂ � +Xb be an associated fundamental domain, we have +Let F be a Hermitian vector bundle on M. If Q is an operator acting on C ∞(� +M, �π∗F) +with a smooth fibrewise kernel Q(�x, �x′) for (�x, �x′) ∈ � +M ×S � +M. If Q commutes with Γ, +then Tr[Q(x, x)] is Γ-invariant and it descends to a function on M. For each b ∈ S, +let �Xb and Xb be the corresponding fibres in � +M and M, then we have an isomorphism +Γ\ �Xb ∼= Xb, and the Von Neumann Γ-trace of Q is given by +TrΓ[Qb] = +� +Xb +Tr[Q(x, x)]dvXb(x). +(5.13) +We will now apply the formalism of the previous sections. +Note that we can lift +geometric data (T HM, gTX, Fp, gFp)p∈N∗ on M to (T H � +M, g�π∗TX, �π∗Fp, g�π∗Fp)p∈N∗ on � +M. +For t > 0, we can define h∧,Γ(A′, gΩ(X,�π∗Fp) +t +) ∈ Ω•(S) by the same formula as in (1.21), +by replacing the supertrace by the corresponding Γ-supertrace TrΓ +s . We assume that the + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +48 +nondegeneracy condition (3.16) holds. As observed by Bismut-Ma-Zhang [9, § 6.6], when +the nondegeneracy condition (3.16) holds, the spectral gap property (3.17) is still valid +over � +M, therefore, for p ∈ N∗ large enough, as t → +∞, |h∧,Γ(A′, gΩ•(X,�π∗Fp) +t +)| = O(e−p2t), +then we can define the following Γ-torsion form similar to (1.24) and (5.10): +T Γ� +T HM, gTX, ∇Fp, gFp� += − +� +∞ +0 +h∧,Γ� +A′, gΩ•(X,�π∗Fp) +t +�dt +t ∈ Ωeven(S). +(5.14) +In particular, if � +X is the universal covering of X, the Γ-torsion is also called the L2- +torsion, denoted by TL2� +T HM, gTX, ∇Fp, gFp� +. For the definition of Γ-torsion forms of +general bundles whose Novikov-Shubin invariant is positive, see [1] for more details. +Theorem 5.2. Under the nondegeneracy condition (3.16), there is a > 0 such that for +ℓ ∈ N, there is C > 0 such that as p → +∞, +���T Γ� +T HM, gTX, ∇Fp, gFp� +− T +� +T HM, gTX, ∇Fp, gFp���� +C ℓ(S) ⩽ Ce−ap. +(5.15) +In particular, the Γ-torsion forms verify the same asymptotic expansion as in (5.8): +����p−n−1ψ1/√pT Γ� +T HM, gTX, ∇Fp, gFp� +− +k +� +i=0 +� +X +W L,ξ +i +p−i +���� +C ℓ(S) +⩽ Cp−k−1. +(5.16) +Proof. For �x0 ∈ � +M, set M Fp +�x0 = �π∗M Fp +�π(�x0) where M Fp +�π(�x0) is defined in (4.46), we clearly +have +exp(−tM �π∗Fp +�x0 +)(0, 0) = �π∗� +exp(−tM Fp +�π(�x0))(0, 0) +� +. +(5.17) +Since the spectral estimate (3.17) holds over � +M, by (4.4) and (4.47), we see that for +ℓ ∈ N, there is C > 0 such that for p ∈ N∗, we have +���Trs +� +exp(−tM�π∗Fp)(�x0, �x0) − exp(−tM �π∗Fp +�x0 +)(0, 0) +���� +C ℓ(� +M) ⩽ Ce−(a−γ)t− c2 +2t −√γc2p. (5.18) +By (4.47), (5.17) and (5.18), we obtain +���Trs +� +exp(−tM�π∗Fp)(�x0, �x0) +� +− Trs +� +exp(−tMFp)(�π(�x0), �π(�x0)) +���� +C ℓ(� +M) +⩽ Ce−(a−γ)t− c2 +2t −√γc2p. +(5.19) +Similar to (4.4), we set +1 +√pψ1/√phΓ� +A′, gΩ•(X,Fp) +4t/p2 +� += +√ +2πiϕ +� +θ−1 +√ +tTrΓ +s +� +exp(−tM�π∗Fp) +��z +. +(5.20) +By (4.4), (5.19) and (5.20), we have +1 +√pψ1/√p +���hΓ� +A′, gΩ•(X,Fp) +4t/p2 +� +− h +� +A′, gΩ•(X,Fp) +4t/p2 +���� +C ℓ(S) ⩽ Ce−(a−γ)t− c2 +2t −√γc2p. +(5.21) +As in the argument before Theorem 1.6, we apply (5.21) to the enlarged manifolds +� +� +M → ��S and � +M → �S, when restricting to s = 1, like (5.1), we get +p−1ψ1/√p +���h∧,Γ� +A′, gΩ•(X,Fp) +4t/p2 +� +− h∧� +A′, gΩ•(X,Fp) +4t/p2 +���� +C ℓ(S) ⩽ Ce−(a−γ)t− c2 +2t −√γc2p. +(5.22) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +49 +From (5.11) and (5.14), we see that for some c > 0, +���ψ1/√pT Γ� +T HM, gTX, ∇Fp, gFp� +− ψ1/√pT +� +T HM, gTX, ∇Fp, gFp���� +C ℓ(S) +⩽ +� +∞ +0 +Cpe−(a−γ)t− c2 +2t −√γc2pdt +t ⩽ Ce−cp, +(5.23) +which gives (5.15). By (5.8) and (5.15), we get (5.16). +□ +5.3. A reduction for W1. In this subsection, we give a reduction formula for W0, W1, +under the assumption that dimC ξ = 1 and F = C (mainly used in (5.36)). By (5.2) and +(5.7), let us compute c0(t) and c1(t) first. +By (4.185), we get +� +X +� +c0(t) + p−1c1(t) +� += (4π) +m +2 +� +X +� � +� +B � +Tr[0] + p−1Tr[1] +�� +T (0) +t,p (0, 0) +�max ++p−1Tr[0] +� +T (1) +t,p (0, 0) +�max +�z +. +(5.24) +Lemma 5.3. We have +� +X +� � +� +B +Tr[0] +� +T (1) +t,p (0, 0) +�max +�z += 0. +(5.25) +Proof. Put +σFp +t += − 1 +4 +� +RTXei, ej +� +�ei ∧ �ej − 1 +4pωFp,2 + +t +4p2 +���ωFp��2 + t +4p�ωFp,2 +− +√ +t +2p ∇ +� +TX⊗Fp,u�ωFp − 1 +2pzωFp. +(5.26) +By (4.178) and (4.179), we get +Tr[0] +� +T (1) +t,p (0, 0) +� += −Tr[0] +� � 1 +0 +� +TX +PFp +1−t1(0, Z)∂ � +L Fp +v,t,Z +∂v +���� +v=0 +PFp +t1 (Z, 0)dZdt1 +� += −Tr[0] +� � 1 +0 +� +TX +(16π2(1 − t1)t1)− m +2 e− +|Z|2 +4(1−t1) −(1−t1)σ +Fp +t ∂ � +L Fp +v,t,Z +∂v +���� +v=0 +e− |Z|2 +4t1 −t1σ +Fp +t dZdt1 +� +. +(5.27) +For 1 ⩽ i, j ⩽ m, we set +at,ij = ZiZj +�� +RTX +x0 ∂i, ∂j +� +− t +� �RTX +x0 ∂i, ∂j +� +− t +2pωFp,2 +x0 +� +∂i, ∂j +�� +, +(5.28) +which is anti-symmetric for i, j. By (4.163), (4.164) and (5.28) , we get +� +i +e− +|Z|2 +4(1−t1)Y (1) +0,t,i∂ie− |Z|2 +4t1 = − 1 +2t1 +� +i,j +e− +|Z|2 +4(1−t1) at,ije− |Z|2 +4t1 = 0. +(5.29) +By (4.164), QFp +t (Z) is a homogeneous polynomial of Z with degree 1, so we have +� +Tx0X +e− +|Z|2 +4(1−t1)QFp +t (Z)e− |Z|2 +4t1 dZ = 0. +(5.30) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +50 +Now we consider the term QFp +t +′, we claim that for 0 ⩽ t1 ⩽ 1, we have +� +X +� � +� +B +Tr[0] +� +e−(1−t1)σ +Fp +t QFp +t +′e−t1σ +Fp +t +�max�z += 0. +(5.31) +Recall QFp′, σFp and θa given in (4.70), (4.137), and (4.156) respectively, then +tθ +−1 +√ +tσFpθ√ +t = σFp +t , +tθ +−1 +√ +tQFp′θ√ +t = QFp +t +′, +(5.32) +which gives +t− m +2 +� +X +� +θ−1 +√ +t +� +� +B +Tr[0] +� +e−(t−t1t)σFptQFp′e−t1tσFp�max�z += +� +X +� � +� +B +Tr[0] +� +e−(1−t1)σ +Fp +t QFp +t +′e−t1σ +Fp +t +�max�z +. +(5.33) +Therefore, instead of (5.31), we only need to prove that for 0 ⩽ t′ ⩽ t, we have +� +X +� � +� +B +Tr[0] +� +e−(t−t′)σFptQFp′e−t′σFp�max�z += 0. +(5.34) +The algebra of Toeplitz operators is non-commutative, while in (5.34), we only need +to get the first term of asymptotic traces, by Theorem 2.2, when dimC ξ = 1, we can +formally treat them as commutative: for k ∈ N∗ and Toeplitz operators T1, · · · , Tk, if +i1, · · · , ik is any permutation of {1, · · · , k}, we have Tr[0][T1 · · · Tk] = Tr[0][Ti1 · · · Tik]. +Then, by (4.67) and (4.70), QFp′ works like an odd derivation: for Toeplitz operator +values differential forms {ωℓ}ℓ=0,1 on M, we have +Tr[0] +� +QFp′(ω0 ∧ ω1) +� += Tr[0] +� +(QFp′ω0) ∧ ω1 +� ++ (−1)deg ω0Tr[0] +� +ω0 ∧ (QFp′ω1) +� +, +(5.35) +therefore, we get +Tr[0] +� +e−(t−t′)σFptQFp′e−t′σFp� += Tr[0] +� +e−(t−t′)σFp · tQFp′(−t′σFp) · e−t′σFp� += Tr[0] +� +tQFp′(−t′σFp) · e−tσFp� += t′Tr[0] +� +QFp′� +e−tσFp�� +. +(5.36) +In (5.34), we only need terms of top degree in Λ(T ∗X)�⊗Λ(� +T ∗X). However, by (4.70), +Q′e−tσFp can never be of top degree in Λ(T ∗X)�⊗Λ(� +T ∗X). We finish the proof. +□ +For any a ∈ R[z]�⊗R[ds]�⊗Λ(T ∗M) with the form a = α1 + zα2 + dsα3 + z ∧ ds ∧ α4 +where αi ∈ Λ(T ∗M), we set [a]z∧ds = α4. Put +ηFp +t +=σFp +t ++ +√ +t +2 ds 1 +2p�ωFp = −1 +4 +� +RTXei, ej +� +�ei ∧ �ej − 1 +4pωFp,2 + +t +4p2 �ωFp(ei)2 ++ t +4p�ωFp,2 − +√ +t +2p ∇ +� +TX⊗Fp,u�ωFp − 1 +2pzωFp + +√ +t +2 ds 1 +2p�ωFp. +(5.37) +Proposition 5.4. For i = 0, 1, we have +W L,ξ +i += +� +∞ +0 +� � +� +B +Tr[i] +� +exp(−ηFp +t ) +�max�z∧dsdt +t . +(5.38) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +51 +Proof. By (5.25), T (1) +t,p (0, 0) does not contribute to the right hand side of (5.24). By +(4.164), (4.178), (4.179), (5.24) and (5.26), we get +c0(t) + p−1c1(t) = (4π) +m +2 +� � +� +B � +Tr[0] + p−1Tr[1] +�� +T (0) +t,p (0, 0) +�max +�z += +� � +� +B � +Tr[0] + p−1Tr[1] +�� +exp(−σFp +t ) +�max +�z +. +(5.39) +By (5.2), (5.5), (5.37) and (5.39), we have +d0(t) + p−1d1(t) = +� � +� +B � +Tr[0] + p−1Tr[1] +�� +exp(−ηFp +t ) +�max +�z∧ds +. +(5.40) +By (5.7) and (5.40), we obtain (5.38). +□ +Remark 5.5. By Theorem 2.4, (5.38) for i = 0 is just the formula for W0 given by +Bismut-Ma-Zhang [9, (9.89)]. By (5.38), we can get an explicit formula for W1 for i = 0 +and the product formula of Toeplitz operators (2.11) for higher order terms (see [25]). +While direct expansion is very complicated and less illuminating. In the next section, +when G is a reductive Lie group, we will rewrite (5.40) in a concise and clear form. +6. The case when the structure group of PG is reductive +In this section, we give an explicit formula for W L,ξ +1 +in (5.8) when the structure group +G of PG is a reductive linear Lie group, dimC ξ = 1 and F = C trivial. +This section is organized as follows. In § 6.1, we introduce the reductive Lie group G, +its compact form U and complexification GC. In § 6.2, we introduce the reduction of the +flat principle bundle PG → M. In § 6.3, we express certain Lie derivative operators as +Toeplitz operators (see Definition 2.1). In § 6.4, for A ∈ u, we get the asymptotics trace +of e +A +p on H(0,0)(N, Lp ⊗ ξ) using the Kirillov formula. In § 6.5, we obtain an explicit +formula for W L,ξ +1 +. In § 6.6, we discuss a special case of W L,ξ +1 +when N is an adjoint orbit. +In § 6.7, we compute W L,ξ +1 +for some concrete examples. +6.1. Reductive Lie groups. Let G be a connected real reductive Lie group with Lie +algebra g, and let Θ ∈ Aut(G) be a Cartan involution of G. Let K ⊂ G be the fixed +point set of Θ in G, which is a maximal compact subgroup of G, and let k be its Lie +algebra. Let p ⊂ g be the eigenspace of Θ associated with the eigenvalue −1. Then we +have the Cartan decomposition +g = p ⊕ k. +(6.1) +Let U be the compact form of G with Lie algebra +u = +√ +−1p ⊕ k. +(6.2) +Let GC be the complexification of G with Lie algebra +gC = g ⊗R C = u ⊗R C = uC +(6.3) +Then GC is also the complexification of U, and U is a maximal compact subgroup of GC. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +52 +6.2. Principle bundles with reductive structure groups. Now we use the same +assumptions as in § 3.1. In particular, p: PG → M is a principal flat G-bundle where +G is a reductive Lie group given in § 6.1. Since G/K is contractible, there are smooth +sections of the fibre bundle PG ×G G/K and each corresponds to a reduction of the +principal G-bundle p: PG → M to a principal K-bundle p: PK → M. +Set gr = PG ×G g ∼= PK ×K g. Then (6.1) gives +gr = pr ⊕ kr. +(6.4) +We denote the flat connection on PG by a g-valued flat connection form θg. +By +projection of θg on p and k with respect to the decomposition (6.1), we get +θg = θp + θk, +(6.5) +then θk can be viewed as a connection form on PK and θp is a section of T ∗X ⊗ pr. +Let Θk be the curvature of θk. Since θg is flat, we get +Θk = −1 +2 +� +θp, θp� +(or Θk = −θp,2), +� +d + θk, θp� += 0. +(6.6) +Let ∇gr,u be the connection on gr induced by θk, which preserves the splitting (6.4). +6.3. Moment map and a class of Toeplitz operators. We use the assumptions and +notation of § 2.1 and § 3.1. In particular, N denotes a compact complex manifold of +complex dimension n and (L, gL) is a holomorphic Hermitian line bundle on N. Let ∇L +be the Chern connection of L with c1(L, gL) = ω. +We assume that the group U acts holomorphically on N. If A ∈ u, let AN be the +corresponding holomorphic vector field on N with its (1, 0) part AN,(1,0) and (0, 1) part +AN,(0,1)), hence A ∈ u → −AN is a morphism of Lie algebras. +We assume further that the action of U on N lifts to a holomorphic unitary action on +L. Then ω is a U-invariant form. If A ∈ u, let LA denote the Lie derivative of A on the +smooth sections of L. If A ∈ u, the Kostant formula [4, Definition 7.5] gives +LA = ∇L +A − 2πi⟨µL, A⟩, +(6.7) +where µL is a moment map µ: N → u∗ such that, if u ∈ U, x ∈ N, +µL(ux) = tAd−1(u)µ(x), +d⟨µL, A⟩ − iANω = 0. +(6.8) +Let (ξ, gξ) be another holomorphic Hermitian line bundle on N and U acts holomorphi- +cally and unitarily on ξ with moment map µξ. +Recall that C ∞(N, Lp ⊗ ξ) is equipped with the L2-product induced by (gL, gξ), the +K¨ahler metric is given by gTRN = ω(·, J·) and Pp is the orthogonal projection operator +from C ∞(N, Lp ⊗ ξ) to H(0,0)(N, Lp ⊗ ξ) as in § 3.2. Then U acts on H(0,0)(N, Lp ⊗ ξ) +unitarily. This action can be extended to a holomorphic GC action. +Let gTN be the metric on TN induced by gTRN with the corresponding Chern connec- +tion ∇TN and its curvature RTN. Denote the (1, 0) part of ∇TN by ∇TN ′. If A ∈ u, +∇TN ′AN,(1,0) is a skew-adjoint endomorphism of TN. +The metric gTN induces a Hermitian metric gdet TN on the line bundle det TN. Let +∇det TN denote the corresponding Chern connection on det TN, then +c1(det TN, gdet TN) = − 1 +2πiTrTN[RTN]. +(6.9) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +53 +The group U acts holomorphically and unitarily on det TN. The associated moment +map µdet TN : N → u∗ is given by +2πi⟨µdet TN, A⟩ = TrTN[∇TN ′AN,(1,0)]. +(6.10) +Theorem 6.1 ([9, Theorem 3.1]). If A ∈ u, the following identity holds: +LA +�� +H(0,0)(N,Lp⊗ξ) = −2πiPp⟨pµL + µξ + µdet TN, A⟩Pp. +(6.11) +Proof. By (3.7) and (6.10), we see that (6.11) is a special case of (3.11). +□ +6.4. The Kirillov formula. For a matrix B, if supλ∈Spec(B)|λ| < 2π, we set +Td(B) = det +� +B +1 − e−B +� +. +(6.12) +If B is self-adjoint, no condition on B is necessary to define Td(B). From the above, for +A ∈ gC such that |A| small, the following form is well defined: +TdA(N) = Td +� +− RTN +2πi + ∇TN ′AN,(1,0)� +. +(6.13) +The following form of the Kirillov formula is given in [9, Theorem 3.5]. +Theorem 6.2. For p large enough, if A ∈ gC and |A| small, +TrH(0,0)(N,Lp⊗ξ)(eA� += +� +N +TdA(N) exp +� +2πi⟨pµL + µξ, A⟩ + pc1(L, gL) + c1(ξ, gξ) +� +. (6.14) +For A ∈ gC, let RL(A) be the integral of Duistermaat-Heckman given by +RL(A) = +� +N +exp +� +2πi⟨µL, A⟩ + c1(L, gL) +� +. +(6.15) +Formally, for p ∈ N∗, let ξ +1 +p be the p-th root of L and (det TN) +1 +2p be the 2p-th root of +det TN at the level of cohomology and moment map, then +c1 +� +ξ +1 +p � += 1 +pc1(ξ, hξ), +c1 +� +(det TN) +1 +2p � += 1 +2pc1(det TN, gdet TN), +µ +ξ +1 +p = 1 +pµξ, +µ +(det TN) +1 +2p = 1 +2pµdet TN. +(6.16) +Proposition 6.3. If A ∈ gC, for p large enough, we have +p−nTrH(0,0)(N,Lp⊗ξ)� +e +A +p � += R +L⊗ξ +1 +p ⊗(det TN) +1 +2p (A) + O +� +p−2� +. +(6.17) +Proof. Let Q(z) be a holomorphic function near 0 given by +Q(z) = ln +z +1 − e−z = z +2 − z2 +24 + · · · . +(6.18) +By (6.12) and (6.18), we get Td(B) = exp +� +Tr +� +Q(B) +�� +. From (6.9) and (6.13), we have +Td A +p +� +N +� += Td0(N) +� +1 + p−1Tr +� +Q′(iRTN/2π)∇TN ′AN,(1,0)�� ++ O +� +p−2� += +� +1 + 1 +2c1(det TN) + · · · +�� +1 + p−1Tr +� +Q′(iRTN/2π)∇TN ′AN,(1,0)�� ++ O +� +p−2� +. +(6.19) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +54 +Note that Q′(0) = 1/2. By (2.5), (6.10), (6.14) and (6.19), we get +p−nTrH(0,0)(N,Lp⊗ξ)� +e +A +p � += +� +N +exp +� +2πi⟨µL, A⟩ +�cn +1(L) +n! ++ 1 +2p +� +2πi⟨µdet TN, A⟩ + 4πi⟨µξ, A⟩ +� +exp +� +2πi⟨µL, A⟩ +�cn +1(L) +n! ++ 1 +2p +� +c1(det TN) + 2c1(ξ) +� +exp +� +2πi⟨µ, A⟩ +�cn−1 +1 +(L) +(n − 1)! + O +� +p−2� +. +(6.20) +By (6.16), we could rewrite the right hand side of (6.20) as +� +N +exp +� +2πi +� +µ +L⊗ξ +1 +p ⊗(det TN) +1 +2p , A +� ++ c1 +� +L ⊗ ξ +1 +p ⊗ (det TN) +1 +2p �� ++ O +� +p−2� +, +(6.21) +which is exactly (6.17). +□ +6.5. An explicit formula for W L,ξ +1 +. For p ∈ N∗, we denote by ρ the holomorphic +representation GC → End +� +H(0,0)(N, Lp ⊗ ξ) +� +. Similar to (0.3), we have +Fp = PK ×K H(0,0)(N, Lp ⊗ ξ). +(6.22) +Let gFp be the Hermitian metric on Fp induced by the L2-metric on H(0,0)(N, Lp ⊗ ξ). +The connection form θg induces a flat connection ∇Fp on Fp, and θk gives a unitary +connection ∇Fp,u on Fp. Recall that θp is a section of T ∗M ⊗ pr. By (6.5), we get +∇Fp,u = ∇Fp − ρθp +(6.23) +where ρθp ∈ Ω1(M, End(Fp)). By (1.6) and (6.23), we have +ω +� +∇Fp,u, gFp� += −2ρθp, +RFp,u = −ρθp,2. +(6.24) +Note that (6.23) and (6.24) are reductive version of Theorem 3.1. Also, by (6.11) and +(6.24), for a local orthonormal frame {ei}m +i=1 of TX, the condition (3.16) becomes +m +� +i=1 +��⟨2πµL, +√ +−1�θp(ei)⟩ +��2 > 0. +(6.25) +Let Sg be the symmetric algebra of g, which can be viewed as the algebra of real +differential operators with constant coefficients on g, and we denote by Sg its formal +completion. Let �θp be the restriction of θp on � +TX. For t ⩾ 0, recall that σt is a section +of Λ(T ∗M)�⊗Λ(� +T ∗X) ⊗ Sgr given in (0.8). +Now we state and prove the main result of this section. +Theorem 6.4. The first two terms of the asymptotic torsion (5.8) are given by +p−n−1ψ1/√pT (T HM, gTX, ∇Fp, gFp) = W L,ξ +0 ++ p−1W L,ξ +1 ++ O +� +p−2� += +√ +2πiϕ +� +∞ +0 +� +� +B θp ∧ �θp +2 +exp(−σt)R +L⊗ξ +1 +p ⊗(det TN) +1 +2p (0) dt +√ +t, +(6.26) +which also could be formally written in the following form: +W L,ξ +0 ++ p−1W L,ξ +1 += W L⊗ξ +1 +p ⊗(det TN) +1 +2p +0 ++ O +� +p−2� +. +(6.27) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +55 +Proof. Let us first explain (6.26) in more detail. Note that θp∧ � +θp +2 +exp(−σt) is a smooth +section of Λ(� +T ∗X)�⊗Λ(T ∗M) ⊗ Sgr, and it acts naturally on R +L⊗ξ +1 +p ⊗(det TN) +1 +2p (·), which +is a function on a neighborhood of 0 ∈ gC (see [9, § 1.4] for more details). +When +evaluating at 0 ∈ gC, we see that θp∧ � +θp +2 +exp(−σt)R +L⊗ξ +1 +p ⊗(det TN) +1 +2p (0) is a smooth section +of Λ(� +T ∗X)�⊗Λ(T ∗M). +Now we come back to the proof of (6.26). We plug (6.23) and (6.24) into (5.37), then +by (5.40) and (6.17), we get +d0(t) + p−1d1(t) = +� +� +B √ +tθp ∧ �θp +2 +exp(−σt)R +L⊗ξ +1 +p ⊗(det TN) +1 +2p (0) + O +� +p−2� +, +(6.28) +which gives (6.26) together with (5.38). +□ +6.6. A special case for coadjoint orbit. Now we discuss a special case of Theorem +6.4 when N is given by a coadjoint orbit. +For the compact connected Lie group U given in § 6.1 with Lie algebra u. For λ ∈ u∗, +denote by Oλ = U · λ the orbit of coadjoint action with the natural symplectic form ωλ: +for A, B ∈ u and f ∈ Oλ, +ωλ(AOµ, BOµ)f = 1 +2π⟨f, [A, B]⟩, +(6.29) +then the associated moment map µλ verifies that 2πµλ is the inclusion +2πµλ: Oλ ֒→ u∗. +(6.30) +Similar to (6.15), for A ∈ u, set +Rλ(A) = +� +Oλ +exp +� +2πi⟨µλ, A⟩ + ωλ +� +. +(6.31) +We fix a maximal torus T of U with Lie algebra t. We note that if λ ∈ u∗ is regular, +we have Oλ ∼= U/T. The integral lattice Λ ⊂ t is defined as the kernel of the exponential +map exp: t → T, and the real weight lattice Λ∗ ⊂ t∗ is defined by Λ∗ = Hom(Λ, 2πZ). +We fix a set of positive roots Φ+ ⊂ Λ∗, a positive open Weyl Chamber C+ and its closure +C+. By the Weyl character formula, finite-dimensional irreducible representations of U +are parameterized by Λ∗ ∩ C+. +Set r = [t, u], then we have u = t ⊕ r, and u∗ = t∗ ⊕ r∗. Hence we identify Λ∗ ∩ C+ +with a subset of u∗. For λ ∈ Λ∗ ∩ C+, let Vλ be the unique finite dimensional irreducible +representation of U with highest weight λ. +If λ ∈ Λ∗ ∩ C+, let (Lλ, gLλ) be the canonical prequantum line bundle on Oλ with +c1(Lλ, gLλ) = ωλ. By the Borel-Weil-Bott theorem [4, Theorem 8.8], we have an isomor- +phism H(0,0)(Oλ, Lλ) ∼= Vλ. +For λ ∈ Λ∗ ∩ C+ and τ ∈ Λ∗, in Theorem 6.4 we take +(N, L, ξ, H(0,0)(N, Lp ⊗ ξ), Fp) = (U/T, Lλ, Lτ, Vpλ+τ, PG ×G Vpλ+τ). +(6.32) +Set ̺u = 1 +2 +� +α∈Φ+ α, then det T(U/T) ∼= L2̺u. Hence, by (6.17) and (6.31) we have +p−nTrVpλ+τ� +e +A +p � += Rλ+p−1(̺u+τ)(A) + O +� +p−2� +. +(6.33) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +56 +Note that we could also derive (6.33) from the Kirillov character formula [4, Theorem +8.4]: for any A ∈ u, let ad(A) be the morphism of u given by ad(A)B = [A, B], then +TrVλ� +eA� += j− 1 +2(A)Rλ+̺u(A), +for j(A) = det +�sinh(ad(A)/2) +ad(A)/2 +� +. +(6.34) +Since ad(A) is anti-symmetric, j− 1 +2(·) is an even function of A ∈ u and j− 1 +2(0) = 1, we +have j− 1 +2(A/p) = 1 + O(p−2). Then by (6.31) and (6.34), we recover (6.33): +p−nTrVpλ+τ� +e +A +p � += p−n� +1 + O(p−2) +� +Rpλ+τ+̺u +�A +p +� += Rλ+p−1(̺u+τ)(A) + O +� +p−2� +. (6.35) +By (6.25) and (6.30), θg is nondegenerated with respect to λ if and only if for u ∈ U, +m +� +i=1 +��⟨u · 2πλ, +√ +−1�θp(ei)⟩ +��2 > 0, +(6.36) +and when (6.36) holds, we set +W λ = +√ +2πiϕ +� +∞ +0 +� +� +B θp ∧ �θp +2 +exp(−σt)Rλ(0) dt +√ +t, +(6.37) +then we have the following corollary from (6.33) exactly as (6.27) follows from (6.17): +Corollary 6.5. If θg is nondegenerated with respect to λ ∈ Λ∗ ∩C+ and τ ∈ Λ∗, we have +p−n−1ψ1/√pT (T HM, gTX, ∇Fp, gFp) = +� +X +W λ+p−1(̺u+τ) + O +� +p−2� +. +(6.38) +6.7. The W1 for SL(2, C). In this subsection, we mainly follow [9, § 8.7]. +Now, we +assume that G = SL(2, C). The Cartan involution is Θ: g → (g∗)−1, then K = SU(2). +Note that U = SU(2) × SU(2) is a compact form of G = SL(2, C), in which K = SU(2) +embeds by the diagonal embedding. +A maximal torus T = S1 in SU(2) is the 1-parameter group exp(2πtk), t ∈ S1 = R/Z +where k is a generator of the Lie algebra of t, and 2πk is a coroot in T. Then +Φ+ = +�k +π +� +, +̺u = k +2π, +(6.39) +and the set Λ∗ ∩ C+ of dominant weights is just given by N/2π · k. +For a ∈ N, the coadjoint orbit Na of ak/2π ∈ t∗ for SU(2) can be identified with a +point for a = 0, with CP1 for a > 0. The orbit Oa carries a canonical line bundle La. +For a, b ̸= 0, put Oa,b = Oa × Ob. Let q1, q2 be the obvious projections from Oa,b on +Oa and Ob. For a, b ∈ N, set La,b = q∗ +1La × q∗ +2Lb. Then g ∈ SL(2, C) acts on Oa,b by the +map (z, z′) → (gz, (Θg)z′), and the action of SL(2, C) lifts to La,b. Moreover, +H(0,0)(Oa,b, La,b) ∼= SymaC2 ⊗ SymbC +2, +(6.40) +where C2 is the tautological representation of SL(2, C), C +2 is its complex conjugate +representation and Symk means the k-th symmetric power. +Now we give the change of notation concerning the previous sections. +Note that +G/K = H3, the hyperbolic space with constant sectional curvature −4. +Let Γ be +a torsion-free discrete cocompact subgroup of SL(2, C). Then Γ\H3 is a compact 3- +dimensional hyperbolic manifold. For a, b ∈ N∗, a′, b′ ∈ Z with a ̸= b, we set +(S, X, PG, PK, N, L, ξ) = ({pt}, Γ\H3, Γ\(H3 × G), Γ\G/K, Oa,b, La,b, La′,b′). +(6.41) + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +57 +As in (6.22), we have +Fp = Γ\G ×K H(0,0)(N, Lp ⊗ ξ) ∼= Γ\G ×K +� +Symap+a′C2 ⊗ Symbp+b′C +2� +, +(6.42) +then we could work under the settings of Corollary 6.5. +Proposition 6.6. The degeneracy condition (6.36) is equivalent to a ∈ N∗, b = 0 or +a, b ∈ N∗, a ̸= b, and we have +W La,0 +0 += 2 +πa2, +W +La,b +0 += +� +2 +3π(3a2b − b3), a > b, +2 +3π(3ab2 − a3), a < b. +(6.43) +The first equality in (6.43) was the main result obtained by M¨uller [34, Theorem 1.1] +and Bismut-Ma-Zhang [9, Theorem 8.1] gave both equalities in (6.43). We note that in +[34], the curvature of H3 is −1 instead of −4. +For a ∈ N∗, a′ ∈ Z, b = b′ = 0 or a, b ∈ N∗, a ̸= b, a′, b′ ∈ Z, set +W +La,0,La′,0 +1 += 4 +πa(a′ + 1), +W +La,b,La′,b′ +1 += +� +2 +π +� +(a2 − b2)(b′ + 1) + 2ab(a′ + 1 +� +, a > b, +2 +π +� +(b2 − a2)(a′ + 1) + 2ab(b′ + 1 +� +, a < b. +(6.44) +Theorem 6.7. For a, b ∈ N∗, a′, b′ ∈ Z, a ̸= b, as p → +∞, we have +p−3T (Γ\H3, Fp) = +� +W +La,b,La′,b′ +0 ++ p−1W +La,b,La′,b′ +1 +� +Vol(Γ\H3) + O +� +p−2� +. +(6.45) +For a ∈ N∗, a′ ∈ Z, b, b′ = 0, as p → +∞, +p−2T (Γ\H3, Fp) = +� +W +La,0,La′,0 +0 ++ p−1W +La,0,La′,0 +1 +� +Vol(Γ\H3) + O +� +p−2� +. +(6.46) +Proof. By (6.43), we get the first order expansion in (6.44). By Corollary 6.5 and (6.39), +we replace a, b with a + p−1(1 + a′), b + p−1(1 + b′) in W La,0 +0 +and W +La,b +0 +and evaluate the +coefficients of p−1, then we get the second orders in the expansions (6.44). +□ +Remark 6.8. By Theorem 5.2, we see that T (Γ\H3, Fp) −TL2(Γ\H3, Fp) = O(e−cp). 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M¨uller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), +no. 3, 233–305. +[33] W. M¨uller, Analytic torsion and R-torsion for unimodular representations, J. Amer. Math. Soc. 6 +(1993), no. 3, 721–753. + +TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS +59 +[34] W. M¨uller, The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric +and differential geometry, Progr. Math., vol. 297, Birkh¨auser/Springer, Basel, 2012, pp. 317–352. +[35] W. M¨uller and J. Pfaff, Analytic torsion and L2-torsion of compact locally symmetric manifolds, J. +Differential Geom. 95 (2013), no. 1, 71–119. +[36] M. Puchol, The asymptotics of the holomorphic torsion forms, to appear at J. London Math. Soc. +[37] D. Quillen, Superconnections and the Chern character, Topology 24 (1985), no. 1, 89–95. +[38] D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in +Math. 7 (1971), 145–210. +[39] K. Reidemeister, Homotopieringe und Linsenr¨aume, Abh. Math. Sem. Univ. Hamburg 11 (1935), +no. 1, 102–109. +[40] N. Savale. Asymptotics of the eta invariant. Comm. Math. Phys., 332(2):847–884, 2014. +[41] N. Savale. A Gutzwiller type trace formula for the magnetic Dirac operator. Geom. Funct. Anal., +28(5):1420–1486, 2018. +[42] M. Schlichenmaier, Deformation quantization of compact K¨ahler manifolds by Berezin-Toeplitz +quantization, Conf´erence Mosh´e Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer +Acad. Publ., Dordrecht, 2000, pp. 289–306. + diff --git a/M9E2T4oBgHgl3EQfqgii/content/tmp_files/load_file.txt b/M9E2T4oBgHgl3EQfqgii/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35ebd44addc0f1c633a0595dfeb7e495552c640a --- /dev/null +++ b/M9E2T4oBgHgl3EQfqgii/content/tmp_files/load_file.txt @@ -0,0 +1,2766 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf,len=2765 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='04040v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='DG] 10 Jan 2023 TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS QIAOCHU MA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This paper aims to study the asymptotic expansion of analytic torsion forms associated with a certain series of flat bundles {Fp}p∈N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We prove the existence of the full expansion and give a formula for the sub-leading term, while Bismut-Ma- Zhang [9] have studied the first order expansion and expressed the leading term as the integral of a locally computable differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Introduction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In the 1930s, the Reidemeister torsion was introduced by Reide- meister [39] and Franz [19] in their study of lens spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The Reidemeister torsion was the first homeomorphic invariant which is not homopoty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let (F, ∇F) be a unitary flat vector bundle over a compact manifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We assume that H•(X, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The Reidemeister torsion is a real number obtained by a simplicial complex with values in F associated with a triangulation of X, which turns out to be independent of the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Ray and Singer asked if there is an analytic interpretation of the Reidemeister torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' They defined their analytic torsion [38] as an alternating product of the regularized de- terminant of Hodge Laplacian and conjectured that it coincides with the Reidemeister torsion for unitary flat bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This conjecture was proved by Cheeger [16] and M¨uller [32] independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Bismut-Zhang [12] and M¨uller [33] simultaneously considered gen- eralizations of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' M¨uller [33] extended it to the case when F is unimodular, X is oriented and odd-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Bismut-Zhang [12] generalized it to any flat vector bundle with a Hermitian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In this paper, we study the asymptotic property of analytic torsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let {Fp}p∈N∗ be a certain family of flat vector bundles over M, we obtain the full asymptotic expansion of the analytic torsion of Fp when p → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let us now give some background on the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In [16, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3], Cheeger observed the relation between the Reidemeister torsion of a simplicial complex and the size of its torsion subgroup of homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By using this rela- tion and discussing the asymptotics of analytic torsions of quotients of symmetric spaces by a decreasing sequence of lattices in an underlying Lie group, Bergeron-Venkatesh [3] studied the growth of the size of torsion elements in the homology of an arithmetic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This was the first application of analytic torsion in arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In [34], M¨uller considered the asymptotics of analytic torsions for a family of flat bundles over a compact 3-dimensional hyperbolic manifold as a real analogue of Bismut- Vasserot’s result on the holomorphic torsion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Γ\\H3 be a compact 3-dimensional hyperbolic manifold with constant sectional curvatures −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For p ∈ N∗, put Fp = SympC2, where C2 is the flat bundle on Γ\\H3 associated with the tautological represen- tation of SL2(C) on C2 and Symp denotes the p-th symmetric power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let T (Γ\\H3, ∇Fp) 1 TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 2 be the the analytic torsion of Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Using Selberg’s trace formula, M¨uller [34] obtained lim p→+∞ p−2T (Γ\\H3, ∇Fp) = 2 πVol(Γ\\H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1) In [8] and [9], Bismut-Ma-Zhang gave a general construction of a family of flat vector bundles {Fp}p∈N∗ on any compact manifold and they expressed the asymptotics of ana- lytic torsions as an integral of a local computable differential form on the base manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Indeed, they worked in a general setting of the analytic torsion forms of Bismut-Lott [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Our main result is to extend Bismut-Ma-Zhang’s work to get a full expansion of torsions and give an explicit formula for the sub-leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we explain in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The existence of full expansion of torsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In [7], Bismut-Lott constructed the analytic torsion form as a family extension of the Ray-Singer torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let π: M → S be a fibration of manifolds with compact fiber X of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We set a metric gTX on the relative tangent bundle TX and a horizontal bundle T HM ⊂ TM such that TM = T HM ⊕ TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) For any flat vector bundle (F, ∇F) over M with a Hermitian metric gF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Bismut-Lott’s torsion form is a differential form T (T HM, gTX, ∇F, gF) ∈ Ωeven(S) and its 0-degree component T0(T HM, gTX, ∇F, gF) is the function which to b ∈ S assigns the Ray-Singer torsion of the fibre (Xb, gTXb) over b, computed using the flat bundle (F|Xb, ∇F|Xb, gF |Xb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let N be a compact K¨ahler manifold with dimC N = n, L a positive holomorphic line bundle and ξ a holomorphic vector bundle on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let G be a Lie group acting holomorphically on N and this action can be lifted to L and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let PG → M be a flat principal G-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set N = PG ×G N, we have a natural projection q: N → M and the real relative tangent bundle TRN = ker q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For p ∈ N∗, let Lp be the p-th tensor power of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let (F, ∇F) be another flat vector bundle on M with metric gF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put Fp = � PG ×G H(0,0)(N, Lp ⊗ ξ) � ⊗ F, (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3) where G acts naturally on H(0,0)(N, Lp⊗ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We summarize geometric settings as follows: We have a flat connection ∇Fp induced by the flat connection on PG and ∇F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We also TRN Lp ⊗ ξ = PG ×G (Lp ⊗ ξ) N N = PG ×G N PG ×G H(0,0)(N, Lp ⊗ ξ) X M S R•q∗ q π Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' denote PG ×G L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' PG ×G ξ) by L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Given a metric gTRN on TRN, it induces a volume form dvN ∈ Ω2n(N ) along the fibre together with PG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let gL (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' gξ) be a metic on L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ξ) over N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then the L2-metric on H(0,0)(N, Lp ⊗ ξ) given by (dvN, gL, gξ) and gF define a metric gFp on Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 3 In [9, Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13], Bismut-Ma-Zhang introduced a non-degeneracy condition for gL (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Under this condition, Bismut-Ma-Zhang [9, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3, § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10] proved that, for p ∈ N∗ large enough, we have H•(M, Fp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In the rest of this section, we always assume that L verifies the non-degeneracy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For a ∈ R, let ψa be the automorphism of Λ• (T ∗S) such that, if α ∈ Λk(T ∗S), then ψaα = akα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let o(TX) be the orientation line of TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Our main result is the following theorem (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' There are locally computable differential forms W L,ξ i ∈ Ω•(M, o(TX)) such that for any k, ℓ ∈ N, there exists C > 0 such that as p → +∞, we have ���p−n−1ψ1/√pT � T HM, gTX, ∇Fp, gFp� − k � i=0 p−i � X W L,ξ i ��� C ℓ(S) ⩽ Cp−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) We note that if dim X = m is odd, T � T HM, gTX, ∇Fp, gFp� ∈ Ωeven(S)/dΩodd(S) is a topological invariant for p ∈ N∗ large, by the anomaly formula of Bismut-Lott [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='24], it is independent on (T HM, gTX, gFp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Hence each W L,ξ i is a topological invariant for i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In [9, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32], Bismut-Ma-Zhang established Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 for k = 0 and gave an explicit formula for W L,ξ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let � M → S be a fibre bundle with fibre � X, and let Γ be a discrete group acting fibrewise freely and properly discontinuously on � M such that Γ\\� M = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have the Γ-torsion form T Γ� T HM, gTX, ∇Fp, gFp� ∈ Ωeven(S) as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) (see also [1], [24] and [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then T Γ� T HM, gTX, ∇Fp, gFp� has the same asymptotic expansion as in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4), indeed, we have the following stronger result (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The asymptotics of the two torsions differ only by an exponentially decay term: there is a > 0 such that for ℓ ∈ N, there is C > 0 that as p → +∞, we have ���T Γ� T HM, gTX, ∇Fp, gFp� − T � T HM, gTX, ∇Fp, gFp���� C ℓ(S) ⩽ Ce−ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5) Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2 was first established by Bismut-Ma-Zhang [9, § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6] when S = {pt}, a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If S = {pt}, X = Γ\\G/K, a compact locally symmetric manifold, and Fp is induced by multiples of the highest weight λ ∈ u∗ of an irreducible U-representation, where U is a compact form of G with lie algebra u, Bismut-Ma-Zhang [9, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12] showed that the nondegeneracy condition is the same to WU · λ ∩ t∗ = ∅, where WU is the Weyl group of U and t is the Lie algebra of a maximal torus in K, which is exactly the strong acyclic condition θΛ ̸= Λ [14, Propostion II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12], M¨uller-Pfaff [35, Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3] gave a new proof of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5) and showed that T Γ� gTX, ∇Fp, gFp� is a polynomial of p ∈ N∗, see also Liu [22, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3] for another proof of this polynomial property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' An explicit formula for W L,ξ 1 for reductive G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we explain another main result, an explicit formula for W L,ξ 1 when G is a connected reductive linear Lie group, rkξ = 1 and F = C trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let G be a connected reductive linear Lie group with Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let K ⊂ G be a maximal compact subgroup of G with Lie algebra k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have the Cartan decomposition g = p ⊕ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let U be a compact form of G with Lie algebra u = √−1p ⊕ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Recall that N is a compact complex manifold with dimC N = n and (L, gL) is a positive line bundle on N with the first Chern form c1(L, gL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We assume further that TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 4 U acts holomorphically on N and this action lifts to a holomorphic unitary action on L, then c1(L, gL) is a U-invariant form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let µL : N → u∗ be the associated moment map obtained from the Kostant formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let θg be the g-valued flat connection form on PG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let PK be a reduction of PG to K-principal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set gr = PK ×K g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By projection of θg on p and k with respect to the Cartan decomposition, we get θg = θp + θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For A ∈ u, let RL(A) be the Duistermaat-Heckman integral RL(A) = � N exp � 2πi⟨µL, A⟩ + c1(L, gL) � , (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) then we naturally extend RL(·) to a holomorphic function on gC ∼= u ⊗R C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Sg be the symmetric algebra of g and we denote by Sg its formal completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Canonically, Sg can be identified with the algebra of real differential operators with constant coefficients on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then Sg naturally acts on RL(·) (see also [9, § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let {ei}m i=1 be a local orthonormal frame of TX with dual frame {ei}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let � TX be another copy of TX and let �θp be the restriction of θp on � TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set |�θp|2 = m � i=1 ��θp(ei) �2 ∈ C ∞(M, S2gr), �θp,2 = 1 2�ei ∧ �ej[�θp(�ei), �θp(�ej)] ∈ C ∞(M, Λ2(� T ∗X) ⊗ gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) Let ∇gr,u be the connection on gr induced by θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For t ⩾ 0, let σt be a section of Λ•(T ∗M)�⊗Λ•(� T ∗X) ⊗ Sgr given by σt = −1 4 � RTXei, ej � �ei ∧ �ej − θp,2 + √ t∇ � T ∗X⊗gr,u�θp + t|�θp|2 + t�θp,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) Denote by � ˆB : Λ•(TM)�⊗Λ•(� TX) → Λ•(TM) the Berezin integral (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='58)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ϕ be the endomorphism of Λ•(T ∗M) ⊗ C which maps α ∈ Λk(T ∗M) ⊗ C to (2πi)−k/2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If ξ is a trivial line bundle, we denote the form W L,ξ i in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) by W L i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Bismut-Ma-Zhang [9, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11] gave the following formula for W L 0 : W L 0 = − √ 2πiϕ � +∞ 0 � � B θp ∧ �θp 2 exp(−σt)RL(0) dt √ t ∈ Ω•(M, o(TX)), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) where the operator θp∧ � θp 2 exp(−σt) ∈ Λ•(T ∗M)�⊗Λ•(� T ∗X) ⊗ Sgr acts on the function RL(·) and we evaluate at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we assume that (ξ, gξ) is a holomorphic Hermitian line bundle on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let (TN, gTN) be the holomorphic tangent bundle of N, where gTN is induced by c1(L, gL): if A ∈ TN, gTN = −ic1(L, gL)(A, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We assume that the action of U on N lifts to holomorphic unitary actions on (ξ, gξ) and (TN, gTN) with moment maps µξ and µdet TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Formally, for p ∈ N∗, let ξ 1 p be the p-th root of L and (det TN) 1 2p be the 2p-th root of det TN at the level of cohomology and moment map (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16)), then R L⊗ξ 1 p ⊗(det TN) 1 2p and W L⊗ξ 1 p ⊗(det TN) 1 2p 0 are well defined even if ξ 1 p ⊗ (det TN) 1 2p may not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have the following formula for W L,ξ 1 (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 5 Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For k = 1 in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4), as p → +∞ we have p−n−1ψ1/√pT � T HM, gTX, ∇Fp, gFp� = � X W L⊗ξ 1 p ⊗(det TN) 1 2p 0 + O � p−2� ∈ Ω•(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10) In other words, W L,ξ 0 + p−1W L,ξ 1 = W L⊗ξ 1 p ⊗(det TN) 1 2p 0 + O � p−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) In § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6 we discuss a special case of Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3 when N is a coadjoint orbit of u∗, and we use this to compute asymptotics of torsion for some concrete examples in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Liu considered the asymptotics of equivariant torsions for compact locally symmet- ric spaces [23] and asymptotic torsions for compact locally symmetric orbifolds [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' There are analogous results under holomorphic settings, Bismut-Vasserot [10] studied the asymptotics of holomorphic torsion associated with increasing powers of a positive line bundle over a K¨ahler manifold and they gave a formula for the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This asymptotic expansion played an important role in the arithmetic Hilbert-Samuel the- orem of Gillet–Soul´e [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' They also extended their results in [11], where the powers of the positive line bundle are replaced by the symmetric powers of a Griffiths positive holomorphic vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In [18], Finski generalized [10] to obtain a full asymptotic expansion and gave a formula for the second term of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Puchol [36] studied asymptotics of holomorphic torsion forms, which extended [10, 11] to family versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Savale [40, 41] obtained the asymptotics of the eta invariants using semiclassical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Main techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we briefly describe the techniques that we will use in the proof of Theorems 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3 as well as the main points in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The theory of Toeplitz operators is recalled (see § 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Espe- cially, one of the key results is the growth of the exponential of a Toeplitz operator (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7), which is used to get uniform estimations for operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Under the nondegeneracy condition (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5), Bismut-Ma- Zhang [9, § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10] showed that the Hodge-de Rham Laplacian (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13)) has a spectral gap: DFp,2 X ⩾ Cp2 for p ∈ N∗ large, this condition is vitally important in analysis, for instance, it ensures that W L,ξ i in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) is well defined (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Analytic localization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The proof of Theorems 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3 relies on a re- finement of Bismut-Ma-Zhang’s argument [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The strategy to get a full expansion is to study the local asymptotics of certain heat kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We will use the analytic localiza- tion method of Bismut-Lebeau [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Bismut’s family local index theory in the context of families [4, 5] also plays an important role, in particular, we apply the rescaling for two Clifford variables as in [12, Chapter 4d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Following [6, Chapter 11] and [9, § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12], in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' we prove that the limit of the trace of certain heat kernels used in the definition of analytic torsion forms (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4)) can be localized, by using the finite propagation speed of the wave operator (see [25, Appendix D]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' And as analysis carried through in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3-§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9, we use the techniques in [25, Chapter 4] to get estimations for high order expansions of resolvents and heat kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Compared with [9] and [36], to get the leading term, one only needs to get a uniform bound and the pointwise convergence for the heat traces, then apply the dominated con- vergence, while to give the full expansion as in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4), we need to give precise estimations for each term in the local expansion of the heat kernel as well as the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' To do TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 6 so, the most important trick is to preserve the spectral gap property for the resulting Laplacian in the localization and rescaling procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Hence we should carefully choose parameters at each step of the analysis, especially for the weighted norm (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='78)) and the corresponding elliptic regularity (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 1, we review the main results of Bismut-Lott [7] of analytic torsion forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 2, we recall some results on the Toeplitz operator following [25, § 7] and analyze the asymptotic behavior of the exponential of a Toeplitz operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 3, we recall Bismut-Ma-Zhang’s definition [9, § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7] for a series of flat bundles {Fp}p∈N∗ over M, which is the main geometric object in the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4, we give the full expansion of the odd superconnection form h � A′, gΩ•(X,Fp)� as p → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 5, we state and prove the main theorem, which gives the existence of the full expansion of the analytic torsions and the Γ-torsions associated with Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 6, we consider the case where G is a reductive Lie group and give an explicit formula for the sub-leading term in the asymptotics of torsion obtained in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In the whole paper, we apply the superconnection formalism of Quillen (see [37] and [4, § 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If E = E+ ⊕ E− is a Z2-vector space, and τ = ±1 defines the Z2-grading, if A ∈ End(E), we denote by Trs[A] the supertrace: Trs[A] = Tr[τA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) For a multi-index α = (α1, · · · , αk) ∈ Nk and a multi-variable Z = (Z1, · · · , Zk), set |α| = k � i=0 αi, Zα = Zα1 1 · · ·Zαk k , ∂α ∂Zα = ∂α1 ∂Zα1 · · · ∂αk ∂Zαk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) We also use the Einstein summation convention, if in a term the same index appears twice, that term is assumed to be summed over all possible values of that index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This work is the main result of our PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' thesis, which was done at Universit´e Paris Cit´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' I am deeply grateful to my thesis supervisor Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Xiaonan Ma for his patient guidance and numerous suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk�lodowska-Curie grant agreement No 754362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The analytic torsion forms In this section, we will summarize the main results on the analytic torsion forms following Bismut-Lott [7], which generalize the classical Ray-Singer analytic torsion [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, we introduce the smooth fibration π: M → S and define some associated tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, given a flat complex vector bundle (F, ∇F) with a Hermitian metric gF, we define a natural unitary connection ∇F,u and the associated odd characteristic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3, we reinterpret the de Rham operator dF as a flat superconnection on the bundle Ω•(X, F) over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4, we get a transgression formula for the odd closed forms h(A′, gΩ(X,F ) t ) ∈ Ω•(S) and define the analytic torsion form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5, we recall Bismut-Lott’s Lichnerowicz formula associated with the odd closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' A smooth fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let π: M → S be a fibration of smooth manifolds with compact fibre X of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let TX ⊂ TM be the tangent bundle to the fibers X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let T HM ⊂ TM be a horizontal subbundle with (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) and P TX : TM → TX the projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Since T HM ∼= π∗TS, for U ∈ TS, let UH ∈ T HM be the horizontal lift of U, such that π∗UH = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let T H(·, ·) ∈ Λ2(TS) ⊗ TX be the curvature of (π, T HM): T H(UH, V H) = −P TX[UH, V H], for U, V ∈ TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1) We also have an identification of bundles through the horizontal lift π∗Λ (T ∗S) �⊗Λ (T ∗X) ∼= Λ (T ∗M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) Let gTX and gTS be Riemannian metrics on TX and TS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We equip TM with the metric gTM = gTX ⊕π∗gTS and the corresponding Levi-Civita connection ∇TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇TX be the connection on TX defined by in [5, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6]: ∇TX = P TX∇TMP TX, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3) and denote its curvature by RTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇TS be the Levi-Civita connection of (TS, gTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇TM,⊕ be the connection on TM given by ∇TM,⊕ = π∗∇TS ⊕ ∇TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put S(·) = ∇TM − ∇TM,⊕ ∈ Ω1(M, End(TM)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' A flat bundle and its odd forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let (F, ∇F) be a complex flat vector bundle on M with a Hermitian metric gF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Following [12, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1], set ω � ∇F, gF� = � gF�−1∇FgF ∈ Ω1(M, End(F)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5) and it takes values in self-adjoint elements of End(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By [12, Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3], we have the following unitary conneciton ∇F,u on F with its curvature: ∇F,u = ∇F + 1 2ω � ∇F, gF� , RF,u = −1 4ω � ∇F, gF�2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) For x ∈ R, set h (x) = x exp � x2� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) Set ϕ the endomorphism of Λ•(T ∗M) ⊗R C sends α ∈ Λk(T ∗M) ⊗R C to (2πi)−k/2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put h � ∇F, gF� = (2πi)1/2 ϕTr � h � ω � ∇F, gF� /2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 ([7, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The odd form h � ∇F, gF� is real and closed and its cohomology class does not depend on gF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' A superconnection and odd forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We make the same assumption as in § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Ω•(X, F) be a formal smooth infinite-dimensional Z-graded vector bundle over S whose fibre over b ∈ S is C∞(Xb, Λ•(T ∗X) ⊗ F|b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) we have an identification of Z-graded vector spaces Ω• (M, F) ∼= Ω•(S, Ω•(X, F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) The exterior differential operator dF acting on Ω• (M, F), then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9), it can be considered as a flat superconnection on Ω• (X, F), we also denote it by A′ = dF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If U ∈ TS, the Lie derivative operator LUH acts naturally on Ω•(X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇Ω•(X,F ) be the connection on Ω• (X, F) such that if U ∈ TM and s ∈ Ω• (X, F), ∇Ω•(X,F ) UH s = LUHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 8 We set iT H = 1 2f α∧f β∧iT H(fα,fβ) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1)), which acts naturally on Λ•(T ∗S)�⊗Λ•(T ∗X)⊗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let dF X be the fibrewise exterior differential operator on Ω•(X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2 ([7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The superconnection A′(= dF) satisfies A′ = dF X + ∇Ω•(X,F ) + iT H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) Let gΩ•(X,F ) be L2-metric on Ω•(X, F) induced by (gTX, gF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then let dF,∗ X be the fibrewise formal adjoint operator of dF X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇Ω•(X,F ),∗ be the adjoint connection of ∇Ω•(X,F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By identifying TX and T ∗X through gTX, we can consider T H as a section of Λ2(T ∗S)�⊗T ∗X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then T H∧ acts naturally on Λ•(T ∗S)�⊗Λ•(T ∗X) ⊗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let A′′ be the adjoint of A′ with respect to the metric gΩ•(X,F ) in the sense of [7, § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4], which is also a flat superconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By [7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7], we have A′′ = dF,∗ X + ∇Ω•(X,F ),∗ − T H ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) Set A = 1 2 (A′ + A′′) , B = 1 2 (A′′ − A′) , DF X = dF X + dF,∗ X , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) then A is a superconnection on Ω•(X, F), B is a section of (π∗Λ• (T ∗S) �⊗End (Ω•(X, F)))odd and DF X is the fiberwise Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ϕ be the action on Λ(T ∗S) ⊗ C as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For B given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13), we set the following odd form similar to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8): h � A′, gΩ•(X,F )� = (2πi)1/2 ϕTrs [h(B)] ∈ Ω•(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The analytic torsion forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Following [9, § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4], for t > 0, we set a rescaling metric gTX t = gTX/t and the associated metric gΩ•(X,F ) t on Ω•(X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For a ∈ R, let ψa be the automorphism of Λ• (T ∗S) such that, if α ∈ Λk(T ∗S) then ψaα = akα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For t > 0, put Ct = ψ−1 √ t √ tAψ√ t, Dt = ψ−1 √ t √ tBψ√ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15) For t > 0, let h � A′, gΩ•(X,F ) t � be the form as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) associated with gΩ•(X,F ) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By [9, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='28)], we have h � A′, gΩ•(X,F ) t � = (2πi)1/2 ϕTrs [h(Dt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15), we have C2 t = −D2 t , together with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16), we get h � A′, gΩ•(X,F ) t � = (2πi)1/2 ϕTrs � Dt exp � − C2 t �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17) Let H• (X, F) = ⊕dim X i=0 Hi(X, F) be the Z-graded vector bundle over S whose fibre over b ∈ S is the cohomology H•(Xb, F|Xb) of the sheaf of locally flat sections of F|Xb on Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For the fiberwise Dirac operator DF X in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13), by the Hodge Theorem, we have Hi (X, F) ∼= ker DF,2 X �� Ωi(X,F ) for any i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) By [7, § 2a], H• (X, F) is canonically equipped with a flat connection ∇H•(X,F ) and a Hermitian metric gH•(X,F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let e(TX, ∇TX) ∈ Ω•(M) be the Euler form of TX (see [9, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='43)]) and we set h(∇H•(X,F ), gH•(X,F )) = � j(−1)jh(∇Hj(X,F ), gHj(X,F )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 9 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4 ([7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The forms h � A′, gΩ•(X,F ) t � are real, odd and closed, and their cohomology class does not depend on t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Moreover, as t → 0, we have h � A′, gΩ•(X,F ) t � = �� X e(TX, ∇TX)h � ∇F, gF� + O( √ t), as t → 0, h � ∇H•(X,F ), gH•(X,F )� + O(1/ √ t), as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19) Now we review the transgression procedure following [7, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We enlarge M, S to � M = M × R∗ +, �S = S × R∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set �π: � M → �S by �π(x, s) = (π(x), s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ρ0 : � M → M and ρ1: � M → R∗ + be the projection maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let � X be the fiber of �π, then we have T �X = ρ∗TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' and we equip T �X with the metric ρ∗gTX/s over M × {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have ∇T � X = ρ∗∇TX + ds � ∂ ∂s − 1 2s � , RT � X = ρ∗ 0RTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20) On �S, we have the odd form h � �A′, �gΩ(X,F ) t � analogous to h � A′, gΩ(X,F ) t � on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let h∧� A′, gΩ(X,F ) t � be the even form on S satisfies (see [7, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='114)]) h � �A′, �gΩ(X,F ) t ��� s=1 = h � A′, gΩ(X,F ) t � + ds ∧ h∧� A′, gΩ(X,F ) t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21) Set χ′(X, F) = m � j=0 (−1)jj dim Hj(X, F), χ(X, F) = m � j=0 (−1)j dim Hj(X, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22) By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21) we have: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6 ([7, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The form h∧� A′, gΩ•(X,F ) t � is even, and ∂ ∂th � A′, gΩ•(X,F ) t � = 1 t h∧� A′, gΩ•(X,F ) t � , h∧� A′, gΩ•(X,F ) t � = � O( √ t), as t → 0, �1 2χ′(X, F) − m 2 χ(X, F) � h′(0) + O(1/ √ t), as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='23) Now we give the definition of the analytic torsion form T � T HM, gTX, ∇F, gF� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7 (see [7, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set T � T HM, gTX, ∇F, gF� = − � +∞ 0 � h∧� A′, gΩ•(X,F ) t � + �m 4 χ(X, F) − 1 2χ′(X, F) �� h′(0) − h′(i √ t/2) ��dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='24) By Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6, the above integral is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8 ([7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The form T � T HM, gTX, ∇F, gF� is even and real, and it verifies the following transgression formula: dT � T HM, gTX, ∇F, gF� = � X e(TX, ∇TX)h � ∇F, gF� − h � ∇H•(X,F ), gH•(X,F )� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='25) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Lichnerowicz type formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let z be an odd Grassmannian variable anticom- mutes with odd variables we used before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If a ∈ R[z]�⊗π∗Λ• (T ∗S) �⊗Λ• (T ∗X), we set [a]z = c, for a = b + zc where b, c ∈ π∗Λ• (T ∗S) �⊗Λ• (T ∗X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26) By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17), we get h � A′, gΩ•(X,F ) t � = (2πi)1/2 ϕTrs � exp � − � C2 t − zDt � ��z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27) From now on, we will always use Latin indices i, j, · · · for vertical variables, and Greek indices α, β, · · · for horizontal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let e1, · · · , em be a local orthonormal frame of TX, and let f1, · · · , fr be a basis of TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The corresponding dual bases are denoted with upper indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We set the following Clifford actions on Λ(T ∗X): c(ej) = ej ∧ −iej, �c(ej) = ej ∧ +iej, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='28) then for 1 ⩽ i, j ⩽ m we have [c(ei), c(ej)] = −2δij, [�c(ei), �c(ej)] = 2δij, [c(ei), �c(ej)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='29) By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='29), the actions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='28) extend to an isomorphism of algebras c�⊗�c: c(TX)�⊗�c(TX) → End(Λ(T ∗X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='30) Put RF = −1 4 � RTXei, ej � �c (ei) �c (ej) − 1 4ω2� ∇F, gF� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='31) Let rX be the scalar curvature of the fibre � X, gTX� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ΛF be a section of R[z]�⊗Λ•(T ∗S)�⊗End(Λ(T ∗X)) ⊗ End(F) by ΛF =1 4rX + 1 2c(ei)c(ej)R (ei, ej) + 1 2fαfβR � f H α , f H β � + c(ei)fαR � ei, f H α � + 1 4ω � ∇F, gF� (ei)2 + 1 8�c(ei)�c(ej)ω � ∇F, gF�2 (ei, ej) − 1 2f H α �c(ei)∇TX⊗Fp,u fH α ω � ∇F, gF� (ei) − 1 2c(ei)�c(ej)∇TX⊗Fp,u ei ω � ∇F, gF� (ej) − 1 2zc(ei)ω � ∇F, gF� (ei) − 1 2zfαω � ∇F, gF�� f H α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) be the fibrewise connection along X by: 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) =∇π∗Λ•(T ∗S)�⊗Λ•(T ∗X) + 1 2 � Sei, f H α � c (ei) f α + 1 4 � Sf H α , f H β � f αf β − z 2ei ∧ �c(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='33) By [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14], the curvature of 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) is given by 0RR[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)(ei, ej) = 1 2 � RTX(ek, eℓ)ei, ej �� c(ek)c(eℓ) − �c(ek)�c(eℓ) � + � RTX(fα, eℓ)ei, ej � f αc(eℓ) + 1 2 � RTX(fα, fβ)ei, ej � f α ∧ f β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='34) On R[z]�⊗π∗Λ• (T ∗S) �⊗Λ• (T ∗X) ⊗ F, let 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗F,u be the fibrewise connection induced by 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X) and ∇F,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 11 Put LF t = C2 t − zDt, which appears in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27) and we set LF = LF 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let N1 be the number operator of the exterior algebra R[z]�⊗π∗Λ•(T ∗S) that acts by multiplication by the total degree of each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15), we have LF 4t = θ−1 √ ttLFθ√ t, where θ√ t = √ t N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='35) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11 ([7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The following Lichnerowicz type formula holds: LF = − �0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗F,u ei �2 + 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗F,u ∇T X ei ei + ΛF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='36) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Toeplitz operators In this section, we describe the formalism of the Toeplitz operator introduced by Berezin [2] and Boutet de Monvel-Guillemin [15], and developed by Bordemann-Meinrenken- Schlichenmaier [13], Schlichenmaier [42] and Ma-Marinescu [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, we introduce the positive line bundle L on a compact K¨ahler manifold N and the Berezin-Toeplitz quantization, then we give some uniform estimations that will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, we investigate the exponential of a Toeplitz operator and give an estimation for each term in the expansion and the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The algebras of Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let N be a compact complex manifold of dimCN = n with the complex structure J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let gTRN be a J-invariant Riemannian metric on TRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Denote the induced Riemannian volume form by dvN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let (L, gL) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (ξ, hξ)) be a holomorphic line bundle (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' holomorphic vector bundle) on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ∇ξ) be the associated Chern connections on L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ξ) with curvature RL (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Rξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We assume that (L, gL, ∇L) is a positive line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then c1(L, gL) = √−1 2π RL defines a K¨ahler form on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We note that gTRN is not necessarily given by the K¨ahler metric induced by c1(L, gL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For p ∈ N∗, put Lp = L⊗p, the p-th tensor power of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have the L2-inner product on C ∞(N, Lp ⊗ξ) induced by (gTRN, gL, hξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We denote the corresponding norm by ∥·∥L2 and by L2(N, Lp ⊗ ξ) the completion of C ∞(N, Lp ⊗ ξ) with respect to the L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We consider the space H(0,0)(N, Lp ⊗ ξ) of holomorphic sections of Lp ⊗ ξ, and let Pp : L2(N, Lp ⊗ ξ) −→ H(0,0)(N, Lp ⊗ ξ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1) be the orthogonal projection with respect to the L2-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 ([25, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The Berezin-Toeplitz quantization of a smooth section H ∈ C ∞(N, End(ξ)) is a sequence of linear operators {TH,p}p∈N∗ given by TH,p : L2(N, Lp ⊗ ξ) → L2(N, Lp ⊗ ξ), TH,p = PpHPp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) The operator TH,p has a smooth kernel TH,p(x, x′) with respect to dvN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' On the diago- nal, TH,p(x, x) ∈ End(Lp ⊗ ξ) = End(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For ℓ ∈ N, let |·|C ℓ(N) be the ℓ-th order smooth norm on C ∞(N, End(ξ)) induced by (∇TRN, ∇E, gTRN, hξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By the proof of [25, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4], we have the following expansion with a uniform estimation for the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' There is a series of differential operators {Ai}+∞ i=0 of order no more than 2i such that for any k, ℓ ∈ N, there exists C > 0 such that for any p ∈ N∗ and TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 12 H ∈ C ∞(N, End(ξ)), ���p−nTH,p(x, x) − k � i=0 p−iAi(H)(x) ��� C ℓ(N) ⩽ C |H|C 2k+2(N) p−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3) And the operators {Ai}+∞ i=0 vary smoothly with respect to (J, gTRN, hL, hξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In particular, A0(H)dvN = c1(L, gL)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) By the Kodaira vanishing Theorem [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4] and the Riemann-Roch-Hirzebruch Theorem [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6], for p ∈ N∗ large enough, dim H(0,0)(N, Lp ⊗ ξ) is a poly- nomial of p ∈ N∗ with leading term dim H(0,0)(N, Lp ⊗ ξ) = dimC ξ · � N c1(L, gL)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' pn + O � pn−1� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5) and clearly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) is a local version of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If A ∈ End (L2 (N, Lp ⊗ ξ)), let ∥A∥ be its operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If A is of trace class, we denote its trace norm by ∥A∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If A = PpAPp, then A is of trace class, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5), there is C > 0 such that for any p ∈ N∗, ∥A∥1 ⩽ ∥A∥ dimC H(0,0)(N, Lp ⊗ ξ) ⩽ C ∥A∥ pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) Following Ma-Marinescu [25, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1], we now define the Toeplitz operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' A Toeplitz operator is a family of operators � Tp | Tp ∈ End(L2 (N, Lp ⊗ ξ)) � p∈N∗ bounded with respect to the L2-norm such that Tp = PpTpPp, and that there exists {Hi | Hi ∈ C ∞(N, End(ξ))}i∈N such that for any k ∈ N, there is ck > 0 such that ���Tp − k � i=0 p−iTHi,p ��� ⩽ ckp−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) As in [25, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5)], we use the notation of formal expansion to denote (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) as Tp = +∞ � i=0 p−iTHi,p + O � p−∞� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) and we replace �+∞ i=0 with �k i=0 if we only refer to the first k terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we introduce the asymptotic trace symbol for ease of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For the Toeplitz operator {Tp}p∈N∗ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7), for k ∈ N, if we set Tr[k][Tp] = � i+j=k � N Trξ[Ai(Hj)]dvN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7), there is Cck,{|Hi|C2i(N)}k+1 i=1 > 0 which depends on ck and {|Hi|C 2i(N)}k+1 i=1 such that ���p−nTrH(0,0)(N,Lp⊗ξ)[Tp] − k � i=0 p−iTr[i][Tp] ��� ⩽ Cck,{|Hi|C2i(N)}k+1 i=1 p−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10) By the proof of [25, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1] and [27, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3], we have the following product formula of Toeplitz operators with estimation for the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 13 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The set of Toeplitz operators is an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In particular, for any k ∈ N, there is C > 0 such that for any H, H′ ∈ C ∞(N, End(ξ)) and p ∈ N∗, we have ��TH,pTH′,p − k � j=0 p−iTCj(H,H′),p �� ⩽ Cp−k−1 |H|C 2k+2(N) · |H′|C 2k+2(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) where Cj(·, ·) is a smooth bidifferential operator of total degree no more than 2j, and C0(H, H′) = HH′, C1(H, H′) − C1(H′, H) = i{H, H′} if H, H′ ∈ C ∞(N, C), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) where {H, H′} is the Poisson bracket of 2πc1(L, gL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The exponential of Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' To discuss the exponential of Toeplitz operators, we first study the inverse of a Toeplitz operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We follow Ma-Zhang [29, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1] where they have proved that if the principal symbol H0 of a Tp as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) is invertible, then T −1 p is also a Toeplitz operator, here we give a uniform estimation for the remainder term in the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Tp be a Toeplitz operator as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If H0(x) is invertible for all x ∈ N, then Tp is invertible for p large, and T −1 p is also a Toeplitz operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Moreover, if we write T −1 p = �∞ i=0 p−iTGi,p + O(p−∞) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8), then we have G0 = H−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We recall a basic fact: for invertible operators A and B, we have A−1 − B = B((1 − (1 − AB))−1 − 1) = B(1 − AB)(1 − (1 − AB))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) For k ∈ N and Ci(·, ·) given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4, we set G0 = H−1 0 , Gk+1 = −H−1 0 � i+j⩽k+1 (i,j)̸=(0,0) Ci � Hj, Gk+1−i−j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14), we can prove inductively that ����Tp � k � i=0 p−iTGi,p � − 1 ���� ⩽ Cp−k−1, ���� � k � i=0 p−iTGi,p � Tp − 1 ���� ⩽ Cp−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15), Tp and �k i=0 p−iTGi,p have both left and right inverse for p ∈ N∗ large, hence they are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We take A = Tp, B = �k i=0 p−iTGi,p in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13), by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15) and the obvious inequality ∥(1 − (1 − AB))−1∥ ⩽ (1 − ∥(1 − AB)∥)−1, we get ���T −1 p − k � i=0 p−iTGi,p ��� ⩽ Cp−k−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By the above proof, we can indeed take C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) as the sum of terms of the following form: C|H−1 0 |α C 0(N) · k � i=0 cβi i |Hi|γi C 2(k+1)(N) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17) where C > 0, α ∈ N and β, γ ∈ Nk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 14 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If Tp is a Toeplitz operator with expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7), exp(−tTp) is also a Toeplitz operator for any t ⩾ 0 with the following expansion in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8): exp(−tTp) = +∞ � i=0 p−iTJi(t),p + O(p−∞), J0(t) = e−tH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) Put h = infx∈N Re � SpecH0(x) � , then for any ε > 0, i, k, ℓ ∈ N, there is C > 0 such that for any p ∈ N∗, |Ji(t)|C k(N) ⩽ C exp(−(h − ε)t), ��� exp(−tTp) − k � i=0 p−iTJi(t),p ��� ⩽ C exp(−(h − ε)t)p−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By the product formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12), for k ∈ N, we have T k p = THk 0,p + O(p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If we ignore the divergence of infinite sums of remainders, we get exp(Tp) = �+∞ k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='THk 0,p + O(p−1) = Texp(H0),p + O(p−1), which is just the first order of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we give rigorous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5, if λ /∈ Spec(H0), then λ − Tp is invertible for p ∈ N∗ large and we have the following expansion in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8): (λ − Tp)−1 = +∞ � i=0 p−iTIi(λ),p (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20) where Ii(λ) ∈ C ∞(N, End(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' As in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4, Ci(·, ·) is a bilinear differential operator for each i ∈ N, then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14), each Ii(λ) for i ∈ N is the sum of terms of the following form: (λ − H0)−n0 f1(λ) (λ − H0)−n1 · · · fj(λ) (λ − H0)−nj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21) where nj ∈ N and fj(λ) ∈ C ∞(N, End(ξ))[λ], the space of polynomials of λ with coeffi- cients in C ∞(N, End(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In particular, the leading term is I0(λ) = (λ − H0)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22) For ε > 0, we choose a bounded curve Γε surrounds Spec(H0) counterclockwise on the complex plane such that infλ∈Γε Re(λ) ⩾ h−ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By the Cauchy integral formula, we have exp(−tTp) = 1 2πi � Γε e−tλ (λ − Tp)−1 dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='23) We denote the C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) associated with Toeplitz operator ( � λ − Tp � )−1 by Cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17), Cλ is uniformly bounded for λ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='23), we have ���� exp(−tTp) − k � i=0 p−iT 1 2πi � Γε e−tλIi(λ)dλ,p ���� ⩽ p−k−1 1 2π � Γε e−tRe(λ)Cλdλ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='24) from which we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) and the first inequality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19) with Ji(t) = 1 2πi � Γε e−tλIi(λ)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='25) And by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22), we get J0(t) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='25), we obtain the second inequality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 15 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In this study, estimations for each term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19) and the remainder term are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19), for any t ⩾ 0, we have ���Tr[k] � exp(−tTp) ���� ⩽ Ce−(h−ǫ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26) Moreover, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19), we get ���p−nTrH(0,0)(N,Lp⊗ξ)[exp(−tTp)] − k � i=0 p−iTr[i][exp(−tTp)] ��� ⩽ Cp−k−1e−(h−ǫ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27) Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27) ensure the exponential decay of each term in the expansion and the remainder as t → +∞, and these play important roles in § 4 and § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The geometry of bundles {Fp}p∈N∗ We make the same assumption as in §§ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The purpose of this section is to review some properties of bundles {Fp}p∈N∗ over M following [9, § 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' This section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, we present the geometry of the fibration N = PG ×G N → M and the line bundle L → N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, we introduce the bundles {Fp}p∈N∗ over M and compute the curvature of the unitary connection ∇Fp,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3, for a smooth family of Toeplitz operators {Tp}p∈N∗, we analyze the asymptotics of TrFp[Tp] when p → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4, we obtain the asymptotics of the odd form h(∇Fp, gFp) for p → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5, we recall the nondegenarated condition of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Geometric settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let N be a compact complex manifold of complex dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let L and ξ be holomorphic bundles on N with dimC(L) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let G be a Lie group acting holomorphically on N, and this action lifts to holomorphic actions on L and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let p: PG → M be a principal flat G-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set N = PG ×G N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1) We denote by q the projection q: N → M with fibre N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We still denote the bundle PG ×G L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' PG ×G ξ) over N by L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let T H 0 N ⊂ TN be the horizontal bundle determined by the flat connection of PG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put TRN = ker q∗, the real relative tangent bundle, and let TN be the holomorphic relative tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let JN be the complex structure on TRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We clearly have T H 0 N ∼= q∗TM, then for U ∈ TM, we denote by UH 0 ∈ T H 0 N its horizontal lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let gL (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' gξ) be a Hermitian metric on L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ξ) over N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ∇ξ) be the fibrewise Chern on L (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Using the flat connection on PG, we can extend ∇L and ∇ξ to connections on L and ξ respectively: for U ∈ TM, put ∇L UH 0 (or ∇ξ UH 0 ) = LUH 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) These connections are in general non-unitary, and we set ω � L, gL� (U) = � gL�−1LUH 0 gL, ω � ξ, gξ� (U) = � gξ�−1LUH 0 gξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3) In what follows, we assume that the first Chern form c1(L, gL) = √−1 2π � ∇L�2 ∈ Ω2(N ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 16 is fibrewisely positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∂N denote the fibrewise Dolbeault operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have the following identities (see [9, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7)]): for U, V ∈ TM and Y ∈ TRN, c1(L, gL)(UH 0 , V H 0 ) = 0, c1(L, gL)(UH 0 , Y ) = √−1 2π ∂N Y ω � L, gL� (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5) Let gTRN be a smooth Hermitian metric on TRN, and let dvN be the corresponding fibrewise volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then we extend dvN to be a form on N that vanishes along the horizontal direction, and still denote it by dvN ∈ Ω•(N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Recall that, if U ∈ TM, the fibrewise Lie derivative operator LUH 0 acts naturally on smooth sections of Λ•(T ∗ RN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We define divN(U) by LUH 0 dvN = divN(U)dvN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5), when gTRN is just the K¨ahler metric gTRN(·, ·) = c1(L, gL)(·, JN·), then for any U ∈ TM, we can prove the following formula: divN(U) = 1 4π∆Nω � L, gL� (U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) where ∆N is the fibrewise (nonnegative) Laplacian operator with respect to the K¨ahler metric which acts on C ∞(N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The curvature of ∇Fp,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Recall the definition of Fp in (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3): Fp = � PG ×G H(0,0)(N, Lp ⊗ ξ) � ⊗ F, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) where the flat bundle (F, ∇F) with metric gF plays the role of a shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let F(F) be the frame bundle of F, which is a GL(dimC(F))-principal bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Since H(0,0)(N, Lp⊗ξ⊗CdimC(F )) ∼= H(0,0)(N, Lp⊗ξ)⊗CdimC(F ), we could always assume in what follows that F = C trivial, or we may replace (G, N, L, ξ) by (G×GL(dimC(F)), N, L, ξ⊗ CdimC(F )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put Fp = PG ×G C ∞(N, Lp ⊗ ξ), which is an infinite dimensional bundle on M and Fp is a subbundle of Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The connection ∇Lp⊗ξ naturally induces a flat connection on ∇Fp: if s is a smooth section of Fp and U ∈ TM, set ∇Fp U s = ∇Lp⊗ξ UH 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) This connection preserves Fp, and it induces a flat connection ∇Fp on Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We equip Fp with the L2-metric gFp induced by (gLp, hξ, dvN), which gives a metric gFp on Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Pp denote the fibrewise orthogonal projection Fp → Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇N be a connection on the infinite dimensional bundle C ∞(N ) → M given as follows: if U ∈ TM and H ∈ C ∞(N ), put ∇N U = UH 0 (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Following [9, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='34)], we set ϑL = −1 2ω � L, gL� , ϑξ = −1 2ω � ξ, gξ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10) We denote by � TX a copy of TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We denote by �ω � ∇F, gF� , �ϑL, � divN the restrictions of ω � ∇F, gF� , ϑL, divN to � TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let {ei}m i=1 and {�ei}m i=1 be a local orthonormal frame of TX and � TX respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We denote the anticommutator by [·, ·]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 ([9, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We have the following identity: 1 2pω � ∇Fp, gFp� = T−ϑL−ϑξ/p+divN/2p,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 17 If H ∈ C ∞(N, End(ξ)) is a smooth section, we have ∇Fp,uTH,p = T∇N H,p + p[TϑL+ϑξ/p−divN/2p,p, TH,p]+ − 2pT(ϑL+ϑξ/p−divN/2p)H,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) Moreover, combined Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4 with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12), we see that 1 4pω(∇Fp, gFp)2, 1 4p2 �ω(∇Fp, gFp)(ei)2 and ∇Fp,uTH,p are Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For a general fibration, Ma-Zhang [30, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19], [28] showed that the curvature operator on Fp is a Toeplitz operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The smooth bundle of Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Analogous to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, we call {Tp | Tp ∈ C ∞� M, End(Fp) � }p∈N∗ a smooth section of Toeplitz operators if there exists {Hi | Hi ∈ C ∞(N , End(ξ))}i∈N such that for any k, ℓ ∈ N, there is C > 0 satisfies ���Tp − k � i=0 p−iTHi,p ��� C ℓ(M,End(Fp)) ⩽ Cp−k−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) where the norm |·|C ℓ(M,End(Fp)) is induced by (∇Fp, gFp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' As all the expansions in § 2 are smoothly dependent on geometric data (J, gTRN, hL, hE), all results in § 2 have a similar version for Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In particular, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11), all the operators in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1 are smooth sections of Toeplitz operators as well as their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Also, for {Tp}p∈N∗ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13), TrFp[Tp] and Tr[k][Tp] are smooth functions on M, and we can replace the absolute values |·| on the left hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27) by a smooth norm |·|C ℓ(M) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The asymptotics of h(∇Fp, gFp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Recall the odd form h(∇Fp, gFp) of Fp as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' As p → +∞, we have smooth odd forms γi ∈ Ωodd(M), i ∈ N such that for any k, ℓ ∈ N, there is C > 0 such that ���p−n 1 √pψ1/√ph � ∇Fp, gFp� − k � i=0 γip−i��� C ℓ(M) ⩽ Cp−k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8), if we take γi = (2πi)1/2ϕTr[k] � exp � 1 4pω � ∇Fp, gFp�2 + 1 2pzω � ∇Fp, gFp���z , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15) then we get the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we recall the non-degeneracy condition [9, Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We say that �ϑL is nondegenerated if there is a > 0 such that m � i=1 �ϑL(ei)2 ⩾ a on N , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) or equivalently, �ϑL ∈ C ∞(N , q∗� T ∗X) is nowhere vanishing on N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6 ([9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If �ϑL is nondegenerate, for any ε > 0 and p ∈ N∗ large, the fibrewise Dirac operator of Fp (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18)) satisfies DFp,2 X ⩾ (a − ε)p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17) In particular, for p ∈ N∗ large enough, DFp,2 X is invertible, thus H•(X, Fp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' by the Hodge Theorem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The asymptotics of the odd superconnection forms The purpose of this section is to obtain the asymptotics of the odd form h � A′, gΩ•(X,Fp) 4t/p2 � as p → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The main technique we use is the analytic localization method by Bismut- Lebeau [6], which is further developed by Dai-Liu-Ma [17] and Ma-Marinescu [25, § 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We treat the two cases 1 ⩽ t < +∞ and 0 ⩽ t ⩽ 1 separately in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1-§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7 and § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The main difficulty is to get an exponential decay term when t → +∞ as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, to do so, we devote a large part of this section to carefully handling the localization procedure to ensure that at each step, the operator we get is always “non-negative” as p−2DFp,2 X in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6 (see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3-§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we give more details on the main points of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, we state the main theorem of this section and express the odd superconnection forms in terms of heat ker- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, by the spectral gap property of LFp, we can use the finite propagation speed of solutions of hyperbolic equations and localize the original question to a problem on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3, we perform the Bismut-Zhang’s rescaling on the operator θ−1 1/√pp−2LFpθ1/√p, then we introduce a smooth family of differential operators � L Fp v |0⩽v⩽1/p as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='66) which links the rescaled operator and a limit operator, and we study the Taylor expansion of � L Fp v with respect to the parameter v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4, we introduce graded Sobolev norms with weights ∥·∥µ,k,p and give some basic elliptic estimations of � L Fp v , an important step is to carefully analyze the structure of � L Fp v and choose a suitable weight µ to preserve the “positivity” of Spec( � L Fp v ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5, we study the Sobolev estimations of the resolvent (λ − � L Fp v )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6, we establish the corresponding convergence of the heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7, we analyze the asymptotic trace of the limit kernel when p → +∞ using results in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2 and properties of Gaussian integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8, we deal with the case 0 < t ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9, we prove the main theorem (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1) of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Asymptotic expansion of the odd forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The constants appearing in the sequel depend on the compact subset of S we are working on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' To simplify the statements in what follows, we always assume that S is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' The following theorem is the main result of this section, and the rest of the section is devoted to its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Recall the odd form h � A′, gΩ•(X,Fp) t � and θa in in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='35) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If �ϑL is nondegenerate as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16), there exist {ci(t) ∈ Ω•(M) | t > 0}i∈N such that, for any γ > 0, k, ℓ ∈ N, we have C > 0 that for p ∈ N∗ and t > 0, ����p−n 1 √pψ1/√ph � A′, gΩ•(X,Fp) 4t/p2 � − k � i=0 � X ci(t)p−i ���� C ℓ(S) ⩽ Ce−(a−γ)tp−k−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1) where a ⩾ 0 is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Moreover, there is C > 0 such that for t > 0, we have ���� � X ck(t) ���� C ℓ(S) ⩽ Ce−(a−γ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2) For p ∈ N∗ and t > 0, set MFp = θ√pp−2LFpθ1/√p, MFp t = θ√p/ √ ttp−2LFpθ√ t/√p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 19 Let exp(−tMFp)(x, x′) and exp(−t′MFp t )(x, x′) be the smooth heat kernel of MFp and MFp t associated with dvX(x′) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3), we have 1 √pψ1/√ph � A′, gΩ•(X,Fp) 4t/p2 � = √ 2πiϕ � θ1/√p � X TrΛ(T ∗X)⊗Fp s � exp(−LFp 4t/p2)(x, x) � dvX(x) �z = √ 2πiϕ � θ−1 √ t � X TrΛ(T ∗X)⊗Fp s � exp(−tMFp)(x, x) � dvX(x) �z = √ 2πiϕ � � X TrΛ(T ∗X)⊗Fp s � exp(−MFp t )(x, x) � dvX(x) �z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4) Notice that to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, we only need to consider the case a = 0, or we may replace LFp with LFp − ap2 so that we get extra e−at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Localization of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For x0 ∈ M, let X be the fibre containing x0, we mainly work along this fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For ε > 0, let BX (x0, ε) and BTx0X (0, ε) be the open balls in X and Tx0X with center x0 and 0 and radius ε respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let expX x0 be the exponential map of (X, gTX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For ε small, expX x0 : BTx0X (0, ε) → BX (x0, ε) is a diffeomorphism, which gives local coordinates by identifying Tx0X with Rm via an orthonormal basis {ei}m i=1 of Tx0X: Z = (Z1, · · · , Zm) ∈ Rm �−→ Ziei ∈ Tx0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5) We will always identify BX (x0, ε) with BTx0X (0, ε) through the isomorphism in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For b ∈ S, let inj(Xb) be the injectivity radius of Xb = π−1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any ε > 0 that 0 < ε < minb∈S inj(Xb)/8, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) (we have minb∈Sinj(Xb) > 0 since S is compact), by the compactness of X, we can choose a finite set {xi}i⩾1 ⊂ X such that � BX (xi, ε) � i⩾1 is an open covering of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For now, ε is not fixed, we will assign it a suitable value after Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For Z ∈ BTxiX (0, ε), we identify Fp,Z and � R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) � Z with Fp,xi and � R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) � xi by parallel transport with respect to the connections ∇Fp and 0∇R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) along the curve s → sZ for s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Γ0 and ΓFp,u be the corresponding connection forms of 0∇R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) and ∇Fp on BTx0X (0, ε) with respect to this trivialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let {fxi}i⩾1 be a partition of unity with respect to {BX (xi, ε)}i⩾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For k ∈ N, we define the Sobolev norm ∥·∥Hk,p on C ∞(X, R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) ⊗ Fp): ∥s∥2 Hk,p = � i � α∈Nm, 0⩽|α|⩽k ∥∂αfxis∥2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7) and we denote by Hk,p its completion with respect to the norm ∥·∥2 Hk,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Note that ∇Fp,u preserves the metric gFp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Hence the two norms ∥·∥H0,p and ∥·∥L2(X,Fp) are equivalent uniformly for p and we will not distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Similar to [25, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2] and [36, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6], we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 20 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For k ∈ N, there is C > 0 such that for p ∈ N∗ and s ∈ H2k,p, we have ∥s∥H2k,p ⩽ Cp2k k � j=0 p−2j��LFp,js �� L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='36), on BX (xi, ε) we have LFp = − � ei + ΓFp,u(ei) + Γ0(ei) �2 + � ∇TX ei ei + ΓFp,u(∇TX ei ei) + Γ0(∇TX ei ei) � + ΛFp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) For any Z ∈ BX (xi, ε), we have a classical relation (see [4, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18]) ΓFp,u Z (∂j) = � 1 0 RFp,u tZ (R, ∂j)dt, for R = Zi∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10) By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10), we see that p−1ΓFp,u Z (∂j) and p−2ΛFp are smooth family of Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For a smooth family of Toeplitz operators Tp, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12), dTp = ∇Fp,uTp − [ΓFp,u, Tp] is also a smooth family of Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9) and the above argument, the operator norms of p−1ΓFp,u and p−2ΛFp are bounded uniformly for p ∈ N∗ as well as their derivatives, then we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) exactly as the proof of [25, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Observe that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) is still true if we replace LFp with LFp,∗ or DFp,2 X since they have the same structure as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Choose f ∈ C ∞(R) even, nonincreasing when t ⩾ 0 and f (t) = � 1 if |t| ⩽ 1 2, 0 if |t| ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For t, h > 0 and a ∈ C, set Ft,h (a) = � R e √ 2iva exp � −v2/2 � f �√ tv/h � dv √ 2π, Gt,h (a) = � R e √ 2iva exp � −v2/2 � � 1 − f �√ tv/h �� dv √ 2π , Ht,h (a) = � R e √ 2iva exp � −v2/2t � � 1 − f (v/h) � dv √ 2πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) The functions Ft,h(a), Gt,h(a), Ht,h(a) are even holomorphic functions, thus there exist holomorphic functions �Ft,h (a), �Gt,h (a) and �Ht,h (a) such that �Ft,h � a2� = Ft,h (a) , �Gt,h � a2� = Gt,h (a) , �Ht,h � a2� = Ht,h (a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) Moreover, the restriction of �Ft and �Gt to R lies in the Schwartz space S (R), and Ft,h (a) + Gt,h (a) = exp � −a2� , Ht,h (a) = Gt,h �√ ta � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) Now we fix a c > 0, let Vc be the following subset of the complex plane: Vc = � a ∈ C: Re (a) ⩾ 1 4c2Im (a)2 − c2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 21 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any h > 0 and k ∈ N, there are C, c1, c2 > 0 such that for t > 0, we have sup a∈Vc ��ak �Ht,h (a) �� ⩽ C exp � c1t − c2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By proceeding as in [25, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5], when |Im (a) | ⩽ c, as ikakeiva = ∂k ∂vk eiva, one can integrate by part the expression of akHt,h (a) given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12) to obtain that sup |Im(a)|⩽c |akHt,h(a)| ⩽ C exp � c1t − c2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17) Note that for c > 0, Vc is just the image of {a ∈ C: |Im(a)| ⩽ c} by the map C → C: a �→ a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='17), we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ For any δ > 0, let Γ be the contour in C defined by {x ± δi | x ⩾ −δ} ∪ {−δ + xi | −δ ⩽ x ⩽ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We choose a suitable δ to make Γ ⊂ Vc and mina∈Γ,b∈Vc |a − b| = c2/2 (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='15)), and we denote this contour Γ by Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' x y Γc Vc Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For k ∈ N, there are C > 0, ℓ ∈ N∗ such that for any p ∈ N∗, λ ∈ Γc, ��� λ − LFp�−1s �� H2k+2,p ⩽ Cpℓ|λ|ℓ∥s∥H2k,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' First, following [36, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='65)] we prove that for some C > 0, i ∈ N∗, ��� λ − LFp�−1s �� L2 ⩽ Cpi|λ|i∥s∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19) By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11, we write LFp = DFp,2 X +G Fp where G Fp is a 1-order fibrewise differential operator with positive degree in π∗Λ (T ∗S), and � λ − LFp�−1 = dim S � i=0 �� λ − DFp,2 X �−1G Fp�i� λ − DFp,2 X �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20) Note the self adjointness of DFp,2 X , for any λ ∈ Γc and x ⩾ 0, we have max{ 1 |λ−x|, x |λ−x|} ⩽ C |λ| |λ−x| + 1 ⩽ C|λ|, so there is C > 0 with ��� λ − DFp,2 X �−1s �� L2 ⩽ C∥s∥L2, ��DFp,2 X � λ − DFp,2 X �−1s �� L2 ⩽ C|λ| · ∥s∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 22 By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='36), there is C > 0 such that for any λ ∈ Γ, ��G Fp� λ − DFp,2 X �−1s �� L2 ⩽ Cp2��� λ − DFp,2 X �−1s �� H2,p ⩽ Cp2���DFp,2 X � λ − DFp,2 X �−1s �� L2 + p2∥ � λ − DFp,2 X �−1s∥L2 � ⩽ Cp4|λ| · ∥s∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22), we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='20) and an argument similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='22), for some C > 0, j ∈ N, we have ��LFp(λ − LFp�−1s �� L2 ⩽ Cpj|λ|j∥s∥L2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='23) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='23), for any k ∈ N, there are C > 0, ℓ ∈ N∗ such that ��LFp,k+1(λ−LFp�−1s �� L2 = ��LFp(λ − LFp�−1LFp,ks �� L2 ⩽ Cpj|λ|j∥LFp,ks∥L2 ⩽ Cpℓ|λ|ℓ∥s∥H2k,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='24) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='24), we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ Set the fibre product M ×S M = {(x1, x2) ∈ M × M | π(x1) = π(x2)} with the projections pri(x1, x2) → xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any two bundles V, V ′ over M, Let V ×S V ′ be the bundle on M ×S M given by pr∗ 1(V ) ⊗ pr∗ 2(V ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' If we have two connections ∇V , ∇V ′ on V and V ′, let ∇V ×SV ′ be the induced connection on V ×S V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We define a bundle Ep = R[z]�⊗π∗Λ (T ∗S) �⊗ �� Λ(T ∗X) ⊗ Fp � ×S � (Λ(T ∗X))∗ ⊗ F ∗ p �� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='25) over M ×S M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇Ep be the connection on Ep induced by 0∇R[z]�⊗π∗Λ•(T ∗S)�⊗Λ•(T ∗X)⊗Fp,u and ∇Fp,u (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='33)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We notice that �Gt,ε(tLFp) is a smooth section of Ep on M ×S M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We also denote the pull back bundle of Ep through the diagonal embedding M → M ×S M given by x0 �→ (x0, x0), then Ep,x0 = R[z]�⊗π∗Λ � T ∗ π(x0)S � �⊗End(Λ(T ∗ x0X)) ⊗ End(Fp,x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any ε > 0 and ℓ ∈ N, there exist c1, c2 > 0 and k ∈ N such that for any t > 0 and p ∈ N∗, we have �� �Gt,ε � tLFp��� C ℓ(M×SM) ⩽ Cpk exp � c1t − c2 t � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27) where the C ℓ (M ×S M)-norm is induced by ∇Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16), for r ∈ N∗, there is a holomorphic function �Ht,h,r(a) defined in a neighborhood of Vc such that (see also [6, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32]) 1 (r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' dr−1 dar−1 �Ht,h,r(a) = �Ht,h(a), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='28) and for any h > 0, there are C, c1, c2 > 0 such that for any t > 0, we have sup a∈Vc |ak �Ht,h,r (a) | ⩽ C exp � c1t − c2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='29) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 23 Let {Ui}dim S i=1 be a set of local vector fields on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let {UH i } denote the set of corresponding horizontal lift local vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any multi-index α = (i1, i2, · · ·) in which 1 ⩽ ij ⩽ dim S, we denote UH α = ∇Ep UH i1 ∇Ep UH i2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='28), we get UH α � �Gt,ε � tLFp�� = 1 2πi � Γ �Ht,ε,r(λ)UH α (λ − LFp�−rdλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='30) Observe that UH α (λ − LFp�−r is a linear combination of operators in the form of (λ − LFp�−r0UH α1 � LFp� (λ − LFp�−r1 · · · UH αℓ � LFp� (λ − LFp�−rℓ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='31) where αj = (j1, j2, · · ·) is multi-index, ℓ ∈ N, ri ⩾ 1, � ri = r + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Note that for any multi-index α, UH α � LFp� is a second order fibrewise differential operator satisfies ��UH α � LFp� s �� Hk,p ⩽ Cp2��s �� Hk+2,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32), for any k ∈ N, there are i, i′ ∈ N∗ such that ��UH α � LFp�� λ − LFp�−1s �� H2k,p ⩽ pi|λ|i′∥s∥H2k,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='33) Let P, Q be differential operators of order 2q and 2q′ with scalar principal symbol and with compact support in BX (x, ε) and BX (x′, ε) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We take r ⩾ (2q + 2q′ + 2)|α| in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='30), then there is a rj ⩾ 2(q + q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='31), we split the operator PUH α � (λ − LFp)−k� Q into the product of two parts P(λ − LFp�−r0UH α1 � LFp� (λ − LFp�−r1 · · · UH αj � LFp� (λ − LFp�−1(λ − LFp�−2q, (λ − LFp�rj−2q′−2q−2(λ − LFp�−2q′−1 · · · UH αℓ � LFp� (λ − LFp�−rℓQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='34) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='33), the first part above is a map from L2-space to itself, and the operator norm is dominated by Cpj0|λ|j′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='5, the adjoint of the second part has the same structure as the first part, and its operator norm is also dominated by Cpj1|λ|j′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Therefore, for some j, j′ ∈ N∗, we have ��PUH α � (λ − LFp)−k� Qs �� L2 ⩽ Cpj|λ|j′∥s∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='35) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='29), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='30), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='35) and the Sobolev inequality, we find that ��UH α � �Gt,ε � tLFp�� (·, ·) �� C q(X×X) ⩽ Cpk exp � c1t − c2 t � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='36) which gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Here we emphasize that the constant in Sobolev inequality is inde- pendent on the dimension of bundle Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Rescaling of the operator LFp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Now we use Bismut-Zhang’s rescaling [12, Chap- ter 4] for two kinds of Clifford variables, which is analogous to Getzler’s rescaling [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Using the same notation as in the beginning of § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2, we fix x0 ∈ M and iden- tify R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) and Fp to � R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) � x0 and Fp,x0 on BTx0X (0, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let Γ0 and ΓFp,u be the corresponding connection forms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put X0 = Tx0X ∼= Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let gTX0 be a Riemannian metric on X0 such that gTX0 = � gTX, on BTx0X (0, inj(X)/2) , gTx0X, outside BTx0X (0, inj(X)) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='37) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 24 and let dvX0 be the associated volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let dvTX be the Riemannian volume form of � Tx0X, gTx0X� , and κ (x) be the smooth positive function defined by dvX0 = κ (Z) dvTX, κ (0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='38) Recall the function f in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any ε > 0 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6), set fε : Rm → R by fε(Z) = f(|Z|/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='39) Put Fp = R[z]�⊗π∗Λ(T ∗S)�⊗Λ(T ∗X) ⊗ Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='40) On the trivial bundle Fp,x0 over X0, we define ∇Fp,x0 = ∇ + fε(Z) � Γ0 Z + ΓFp,u Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='41) Let ∆Fp,x0 be the Laplacian associated with ∇Fp,x0 and gTX0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let ∇TX0 be the Levi-Civita connection of � X0, gTX0� and let (gij) be the inverse of (gij) = � gTX0(∂i, ∂j) � , then ∆Fp,x0 = −gij� ∇ Ep,x0 ∂i ∇ Ep,x0 ∂j − ∇ Ep,x0 ∇T X0 ∂i ∂j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='42) Recall ΛFp in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='32), locally we view ΛFp as an element in C ∞(BTx0X(0, ε), Ep,x0) (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set Λx0,p = fε(Z)ΛFp ∈ C ∞ 0 (Tx0X, Ep,x0), L Fp x0 = ∆Ep,x0 + Λx0,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='43) Then L Fp x0 is a family of differential operators acting on C ∞(Tx0X, Fp,x0) for x0 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Let exp(−tLFp) (x, x′) , (x, x′) be the smooth kernel of exp(−tLFp) with respect to dvX for ∈ M ×S M, and exp(−tL Fp x0 ) (Z, Z′) the smooth kernel of exp(−tL Fp x0 ) with respect to dvX0 for (Z, Z′) ∈ Tx0X × Tx0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' As in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='26), we still denote the pull back bundle of Ep through the projection TX×M TX → M by Ep, then we can view exp(−tL Fp x0 ) (Z, Z′) as a smooth section of Ep on TX ×M TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any ε > 0 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6) and ℓ ∈ N, there exist C, c1, c2 > 0 and k ∈ N such that for any p ∈ N∗, t > 0, we have ��� exp(−tLFp)(x0, x0) − exp(−tL Fp x0 )(0, 0) ��� C ℓ(M) ⩽ Cpkec1t− c2 t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='44) where |·|C ℓ(M) is the C ℓ norm with respect to the parameter x0 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='41), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='42) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='43), L Fp x0 and LFp coincide over BTx0X (0, ε), then from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='13) and the finite propagation speed of wave operator (see [25, Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='1]), we obtain that �Ft,ε � tLFp� (x0, ·) = �Ft,ε � tL Fp x0 � (0, ·) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='45) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='43), L Fp x0 has the same structure as LFp, especially, Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='9 are still true if we replace LFp with L Fp x0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='44) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ Put M Fp x0 = θ√pp−2L Fp x0 θ1/√p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='46) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 25 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any ε > 0 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6), γ > 0 and ℓ ∈ N, there exist C, c1, c2 > 0 such that for any t > 0 and p ∈ N∗ large enough, we have ���θ−1 √ tTrΛ(T ∗X)⊗Fp s � exp(−tMFp)(x0, x0) − exp(−tM Fp x0 )(0, 0) ���� C ℓ(M) ⩽ Ceγt− c2 4t −√γc2p, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='47) where |·|C ℓ(M) is the C ℓ norm with respect to x0 ∈ M, and c1 and c2 are given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='46) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='44) by replacing t with t/p2, there is C > 0 with ���θ−1 √ tTrΛ(T ∗X)⊗Fp s � exp (−tMFp) (x0, x0) − exp (−tM Fp x0 ) (0, 0) ���� C ℓ(M) = ���θ√ p t TrΛ(T ∗X)⊗Fp s � exp (−tp−2LFp) (x0, x0) − exp (−tp−2L Fp x0 ) (0, 0) ���� C ℓ(M) ⩽C(1 + t− dim S/2)pdim S/2 exp � c1tp−2 − c2p2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='48) By the mean value inequality, we have c1tp−2 − c2p2 t ⩽ γt − c2 2t − �γt 2 + c2p2 2t � ⩽ γt − c2 2t − √γc2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='49) It is obvious that pdim S/2e−√γc2p and (1+t− dim S)e−c2/4t are uniformly bounded for p ∈ N∗ and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Therefore, we get (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='47) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='48) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' □ For s ∈ C ∞� Tx0X, Ep,x0 � , Z ∈ Rm, u > 0, we set (Kus) (Z) = s (uZ) and N Fp x0 = K1/pM Fp x0 Kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='50) Let exp(−tN Fp x0 ) (Z, Z′) be the kernel of exp(−tN Fp x0 ) with respect to dvTX, then � exp(−tN Fp x0 )s � (Z) = � K1/p exp(−tM Fp x0 )Kps � (Z) = � Rm exp(−tM Fp x0 )(p−1Z, Z′)s(pZ′)κ(Z′)dvTX(Z′) = p−m � Rm exp(−tM Fp x0 )(p−1Z, p−1Z′)κ(p−1Z′)s(Z′)dvTX(Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='51) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='38) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='51), we get exp(−tN Fp x0 )(0, 0) = p−m exp(−tM Fp x0 )(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='52) We introduce another copy � Tx0X of Tx0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We will add an extra hat when we refer to an element in � Tx0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For s > 0, put cs(ej) = 1 √sej ∧ −√siej �cs(ej) = 1 √sej ∧ +√siej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='53) We denote by � L Fp x0 the operator obtained from N Fp x0 by replacing c(ei), �c(ei) with c1/p(ei) and �c1/p(�ei) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Set �Fp = R[z]�⊗π∗Λ (T ∗S) �⊗Λ (T ∗X) �⊗Λ(� T ∗X) ⊗ Fp, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='54) then � L Fp x0 acts naturally on C ∞(Tx0X, �Fp,x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' We denote by exp(−t � L Fp x0 ) (Z, Z′) the smooth kernels of exp(−t � L Fp x0 ) with respect to dvTX(Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' Put �Ep = R[z]�⊗π∗Λ(T ∗S)�⊗End � Λ(T ∗X) ��⊗End � Λ(� T ∗X) � ⊗ End(Fp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content='55) TOEPLITZ OPERATORS AND THE FULL ASYMPTOTIC TORSION FORMS 26 and denote by π: TX ×M TX → M the natural projection, then exp(−t � L Fp x0 ) (Z, Z′) is a smooth section of π∗�Ep over TX ×M TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E2T4oBgHgl3EQfqgii/content/2301.04040v1.pdf'} +page_content=' For any H ∈ End � Λ(T ∗X) ��⊗End � Λ(� T ∗X) � , it can be expanded uniquely in the fol- lowing form: H = � 1⩽i1<···500% +3 layers +105% +115% +>500% +In both Table 1 and Table 2, lower percentage indicates better +model performance. Although embedding model with BERT is the +best one in terms of model quality, its inference latency increases by +4 times compared to baseline, which is very slow. Considering we +need to process hundreds of millions of customers offline, the total +running time of our system using BERT is not ideal. Embedding +model with word embedding strikes a good balance between quality +and inference latency. +6 +CONCLUSIONS +In this paper, we propose an effective and efficient marketing au- +dience expansion system, which can handle hundreds of millions +of customers. We use a deep learning model to generate customer +embeddings. The deep learning model can handle both dense nu- +merical features and sparse categorical features. And the model +encodes location embedding using transfer learning Our system +has great adaptability by constructing different similarity metrics +for different campaigns. The similarity metrics are interpretable +and meaningful and can reflect different business metrics. Extensive +experiments have been conducted to demonstrate the scalability, +efficiency and quality of our system and embedding model. +There are several directions for the future work in our market- +ing audience expansion system. First, we can explore the direction +of combining both similarity-based and classification-based ap- +proaches for searching lookalike customers. Second, we can study +how to filter and rank the lookalike customers using machine learn- +ing models to improve ranking quality. +REFERENCES +[1] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. BERT: +Pre-training of Deep Bidirectional Transformers for Language Understanding. +https://doi.org/10.48550/ARXIV.1810.04805 +[2] Stephanie deWet and Jiafan Ou. 2019. Finding users who act alike: transfer +learning for expanding advertiser audiences. 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Query-Driven Knowledge Base Com- +pletion using Multimodal Path Fusion over Multimodal Knowledge Graph. +https://doi.org/10.48550/ARXIV.2212.01923 +[19] Yang Peng, Daisy Zhe Wang, Ishan Patwa, Dihong Gong, and Chunsheng Victor +Fang. 2015. Probabilistic Ensemble Fusion for Multimodal Word Sense Disam- +biguation. In Multimedia (ISM), 2015 IEEE International Symposium on. IEEE, +172–177. +[20] Yang Peng, Xiaofeng Zhou, Daisy Zhe Wang, and Chunsheng Victor Fang. 2016. +Scalable image retrieval with multimodal fusion. In The Twenty-Ninth Interna- +tional Flairs Conference. +[21] Yang Peng, Xiaofeng Zhou, Daisy Zhe Wang, Ishan Patwa, Dihong Gong, and +Chunsheng Fang. 2016. Multimodal Ensemble Fusion for Disambiguation and +Retrieval. IEEE MultiMedia (2016). +5 + diff --git a/MNE1T4oBgHgl3EQfZQQr/content/tmp_files/load_file.txt b/MNE1T4oBgHgl3EQfZQQr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d06ea6c81030ef90255007db12de896e1e62cf1 --- /dev/null +++ b/MNE1T4oBgHgl3EQfZQQr/content/tmp_files/load_file.txt @@ -0,0 +1,300 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf,len=299 +page_content='Finding Lookalike Customers for E-Commerce Marketing Yang Peng yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='peng@walmart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='com Walmart Global Tech Changzheng Liu changzheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='liu@walmart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='com Walmart Global Tech Wei Shen wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='shen@walmart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='com Walmart Global Tech ABSTRACT Customer-centric marketing campaigns generate a large portion of e-commerce website traffic for Walmart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' As the scale of customer data grows larger, expanding the marketing audience to reach more customers is becoming more critical for e-commerce companies to drive business growth and bring more value to customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In this paper, we present a scalable and efficient system to expand targeted audience of marketing campaigns, which can handle hundreds of millions of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We use a deep learning based embedding model to represent customers and an approximate nearest neighbor search method to quickly find lookalike customers of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The model can deal with various business interests by constructing interpretable and meaningful customer similarity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We con- duct extensive experiments to demonstrate the great performance of our system and customer embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 1 INTRODUCTION In customer relationship management (CRM) systems, customer acquiring and retention are crucial for marketing success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Expand- ing the set of targeted customers is a very important component in CRM systems for both customer acquiring and retention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In this article, we consider the problem of building a large scale marketing audience expansion system, aiming at driving e-commerce growth by finding more customers for CRM email and push marketing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Formally speaking, the problem to solve in this paper is: given a set of existing customers (seed customer set) for a marketing campaign, how to find more customers that are similar to these seed customers, so that we can increase revenue and drive more traffic for Walmart e-commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' One of the biggest challenges we face in this problem is the scale of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Walmart has hundreds of millions of active customers in the US market alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Each customer could have hundreds or even thousands of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' And customer data is increasing rapidly year by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Thus building a scalable and efficient system to handle ever- increasing big customer data is our top priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Besides scalability, we need to take into account the interpretability of our method when finding lookalike customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Model interpretability not only helps explain how our method works to business partners, but also provides an intuitive way for examining system quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In this article, we propose a scalable and efficient audience expan- sion system for marketing campaigns, which can handle hundreds of millions of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our system can generate low dimensional dense embeddings to represent customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In the customer embedding space, we use the cosine dis- tance between the two customer embedding vectors as the estimation of the similarity between two customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The similarity metric we design can measure different busi- ness interests (such as purchases, visits and engagements), which are interpretable and meaningful business goals for marketing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We use approximate nearest neighbor search to quickly find lookalike customers to the seed customers in the customer embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' To improve the quality of the customer embedding model, our model ingests multimodal features from different data sources, such as transactions, visits, engagements and customer metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Multi- modal fusion techniques have demonstrated great benefits for tasks such as information retrieval, information extraction and classi- fication [3, 13, 14, 16–21] by leveraging the complementary and correlative relations between different types of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our embed- ding model can also achieve better quality for audience expansion by combining different types of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our contributions are shown below: We propose an effective and efficient marketing audience expansion system, which can handle hundreds of millions of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We use a deep learning model to generate customer em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The deep learning model can handle both dense numerical features and sparse categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' And the model encodes location embedding using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our system has great adaptability by constructing different similarity metrics for different campaigns and business goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The similarity metrics are both interpretable and meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We develop a scalable and efficient approximate nearest neighbor search method based on FAISS to quickly find sim- ilar customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We design multimodal features from various data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Extensive experiments have been conducted to demonstrate the scalability, efficiency and quality of our system and em- bedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Overview Related work on lookalike modeling and audience ex- pansion systems is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The overview of our system is presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The embedding model is illus- trated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We demonstrate the effectiveness and efficiency of our system through extensive experiments in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The conclusions and future work of our lookalike model are discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 2 RELATED WORK In this section, we will discuss the previous work on audience expansion systems, use representation models and fast similarity search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' For the literature review, we mostly focus on re- lated work in industry solutions, since we are tacking large-scale or even web-scale datasets which are very rarely studied in academic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='03147v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='LG] 9 Jan 2023 Figure 1: Audience Expansion System in Walmart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='1 Audience Expansion In [12], Ma et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' discussed three types of approaches for audience expansion for marketing campaigns: similarity-based, regression- based and segmentation-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' They proposed a graph- based lookalike system in Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' advertising platform, which takes advantages of both simple similarity and regression-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In [11], Liu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' developed two methods to achieve audience ex- pansion in LinkedIn: campaign-agnostic expansion based on user attributes and campaign-aware expansion using nearest neighbor search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In [6], Jiang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' discussed rule-based, similarity-based and model-based methods for finding lookalike users and proposed a deep neural network classification model for audience expansion in MiningLamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In [2], deWet et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' proposed a two-stage embedding- based audience expansion model that is deployed in production at Pinterest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' For the first stage, they trained a global user embedding model on sitewide user activity logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In the second stage, they used statistical techniques to create lightweight seed list representations in the embedding space for each advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In our system, we first use a similarity-based approach to search lookalike customers from the whole customer universe and then rank these new customers based on their similarity scores or a sepa- rate classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We choose the similarity-based approach for the first step because of its great scalability and low search latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Similarity-based approaches usually require building user representations first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In the next section, we will discuss embedding models for representing customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='2 User Representation Embedding models are very useful in terms of transforming high dimensional sparse feature vectors of customers to low dimensional dense representations of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Embedding models have been widely used in industry, for example, for search engine marketing at Walmart [7–9], search ranking at Airbnb [4], and recommendation at Pinterest [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Embedding models can be trained in a way to capture similarity between customers, so that we can use approx- imate nearest neighbor search methods to quickly find lookalike customers of seed customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In our case, we use a two-tower ar- chitecture to train the customer embedding model, which is well recognized in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our novelty in user representation is modeling business metrics using the cosine similarity between two embedding vectors, which has great interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='3 Similarity Search After user representation, we need a fast approach to search looka- like customers from a very large customer universe with potential size of hundreds of millions of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Scanning the whole cus- tomer universe is not scalable or efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Approximate nearest neighbor search is the most popular approach used in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' For example, locality sensitive hashing (LSH) has been used in previous work [2, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' There are several good open-source tools for approximate nearest neighbor search, such as ScaNN [5] by Google and FAISS [10] by Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We choose FAISS for lookalike customer search for its scalability of handling billions of vectors and support of various types of distance measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' dot-product, cosine, Euclidean distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 2 Online Ranking Embedding Search Embedding Indexing Offline3 SYSTEM OVERVIEW In this section, we explain our audience expansion system pipeline for finding lookalike customers for e-commerce marketing in Wal- mart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The system diagram is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our audience expansion system has an online stage and an offline stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In the offline stage, we generate customer embeddings for all the cus- tomers in the customer universe and then build indexing on the customer embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In the online stage, the seed customers are transformed into embeddings and then we search for lookalike customers using the pre-built indexes from the offline stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' After getting the lookalike customers, we will conduct filtering and rank- ing on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The ranking approach can be based on their similarity scores or a different classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In this article, we will mostly focus on the embedding model and explain how we build this model in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The customer embedding model yields unified dense representations of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our customer embedding model can be utilized in a lot of use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Besides finding lookalike customers, customer embeddings can be employed as input features in other customer models, such as purchase propensity models and life-time value models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Ranking method is not a focus in this paper and will be studied in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The scalability issue in this system is how to quickly search for lookalike customers from a pool of hundreds of millions of candidate customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' To narrow down the customers for consideration, we build indexes using FAISS [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We choose FAISS for a few reasons as listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' FAISS supports a wide range of different indexes and provides both CPU and GPU implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' FAISS also supports different types of distance measures, such as Euclidean distances, dot products and cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' And we can utilize compression techniques in FAISS to process large datasets that cannot fit in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' How we implement indexing and search is not the main focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' If you are interested in more details about FAISS, please visit their Github project page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 4 EMBEDDING MODEL In this section, we present our deep learning based embedding model, which can capture the similarity between customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' To design this model, we need to first define the similarity metric be- tween two customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' While some models in previous work [2] learned relative similarity scores between positive and negative customer pairs, we use direct similarity metrics between two cus- tomers, which are interpretable, meaningful and reflecting business metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The similarity metrics we define can be very useful in improving and explaining marketing campaign performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We use a two-tower architecture to calculate the cosine distance between two customer embedding vectors and use the cosine dis- tance as the estimate of similarity metric between two customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The customer embedding model is trained to minimize the total loss between cosine distances and true similarity scores in the training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='1 Similarity Metric Definition In Walmart, we care about a lot of different business metrics for marketing campaigns, such as transactions, website visits, campaign engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' When expanding audience for marketing campaigns, it’s a business advantage to find new customers that have similar behavior on Walmart e-commerce website as the existing seed customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Finding new customers with similar business metrics can allow marketing campaigns to maintain a similar customer distribution after expansion, which is very beneficial for cold-start campaigns or conversion campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' There are three types of business metrics of particular interest to our marketing campaigns: transactions, visits and engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Let’s take transactions as an example to illustrate how we define the similarity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Let’s say there are a list of product categories in Walmart catalog, 𝑐1,𝑐2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=',𝑐𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Customer A has made purchase orders in these categories, 𝑂𝐴 = (𝑎1,𝑎2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=',𝑎𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Customer B has also made purchase orders in these categories, 𝑂𝐵 = (𝑏1,𝑏2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=',𝑏𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The similarity between A and B is defined as: 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝐴, 𝐵) = 𝑐𝑜𝑠𝑖𝑛𝑒_𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦(𝑂𝐴,𝑂𝐵) (1) We can also use other similarity distance functions, such as Jac- card similarity and Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' This method of calculating similarity metrics can also be applied for visits and engagements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Figure 2: Two-Tower Model Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='2 Two-Tower Architecture After defining the similarity metric between customers, the next task is to build a machine learning model to predict the similarity metric given customer features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Our approach is: (1) extract raw features for customers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' (2) transform customer features into customer embeddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' (3) calculate the cosine distances of customer embedding pairs in the embedding space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' (4) use the cosine distances as the estimates of true similarity scores of customer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The process is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The loss function for optimization is the L1 loss between cosine distance prediction and true similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The multimodal customer features are extracted from multiple data sources, including transaction data, visit data, engagement data and demographics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The multimodal customer features 3 Similarity Embedding Embedding Feature Feature Extraction ExtractionFigure 3: The Embedding Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' are composed of dense numerical features (such as number of or- ders, GMV, number of visits) and sparse categorical features (such as gender, education, occupation, location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' For low dimensional categorical features, we can use one-hot encoding to transform them into numbers, for example gender and education level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The location feature contains street address, city name, state name and zip code, so it’s a very high dimensional categorical feature, which is too inefficient to use one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We convert the location feature into location embedding using transfer learning and then concatenate it with other features together, which is explained in the next part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='3 Embedding Model Structure The embedding model is described in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We have the trans- action, visit, engagement, demographics and location features as input to the embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The location features are treated as textual sentences and then transformed to location embeddings using transfer learning of pre-trained text embedding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Then we concatenate the numerical features and location embeddings as input to the final feed-forward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The feed-forward network is composed of multiple fully connected ReLU layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The output of the feed-forward network is a 128 dimensional embedding as customer representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content='1 Location Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We tried a few different approaches to convert location text to location embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' One approach is to use a pre-trained word embedding model in PyTorch (GloVe), which is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Another approach is to use the state-of-the-art BERT model [1] for text representation learning, which is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We also fine tune the pre-trained BERT model in our training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Figure 4: Location Embedding using Word Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Figure 5: Location Embedding using BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 4 Fully Connected ReLU Layer Fully Connected ReLU Layer Concatenation Text EmbeddingAveraging Word Embedding TokenizationLinear Layer with ReLu Activation BERT Tokenization5 EXPERIMENTAL RESULTS For evaluation, we setup the training, validation and testing datasets by different time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' For example, we can use data of last 𝑛 years as training data, data of next one month or one quarter as validation and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The evaluation metric for model quality is mean absolute error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Due to Walmart’s privacy policy, the results are presented as percentage proportions to the baseline embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' In this section, we compare the quality and inference time of different model setups, from different numbers of fully connected layers to different location embedding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The baseline model is using two fully connected ReLU layers and no location embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The results are shown in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Table 1: Quality (MAE) of Embedding Model with Different Setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' No Location Word Embedding BERT 2 layers 100% 97% 91% 3 layers 94% 87% 86% Table 2: Inference Time of Embedding Model with Different Setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' No Location Word Embedding BERT 2 layers 100% 110% >500% 3 layers 105% 115% >500% In both Table 1 and Table 2, lower percentage indicates better model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Although embedding model with BERT is the best one in terms of model quality, its inference latency increases by 4 times compared to baseline, which is very slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Considering we need to process hundreds of millions of customers offline, the total running time of our system using BERT is not ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Embedding model with word embedding strikes a good balance between quality and inference latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 6 CONCLUSIONS In this paper, we propose an effective and efficient marketing au- dience expansion system, which can handle hundreds of millions of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' We use a deep learning model to generate customer embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The deep learning model can handle both dense nu- merical features and sparse categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' And the model encodes location embedding using transfer learning Our system has great adaptability by constructing different similarity metrics for different campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' The similarity metrics are interpretable and meaningful and can reflect different business metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Extensive experiments have been conducted to demonstrate the scalability, efficiency and quality of our system and embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' Multimodal Ensemble Fusion for Disambiguation and Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' IEEE MultiMedia (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE1T4oBgHgl3EQfZQQr/content/2301.03147v1.pdf'} diff --git a/N9AyT4oBgHgl3EQfgvi_/content/tmp_files/2301.00365v1.pdf.txt b/N9AyT4oBgHgl3EQfgvi_/content/tmp_files/2301.00365v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e23c84165c65adaff38f76f58f65f84fd86ecd5 --- /dev/null +++ b/N9AyT4oBgHgl3EQfgvi_/content/tmp_files/2301.00365v1.pdf.txt @@ -0,0 +1,850 @@ +arXiv:2301.00365v1 [math.NT] 1 Jan 2023 +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS +DEFINED BY x7 + ax + b +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +Abstract. Let f(x) = x7 + ax + b be an irreducible polynomial having integer coeffi- +cients and K = Q(θ) be an algebraic number field generated by a root θ of f(x). In the +present paper, for every rational prime p, our objective is to determine the necessary and +sufficient conditions involving only a, b so that p is a divisor of the index of the field K. +In particular, we provide sufficient conditions on a and b, for which K is non-monogenic. +In a special case, we show that if either 8 divides both a ± 1, b or 32 divides both a + 4, +b, then K is non-monogenic. We illustrate our results through examples. +1. Introduction and statements of results +Let K = Q(θ) be an algebraic number field with θ in the ring OK of algebraic integers +of K. Let f(x) ∈ Z[x] be the minimal polynomial of θ having degree n over the field +Q of rational numbers. +It is a basic result in algebraic number theory that OK is a +free abelian group of rank n. An algebraic number field K is said to be monogenic if +there exists some α ∈ OK such that {1, α, · · · , αn−1} is an integral basis of K. In this +case OK = Z[α], i.e., [OK : Z[α]] = 1. If K does not have any such α, then the field +K is said to be non-monogenic. It is well-known that every quadratic and cyclotomic +field is monogenic. It is important to know that whether a number field is monogenic +or not. It was Dedekind, who gave the first non-monogenic number field K = Q(ξ), +where ξ is a root of the polynomial x3 − x2 − 2x − 8 (cf. [15, page 64]). The problem +of testing the monogenity of number fields and constructing power integral bases have +been intensively studied (cf. [1], [2], [6], [9], [10], [11], [14], [18]). In 1984, Funakura [5] +gave necessary and sufficient conditions on those integers m for which the quartic field +Q(m1/4) is monogenic. Ahmad, Nakahara and Husnine in [1], [2] proved that for a square +free integer m not congruent to ±1 mod 9, a pure field Q(m1/6) having degree six over +Q is monogenic when m ≡ 2 or 3 mod 4 and it is non-monogenic when m ≡ 1 mod 4. +2010 Mathematics Subject Classification. 11R04, 11R21. +Key words and phrases. Monogenity, Theorem of Ore, prime ideal factorization. +The first author is thankful to SERB grant SRG/2021/000393 and IIT Madras for NFIG RF/22- +23/1035/MA/NFIG/009034. The second author is grateful to the Council of Scientific and Industrial +Research, New Delhi for providing financial support in the form of Senior Research Fellowship through +Grant No. 09/135(0878)/2019-EMR-1. The third author is grateful to the University Grants Commi- +sion, New Delhi for providing financial support in the form of Junior Research Fellowship through Ref +No.1129/(CSIR-NET JUNE 2019). +1 + +2 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +In 2017, Ga´al and Remete [6] studied monogenity of algebraic number fields of the type +Q(m1/n) where 3 ≤ n ≤ 9 and m is square free. In [9], A. Jakhar and S. Kumar provides +some sufficient conditions for which the number field K defiend by the sextic trinomial +x6 + ax + b ∈ Z[x], is non-monogenic. In [7], Ga´al studied monogenity of number fields +defined by some sextic irreducible trinomials. +Recall that for an algebraic number field L = Q(ξ) with ξ an algebraic integer satis- +fying a monic irreducible polynomial g(x) over Q, the discriminant D of g(x) and the +discriminant dL of L are related by the formula +D = (ind ξ)2 · dL. +(1.1) +Throughout this paper, ind θ will denote the index of the subgroup Z[θ] in OK and +i(K) will stand for the index of the field K defined by +i(K) = gcd{ind α | K = Q(α) and α ∈ OK}. +A prime divisor of i(K) is called a common index divisor of K. Note that i(K) = 1, for +every monogenic number field K. But there exist non-monogenic number fields having +i(K) = 1, e.g., K = Q( +3√ +175) is non-monogenic with i(K) = 1. +In what follows, let K = Q(θ) be an algebraic number field with θ a root of an irre- +ducible trinomial f(x) = x7 + ax + b ∈ Z[x], then for every rational prime p, we provide +necessary and sufficient conditions on a, b, so that p is a common index divisor of K. In +particular, under these conditions K is non-monogenic. Our method is based on the the- +ory of Newton polygons and theorem of Ore. In 1878, (see [ [4], Theorem 6.1.4]) Dedekind +gave a criterion to determine whether p divides the index [OK : Z[θ]]. When p divides +the index [OK : Z[θ]], then a method of Ore 1928, can be used in order to evaluate the +prime ideal factorization of pOK (see [19], [20]). If Ore’s method doesn’t work, then an +algorithm developed by Guardia, Montes, and Nart [21], based on higher order Newton +polygons can be used to determine the prime ideal factorization of pOK. +For a prime p and a non-zero m belonging to the ring Zp of p-adic integers, vp(m) will +denote the highest power of p dividing m. For a non-zero integer l, let lp denote +l +pvp(l). +If a rational prime p is such that p6 divides a and p7 divides b, then θ/p is a root of the +polynomial x7 + (a/p6)x + (b/p7) having integer coefficients. So we may assume that for +each prime p +either vp(a) ≤ 5 or vp(b) ≤ 6. +(1.2) +Also, D will stand for the discriminant of f(x). One can check that +D = −77b6 − 66a7. +(1.3) +With the above notations and assumption (1.2), we prove +Theorem 1.1. Let K = Q(θ) be an algebraic number field with θ a root of an irreducible +polynomial f(x) = x7 + ax + b. If u = v2(D)−6 +2 +and ρ = 2u + 7b +6a, then 2 | i(K) if and only +if one of the following hold: + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +3 +(1) a ≡ 3 mod 4 and b ≡ 0 mod 8. +(2) a ≡ 3 mod 8 and b ≡ 4 mod 8. +(3) a ≡ 1 mod 4 and b ≡ 0 mod 4. +(4) a ≡ 1 mod 4, b ≡ 2 mod 4, v2(D) is odd. +(5) a ≡ 1 mod 4, b ≡ 2 mod 4, v2(D) is even and v2(f(ρ)) = 2u + 1. +(6) a ≡ 1 mod 4, b ≡ 2 mod 4, v2(D) is even and v2(f(ρ)) ≥ 2u + 3. +(7) a ≡ 28 mod 32 and b ≡ 0 mod 32. +(8) a ≡ 48 mod 64 and b ≡ 0 mod 128. +The next three corollaries follow immediately from the above theorem. +Corollary 1.2. Let K = Q(θ) be an algebraic number field, where θ satisfies the ir- +reducible polynomial x7 + ax + b ∈ Z[x]. +If 8 divides both a ± 1 and b, then K is +non-monogenic. +Corollary 1.3. Let K = Q(θ) be an algebraic number field generated by a root θ of an +irreducible polynomial x7 + ax + b belonging to Z[x]. If both a + 4 and b are divisible by +32, then K is non-monogenic. +Corollary 1.4. Let K = Q(θ) be an algebraic number field generated by a root θ of an +irreducible polynomial x7 + ax + b belonging to Z[x]. Then K is non-monogenic, if both +a + 16 and b are divisible by 128. +Theorem 1.5. Let K = Q(θ) be an algebraic number field with θ a root of an irreducible +polynomial f(x) = x7 + ax + b. Then 3 | i(K) if and only if one of the following hold: +(1) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ 1 mod 3, v3(a + 7) = 2 and v3(b − a − 1) ≥ 3. +(2) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3, +2v3(a + 7) > v3(b − a − 1) + 1 and (b − a − 1)3 ≡ −1 mod 3. +(3) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and +2v3(a + 7) < v3(b − a − 1) + 1. +(4) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ −1 mod 3, v3(a + 7) = 2 and v3(b + a + 1) ≥ 3. +(5) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3, +2v3(a + 7) > v3(a + b + 1) + 1 and (b + a + 1)3 ≡ −1 mod 3. +(6) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and +2v3(a + 7) = v3(a + b + 1) + 1. +(7) a ≡ 5 mod 9 and v3(b) = 1. +(8) v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, 1), (1, −1)} mod 3. +(9) v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, −1), (1, 1)} mod 3. +The following corollary follow immediately from the above theorem. +Corollary 1.6. Let K = Q(θ) where θ satisfies the irreducible polynomial x7 + ax + b. +Then K is non-monogenic, if one of the following hold: +(1) a ≡ 5 mod 9 and b ≡ 3 mod 9. + +4 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +(2) a ≡ 5 mod 9 and b ≡ 6 mod 9. +Theorem 1.7. Let K = Q(θ) be an algebraic number field with θ a root of an irreducible +polynomial f(x) = x7 + ax + b ∈ Z[x]. Let p ≥ 5 be a rational prime, then p ∤ i(K). +Remark 1.8. Note that when a = 0 and b is a non-zero odd integer, then the index of K +is 1. +Corollary 1.9. Let K = Q(θ) be an algebraic number field generated by a root θ of an +irreducible polynomial f(x) = x7 + ax+ b ∈ Z[x]. If a ≡ 15 mod 72 and b ≡ 12 mod 72, +then in view of Theorem 1.1, 1.5 and 1.7, we have i(K) = 1 +We now provide some examples of non-monogenic number fields. +Example 1.10. Let K = Q(θ) where θ is a root of f(x) = x7 + 17x+ 51. Note that f(x) +satisfies Eisenstein’s criterion with respect to 17, hence it is irreducible over Q. So, by +Corollary 1.6 K is non-monogenic. +Example 1.11. Let K = Q(θ) with θ satisfying f(x) = x7 + 7x + 56. It can be easily +seen that f(x) is 7-Eisenstein. Hence, in view of Corollary 1.4 K is non-monogenic. +2. Preliminary Results +Let K = Q(θ) be an algebraic number field with θ a root of a monic irreducible +polynomial f(x) belonging to Z[x]. In what follows, OK will stand for the ring of algebraic +integers of K. For a rational prime p, let Fp denote the finite field with p elements. +The following lemma (cf. [17, Theorem 2.2]) will play an important role in the proof of +Theorems 1.1, 1.5 and 1.7. +Lemma 2.1. Let K be an algebraic number field and p be a rational prime. Then p is +a common index divisor of K if and only if for some positive integer h, the number of +distinct prime ideals of OK lying above p having residual degree h is greater than the +number of monic irreducible polynomials of degree h in Fp[x]. +We shall first introduce the notion of Gauss valuation, φ-Newton polygon and Newton +polygon of second order, where φ(x) belonging to Zp[x] is a monic polynomial with φ(x) +irreducible over Fp. +Definition 2.2. The Gauss valuation of the field Qp(x) of rational functions in an inde- +terminate x which extends the valuation vp of Qp and is defined on Qp[x] by +vp,x( a0 + a1x + a2x2 + ..... + asxs) = min{vp(ai), 1 ≤ i ≤ s}, ai ∈ Qp. +Definition 2.3. Let p be a rational prime. Let φ(x) ∈ Zp[x] be a monic polynomial +which is irreducible modulo p and f(x) ∈ Zp[x] be a monic polynomial not divisible by +φ(x). Let +n +� +i=0 +ai(x)φ(x)i with deg ai(x) < deg φ(x), an(x) ̸= 0 be the φ(x)-expansion of + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +5 +f(x) obtained on dividing it by the successive powers of φ(x). Let Pi stand for the point +in the plane having coordinates (i, vp,x(an−i(x))) when an−i(x) ̸= 0, 0 ≤ i ≤ n. Let µij +denote the slope of the line joining the point Pi with Pj if an−i(x)an−j(x) ̸= 0. Let i1 be +the largest positive index not exceeding n such that +µ0i1 = min{ µ0j | 0 < j ≤ n, an−j(x) ̸= 0}. +If i1 < n, let i2 be the largest index such that i1 < i2 ≤ n with +µi1i2 = min{ µi1j | i1 < j ≤ n, an−j(x) ̸= 0} +and so on. The φ-Newton polygon of f(x) with respect to p is the polygonal path having +segments P0Pi1, Pi1Pi2, . . . , Pik−1Pik with ik = n. These segments are called the edges of +the φ-Newton polygon and their slopes form a strictly increasing sequence; these slopes +are non-negative as f(x) is a monic polynomial with coefficients in Zp. +Definition 2.4. Let φ(x) ∈ Zp[x] be a monic polynomial which is irreducible modulo a +rational prime p having a root α in the algebraic closure �Qp of Qp. Let f(x) ∈ Zp[x] be +a monic polynomial not divisible by φ(x) with φ(x)-expansion φ(x)n + an−1(x)φ(x)n−1 + +· · ·+a0(x) such that f(x) is a power of φ(x). Suppose that the φ-Newton polygon of f(x) +consists of a single edge, say S, having positive slope denoted by l +e with l, e coprime, i.e., +min +�vp,x(an−i(x)) +i +| 1 ≤ i ≤ n +� += vp,x(a0(x)) +n += l +e +so that n is divisible by e, say n = et and vp,x(an−ej(x)) ≥ lj for 1 ≤ j ≤ t. Thus the +polynomial bj(x) := an−ej(x) +plj +has coefficients in Zp and hence bj(α) ∈ Zp[α] for 1 ≤ j ≤ t. +The polynomial T(Y ) in an indeterminate Y defined by T(Y ) = Y t+ +t� +j=1 +bj(α)Y t−j having +coefficients in Fp[α] ∼= Fp[x] +⟨φ(x)⟩ is called residual polynomial of f(x) with respect to (φ, S). +The following definition gives the notion of residual polynomial when f(x) is more +general. +Definition 2.5. Let φ(x), α be as in Definition 2.4. Let g(x) ∈ Zp[x] be a monic poly- +nomial not divisible by φ(x) such that g(x) is a power of φ(x). Let λ1 < · · · < λk be +the slopes of the edges of the φ-Newton polygon of g(x) and Si denote the edge with +slope λi. In view of a classical result proved by Ore (cf. [3, Theorem 1.5], [13, Theorem +1.1]), we can write g(x) = g1(x) · · · gk(x), where the φ-Newton polygon of gi(x) ∈ Zp[x] +has a single edge, say S′ +i, which is a translate of Si. Let Ti(Y ) belonging to Fp[α][Y ] +denote the residual polynomial of gi(x) with respect to (φ, S′ +i) described as in Definition +2.4. For convenience, the polynomial Ti(Y ) will be referred to as the residual polyno- +mial of g(x) with respect to (φ, Si). The polynomial g(x) is said to be p-regular with +respect to φ if none of the polynomials Ti(Y ) has a repeated root in the algebraic closure +of Fp, 1 ≤ i ≤ k. In general, if F(x) belonging to Zp[x] is a monic polynomial and +f(x) = φ1(x)e1 · · · φr(x)er is its factorization modulo p into irreducible polynomials with + +6 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +each φi(x) belonging to Zp[x] monic and ei > 0, then by Hensel’s Lemma there exist +monic polynomials f1(x), · · · , fr(x) belonging to Zp[x] such that f(x) = f1(x) · · · fr(x) +and f i(x) = φi(x)ei for each i. The polynomial f(x) is said to be p-regular (with respect +to φ1, · · · , φr) if each fi(x) is p-regular with respect to φi. +To determine the number of distinct prime ideals of OK lying above a rational prime +p, we will use the Newton polygon of second order and the following theorem which is a +weaker version of Theorem 1.2 of [13]. +Theorem 2.6. Let L = Q(ξ) be an algebraic number field with ξ satisfying an irre- +ducible polynomial g(x) ∈ Z[x] and p be a rational prime. Let φ1(x)e1 · · · φr(x)er be the +factorization of g(x) modulo p into powers of distinct irreducible polynomials over Fp +with each φi(x) ̸= g(x) belonging to Z[x] monic. Suppose that the φi-Newton polygon of +g(x) has ki edges, say Sij having slopes λij = lij +eij with gcd (lij, eij) = 1 for 1 ≤ j ≤ ki. +If Tij(Y ) = +sij +� +s=1 +Uijs(Y ) is the factorization of the residual polynomial Tij(Y ) into distinct +irreducible factors over Fp with respect to (φi, Sij) for 1 ≤ j ≤ ki, then +pOL = +r� +i=1 +ki +� +j=1 +sij +� +s=1 +p +eij +ijs, +where pijs are distinct prime ideals of OL having residual degree deg φi(x) × deg Uijs(Y ). +Let L = Q(γ) where γ is a root of a monic polynomial g(x) = anxn + · · · + a0 ∈ +Z[x], a0 ̸= 0. +Let p be a prime number such that g(x) ≡ xn(p). +Suppose that the +p-Newton polygon of g(x) consists of a single edge with positive slope λ = +l +e, where +gcd(l, e) = 1. Let the residual polynomial Tg(Y ) ∈ Fp[Y ] of g(x) is a power of monic +irreducible polynomial ψ(Y ) over Fp, i.e., Tg(Y ) = ψ(Y )s in Fp[Y ], where s ≥ 2. In +this case, we construct a key polynomial Φ(x) attached with the slope λ such that the +following hold: +(i) Φ(x) is congruent to a power of x modulo p. +(ii) The p-Newton polygon of Φ(x) of first order is one-sided with slope λ. +(iii) The residual polynomial of Φ(x) with respect to p is ψ(Y ) in Fp[Y ]. +(iv) deg Φ(x) = e deg ψ(Y ). +As described in [8, Section 2.2], the data (x; λ, ψ(Y )) determines a p-adic valuation V +of the field Qp(x) which satisfies the following properties: +(i) V (x) = l where λ = l +e with gcd(l, e) = 1. +(ii) If p(x) = +� +0≤i +bixi ∈ Zp[x] is any polynomial, then +V (p(x)) = e min +0≤i {vp(bi) + iλ} +(2.1) + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +7 +We define the above valuation V to the valuation of second order. +If g(x) = +u +� +i=0 +ai(x)Φ(x)i ∈ Zp[x] is a Φ-adic expansion of g(x), then the Newton polygon +of g(x) with respect to V (also called V -Newton polygon of g(x) of second order) is the +lower convex hull of the set of the points (i, V (au−i(x)Φ(x)i)) of the Euclidean plane. +Let the V -Newton polygon of g(x) of second order has k-sides E1, · · · , Ek with positive +slopes λ1, · · · , λk. Let λt = lt +et with gcd(lt, et) = 1 and [at, bt] denote the projection to +the horizontal axis of the side of slope λt for 1 ≤ t ≤ k. Then, there is a natural residual +polynomial ψt(Y ) of second order attached to each side Et, whose degree coincides with +the degree of the side (i.e. +bt−at +et ) [8, Section 2.5]. Only those integral points of the V - +Newton polygon of g(x) which lie on the side, determine a non-zero coefficient of this +second order residual polynomial. We define g(x) to be ψt-regular when the second order +residual polynomial ψt(Y ) attached to the side Et of the V -Newton polygon of g(x) of +second order is separable in Fp[Y ] +⟨ψ(Y )⟩. We define g(x) to be V -regular if g(x) is ψt-regular +for each t, 1 ≤ t ≤ k. Further, if each residual polynomial ψt(Y ), t ∈ {1, 2, · · · , k}, +is irreducible in Fp[Y ] +⟨ψ(Y )⟩ , then each ψt(Y ) provides a prime ideal having residual degree +deg ψ · deg ψt and ramification index e · et. +3. Proof of Theorems 1.1, 1.5 and 1.7. +Proof of Theorem 1.1. If 2 is a common index divisor of K, then 2 | D, i.e. 2 | b. +Case A1: a ≡ 1 mod 2, b ≡ 0 mod 2. In this case, f(x) ≡ x(1 + x + x2)2(x + 1)2 +mod 2. Let φ1(x) = x, φ2(x) = 1 + x + x2 and φ3(x) = x + 1. The φ1-Newton polygon of +f(x) has a single edge joining the points (0, 0) and (7, v2(b)). The residual polynomial +associated with this edge is linear. Therefore φ provides one prime ideal say p1 of residual +degree 1. So +2OK = p1P, where P is an ideal of OK +(3.1) +Using Theorem 2.6, we see that φ2 provides prime ideals of residual degree multiple of 2. +Therefore keeping in mind Lemma 2.1, 2 | i(K) if and only if φ2 and φ3 provides atleast +two prime ideals of residual degree t, where t ∈ {1, 2}. The φi, for i = 2, 3 expansion of +f(x) is +(x − 3)φ3 +2 + (3x + 5)φ2 +2 − (4x + 2)φ2 + b + x(a + 1) +(3.2) +φ7 +3 − 7φ6 +3 + 21φ5 +3 − 35φ4 +3 + 35φ3 +3 − 21φ2 +3 + (a + 7)φ3 + (b − a − 1) +(3.3) +The φ2-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (1, 0), (2, 1) +and (3, min{v2(b), v2(a + 1)}) and φ3-Newton polygon of f(x) is the lower convex hull of +the points (0, 0), (1, 0), (2, 0), (3, 0) , (4, 0), (5, 0), (6, v2(a+7)) and (7, v2(b−a−1)). +Let a ≡ 3 mod 4 and b ≡ 2 mod 4, then v2(a + 1) ≥ 2. For each i = 2, 3, φi-Newton +polygon of f(x) has a single edge of slope 1 +2. The residual polynomial attached to this +edge is linear. Thus φ2 provides one prime ideal say p2 of residual degree 2 and φ3 provides +one prime ideal say p3 of residual degree 1. So by Theorem 2.6, P = p2 +2p2 +3. Thus 2 ∤ i(K). + +8 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +Let a ≡ 3 mod 8 and b ≡ 0 mod 8 , then for each i = 2, 3, the φi-Newton polygon +of f(x) has a single edge of positive slope. The residual polynomial of f(x) associated to +this edge with respect to φ2 is (Y − (x + 1))(Y − 1) over F2[x] +⟨φ2(x)⟩ and with respect to φ3 is +Y 2 +Y +¯1 ∈ F2[Y ]. Therefore P = p2p3p4, where residual degree of each pi, for i = 2, 3, 4 +is 2. Thus 2 | i(K). +Let a ≡ 7 mod 8 and b ≡ 0 mod 8. Then for each i = 2, 3, the φi-Newton polygon +of f(x) has a two edges of positive slope. The residual polynomial of f(x) associated to +each edge is linear. Therefore P = p2p3p4p5, where residual degree of each p2, p3 is 2 and +of p4, p5 is 1. Thus 2 | i(K). +Let a ≡ 3 mod 8 and b ≡ 4 mod 8. Then the φ2-Newton polygon of f(x) has a +single edge of positive slope. The residual polynomial of f(x) associated to this edge with +respect to φ2 is Y 2 + xY + 1 over F2[x] +⟨φ2(x)⟩. The φ3-Newton polygon of f(x) has two edges +of positive slope. The residual polynomial of f(x) associated to each edge is linear. So +P = p2p3p4, where residual degree of p2 is 4 and of p3, p4 is 1. Hence 2 | i(K). +Let a ≡ 7 mod 8 and b ≡ 4 mod 8. Then the φ2-Newton polygon of f(x) has a +single edge of positive slope. The residual polynomial of f(x) associated to this edge with +respect to φ2 is Y 2+xY +x over F2[x] +⟨φ2(x)⟩. The φ3-Newton polygon of f(x) has a single edge +of positive slope. The residual polynomial of f(x) associated to the edge is Y 2 + Y + ¯1. +So P = p2p3, where residual degree of p2 and p3 is 4 and 2 respectively. Hence 2 ∤ i(K). +Let a ≡ 1 mod 4 and b ≡ 0 mod 2. Then φ2-Newton polygon of f(x) has a single +edge of slope 1 +2. Thus φ2 provide one prime ideal say p2 of residual degree 1. Therefore +by using (3.1), we have +2OK = p1p2 +2I, +where I is an ideal of OK. +(3.4) +It is clear that 2 | i(K) if and only if φ3 provides either two prime ideals of residual degree +2 each or atleast one prime ideal of residual degree 1. +If a ≡ 1 mod 4 and b ≡ 0 mod 4, then φ3 provides one prime ideal say p3 of residual +degree 1. Thus I = p2 +3. So, 2 | i(K). +One can observe that if a ≡ 1 mod 4, b ≡ 2 mod 4, v2(b − a − 1) < 2v2(a + 7) and +v2(b−a−1) is even, then the φ3-Newton polygon has a single edge of positive slope whose +residual polynomial is not square-free, therefore we find some rational µ such that f(x) +is x − µ regular. +Let a ≡ 1 mod 4 and b ≡ 2 mod 4 and take µ = −7b +6a , then v2(µ) = 0. Let φ3(x) = +x − µ, then the φ3 expansion of f(x) is +φ7 +3 + 7µφ6 +3 + 21µ2φ5 +3 + 35µ3φ4 +3 + 35µ4φ3 +3 + 21µ5φ2 +3 + f ′(µ)φ3 + f(µ) +(3.5) +A simple calculation shows that f(µ) = +−Db +67a7 and f ′(µ) = +D +66a6. +Clearly v2(f(µ)) = +v2(f ′(µ)) = v2(D) − 6. It is easy to verify that v2(D) − 6 ≥ 1. If v2(D) is odd, then +φ3-Newton polygon of f(x) has one edge of positive slope joining the points (5, 0) and +(7, v2(D) − 6). So φ3 provides one prime ideal say p3 of residual degree 1. Therefore +I = p2 +3 and so 2 | i(K). + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +9 +If v2(D) is even, then f(x) is not x − µ regular. So we choose another rational number. +Here we have a ≡ 1 mod 4, b ≡ 2 mod 4 and v2(D) is even. Take u = v2(D)−6 +2 +and +define ρ = 2u − µ, then v2(ρ) = 0. Let φ3(x) = x − ρ. The φ3 expansion of f(x) is +φ7 +3 + 7ρφ6 +3 + 21ρ2φ5 +3 + 35ρ3φ4 +3 + 35ρ4φ3 +3 + 21ρ5φ2 +3 + f ′(ρ)φ3 + f(ρ) +(3.6) +One can verify that v2(f(ρ)) ≥ 2u + 1 and v2(f ′(ρ)) = u + 1. If v2(f(ρ)) = 2u + 1, then +φ3 provides one prime ideal say p3 of residual degree 1. Therefore I = p2 +3. So 2 | i(K). If +v2(f(ρ)) = 2u+2, then φ3 provides one prime ideal say p3 of residual degree 2. Therefore +I = p3. So 2 ∤ i(K). If v2(f(ρ)) ≥ 2u + 3, then φ3 provides two prime ideal say p3 and p4 +of residual degree 1 each. Therefore I = p3p4. So 2 | i(K) +Case A2: a ≡ 0 mod 2, b ≡ 0 mod 2. Here f(x) ≡ x7 mod 2. The x-Newton polygon +of f(x) is the lower convex hull of the points (0, 0), (6, v2(a)) and (7, v2(b)). +Let 7v2(a) > 6v2(b), then by (1.2), v2(b) < 7. The x-Newton polygon of f(x) has +a single edge of positive slope v2(b) +7 . The residual polynomial associated to this edge is +linear. Therefore 2OK = p7. Thus 2 ∤ i(K). +For the remaining case, let 7v2(a) < 6v2(b). Then by (1.2), we have 1 ≤ v2(a) ≤ 5. The +x-Newton polygon of f(x) has two edges of positive slope. The first edge say S1 is line +segment joining the points (0, 0) and (6, v2(a)) with slope v2(a) +6 . The second edge say S2 +is the line segment joining the points (6, v2(a)) and (7, v2(b)). The residual polynomial +associated to the edge S2 is linear. Therefore S2 provides one prime ideal say q of residual +degree 1. Thus +2OK = Rq, where R is an ideal of OK +(3.7) +If v2(a) ∈ {1, 5}, then S1 provides one prime ideal say p1 of residual degree 1. So R = p6 +1. +Hence 2 ∤ i(K). +If v2(a) = 3, then the residual polynomial associated to S1 is (Y + ¯1)(Y 2 + Y + ¯1). +Therefore R = p2 +1p3 +2, where residual degree of p1 and p2 is 1 and 2 respectively. Thus +2 ∤ i(K). +If v2(a) = 2, then λ = 1 +3 and the residual polynomial associated to S1 is (Y + ¯1)2, i.e. +residual polynomial is not square-free. Let ψ(Y ) = Y + ¯1, h = 1 and e = 3. Since e > 1, +for the prime ideal factorization of 2OK, we shall use higher order Newton polygons. Take +Φ(x) = x3 + 2, then Φ(x) is the key polynomial attached to the data (x, λ, ψ(Y )). As +in (2.1), we can define valuation V of second order such that V (Φ) = 3, V (x) = 1 and +V (2) = 3. The Φ(x) expansion of f(x) is +f(x) = xΦ2(x) − 4xΦ(x) + (a + 4)x + b +(3.8) +The Φ-Newton polygon of f(x) of second order is the lower convex hull of the points +(0, 7), (1, 10) and (2, v), where v = min{3v2(b), 3v2(a + 4) + 1}. Since v2(a) = 2, we +have v2(b) ≥ 3. +If a ≡ 4 mod 16 and b ≡ 0 mod 16, then v = 10. The Φ-Newton polygon of f(x) of +second order has a single edge and the residual polynomial attached to this edge is linear. + +10 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +So, R = p6 +1, where residual degree of p1 is 1. Thus 2 ∤ i(K). +If a ≡ 12 mod 16 and b ≡ 16 mod 32, then v = 12. The Φ-Newton polygon of f(x) of +second order has a single edge whose residual polynomial is linear. So, R = p6 +1, where +residual degree of p1 is 1. Thus 2 ∤ i(K) +If a ≡ 12 mod 32 and b ≡ 0 mod 32, Then v = 13, then Φ-Newton polygon of f(x) of +second order has a single edge and residual polynomial attached to this edge is Y 2+Y +¯1. +So, R = p3 +2,where residual degree of p2 is 2. Thus 2 ∤ i(K). +If a ≡ 28 mod 32 and b ≡ 0 mod 32, then v > 13, then Φ-Newton polygon of f(x) of +second order has two edges of positive slope. The residual polynomial attached to each +edge is linear. So, Φ provides two prime ideals say p1 and p2 of residual degree 1 each. +Therefore R = p3 +1p3 +2. hence 2 | i(K). +Note that if v2(b) = 3, then the Φ-Newton polygon of f(x) of second order has a single +edge. The residual polynomial attached to this edge is not square-free. So we shall use +another key polynomial. +Let v2(b) = 3 and take Φ(x) = x3 + 2x + 2. Then Φ(x) is key polynomial attached to +(x, 1 +3, ψ(Y )). Let V be same as given above. The Φ(x) expansion of f(x) is +f(x) = xΦ2(x) + (4 − 4x − 4x2)Φ(x) + (b − 8 + (a − 4)x + 8x2) +(3.9) +The Φ-Newton polygon of f(x) of second order being the lower convex hull of the points +(0, 7), (1, 9) and (2, w), where w = V (b − 8 + (a − 4)x + 8x2). +If a ≡ 12 mod 16 and b ≡ 8 mod 16, then w = 10. The Φ provide one prime ideal say +p1 of residual degree 1. So, R = p6 +1. Thus 2 ∤ i(K). +If a ≡ 4 mod 16 and b ≡ 8 mod 16, then w = 11, then Φ provides one prime ideal say +p1 of residual degree 2. So, R = p3 +1. Thus 2 ∤ i(K). +Let v2(a) = 4, then v2(b) ≥ 5, λ = +2 +3, h = 2 and e = 3. +Take Φ(x) = x3 + 4. +The valuation V of second order defined in (2.1) is such that V (Φ) = 6, V (x) = 2 and +V (2) = 3. The Φ(x) expansion of f(x) is +xΦ2(x) − 8xΦ(x) + (a + 16)x + b +(3.10) +Let w′ = min{3v2(b), 3v2(a + 16) + 2}), then w′ ≥ 15. The Φ-Newton polygon of f(x) of +second order is the lower convex hull of the points (0, 14), (1, 17) and (2, w′). +If a ≡ 16 mod 32 and b ≡ 32 mod 64, then w′ = 15. The Φ provide one prime ideal say +p1 of residual degree 1. So, R = p6 +1. Therefore 2 ∤ i(K). +If a ≡ 16 mod 64 and b ≡ 0 mod 128, then w′ = 17. So, the Φ-Newton polygon of f(x) +of second order has a single edge of positive slope. The residual polynomial attached to +this edge is linear. Therefore R = p6 +1. Thus 2 ∤ i(K). +If a ≡ 48 mod 64 and b ≡ 0 mod 128, then w′ = 20. So, the Φ-Newton polygon of f(x) +of second order has a single edge of positive slope. The residual polynomial attached to +this edge is Y 2 + Y + ¯1. Therefore R = p3 +1. Thus 2 ∤ i(K). +If a ≡ 16 mod 64 and b ≡ 64 mod 128, then w′ = 17. Hence R = p6 +1. Thus 2 ∤ i(K). +If a ≡ 48 mod 64 and b ≡ 64 mod 128, then w′ = 18. So, Take Φ(x) = x3 + 4x+ 4. The + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +11 +Φ expansion of f(x) is +xΦ2(x) − (8x + 4x2)Φ(x) + b − 64 + x(a − 48) + 32x2 +(3.11) +The Φ-Newton polygon of f(x) of second order is the lower convex hull of the points +(0, 14), (1, 16) and (2, 19). Thus Φ provide two prime ideals say p1 and p2 of residual +degree 1 each. +So R = p3 +1p3 +2. +Therefore 2 | i(K). +This completes the proof of the +theorem. +□ +Proof of Theorem 1.5. If 3 | i(K), then by using (1.1) and (1.3), we have 3 | b. +Case B1: Let a ≡ 0 mod 3 and b ≡ 0 mod 3, then f(x) ≡ x7 mod 3. The x-Newton +polygon of f(x) is the lower convex hull of the points (0, 0), (6, v3(a)) and (7, v3(b)). If +7v3(a) > 6v3(b), then x provides one prime ideal of residual degree 1. So 3OK = p6. +Let 7v3(a) < 6v3(b). Then the x-Newton polygon of f(x) being the lower convex hull of +the points (0, 0), (6, v3(a)) and (7, v3(b)) has two edges of positive slopes. The first say +S1 is the line segment joining the points (0, 0) and (6, v3(a)). The second edge say S2 +is the line segment joining the points (6, v3(a)) and (7, v3(b)). The residual polynomial +associated with the edge S2 is linear. Thus 3OK = Lq, for some ideal L of OK. Using +(1.2) and 7v3(a) > 6v3(b), we have v3(b) ∈ {1, 2, 3, 4, 5}. +If v3(a) ∈ {1, 5}, then S1 provides one prime ideal say p1 of residual degree 1. So L = p6 +1. +If v3(a) ∈ {2, 4}, then the residual polynomial associated with S1 is (Y − ¯1)(Y + ¯1). So +L = p3 +1p3 +2. +Let v2(a) = 3, then the residual polynimial associated with S1 is (Y + δ)2, where δ ∈ +{1, −1}. Let ψ(Y ) = Y + δ, λ = 1 +2. Take φ(x) = x2 + 3δ, h = 1 and e = 2, then Φ(x) +is key polynomial attached to the data (x, λ, ψ(Y )). The valuation V on Q2[x], given in +(2.1) is such that V (Φ) = 2, V (x) = 1 and V (3) = 2. The Φ(x) expansion of f(x) is +xΦ3(x) − 9δΦ2(x) + 27xΦ(x) + b + x(a − 27δ) +(3.12) +The Φ(x)-Newton polygon of second order is the lower convex hull of the points (0, 7), +(1, 9), (2, 9) and (3, min{2v3(b), 2v3(a − 27δ) + 1}). As v3(a) = 3, so by using (1.2), we +have v3(b) ≥ 4. +If v3(b) = 4, then Φ-Newton polygon of f(x) has a single edge of slope 1 +3. The residual +polynomial associated to this edge is linear. Therefore L = p6 +1, where degree of p1 is 1. +If v3(b) > 4 and v3(a − 27δ) = 4, then Φ-Newton polygon of f(x) has a single edge of +slope 2 +3. The residual polynomial associated to this edge is linear. Therefore L = p6 +1, +where degree of p1 is 1. +If v3(b) = 5 and v3(a−27δ) > 4, then Φ-Newton polygon of f(x) has a single edge joining +the points (0, 7) and (3, 10) with a lattice point (2, 9) on it. The residual polynomial +associated to this edge is Y 3 ± Y ± ¯1, which is separable. Therefore L = p2 +1, where degree +of p1 is 3. +If v3(b) > 5 and v3(a−27δ) > 4, then Φ-Newton polygon of f(x) has two edges of positive +slopes. The residual polynomial associated to the first edge is Y 2+¯1. Therefore L = p2 +1p2 +2. +Hence we conclude that when a ≡ 0 mod 3 and b ≡ 0 mod 3, 3 ∤ i(K). + +12 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +Case B2: a ≡ −1 mod 3 and b ≡ 0 mod 3. In this case f(x) ≡ x(x − 1)3(x + 1)3 +mod 3. Clearly x-Newton polygon of f(x) provides one prime ideal say q of residual +degree 1. Therefore +3OK = qJ, where J is an ideal of OK +(3.13) +Keepind in mind Lemma 2.1, 3 | i(K) if and only if x−1 and x+ 1 provides atleast three +prime ideals of residual degree 1 each. Let φδ(x) = (x + δ), where δ ∈ {−1, 1}, then the +φδ(x) expansion of f(x) is +φ7 +δ(x)−7δφ6 +δ(x)+21φ5 +δ(x)−35δφ4 +δ(x)+35δφ3 +δ(x)−21δφ2 +δ(x)+(a+7)φδ(x)+(b−δ(a+1)) +(3.14) +The φδ(x)-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (1, 0), +(2, 0), (3, 0), (4, 0), (5, 1), (6, v3(a + 7)) and (7, v3(b − δ(a + 1)). If v3(a + 1) = 1 and +v3(b) > 1, then for each δ ∈ {−1, 1}, φδ provides one prime ideal of residual degree 1. +Therefore J = p3 +1p3 +2. Hence 3 ∤ i(K). +Note that when v3(a + 1) = 1 and v3(b) = 1, then either a ≡ 2 mod 9 or a ≡ 5 mod 9. +Let a ≡ 2 mod 9, b3 ≡ 1 mod 3 and v3(b − a − 1) = 2, then for each δ ∈ {1, −1}, +φδ-Newton polygon of f(x) has a single edge of positive slope and the residual polynomial +attached to this edge is linear. Thus J = p3 +1p3 +2, where residual degree of p1 and p2 is 1. +Hence 3 ∤ i(K). +Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) = 2 and v3(b − a − 1) ≥ 3. Then φ−1 +provides one prime ideal of degree 1. The φ1-Newton polygon of f(x) has two edges of +positive slope. The first edge is the line segment joining the points (4, 0) and (6, 2). +The second edge is the line segment joining the points (6, 2) and (7, v3(b − a − 1)). The +residual polynomial attached to each edge is linear. Therefore J = p3 +1p2 +2p3, where for each +i = 1, 2, 3 the residual degree of pi is 1. Hence 3 | i(K). +Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) > +v3(b − a − 1) + 1. Then φ−1 provides one prime ideal of residual degree 1 and φ1-Newton +polygon of f(x) has two edges of positive slope. The first edge is the line segment joinig +the points (4, 0) and (5, 1) and the second edge join (5, 1) with (7, v3(b − a − 1)). The +residual polynomial attached to the first edge is linear and to second edge is (Y −¯1)(Y +¯1) +or Y 2 + ¯1 (according as (b − a − 1)3 ≡ −1 mod 3 or (b − a − 1)3 ≡ 1 mod 3). Thus +either J = p3 +1p2 +2p3, where the residual degree of pi = 1, for each i = 1, 2, 3 or J = p3 +1p2 +2p3, +where the residual degree of p1, p2 is 1 and of p3 is 2. +Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) = +v3(b − a − 1) + 1. Then φ−1 provides one prime ideal of residual degree 1 and φ1 has two +edges of positive slope. The residual polynomial attached to the first edge is linear and +to the second edge is Y 2 − Y − 1 or Y 2 + Y − 1. Thus J = p3 +1p2 +2p3, where the residual +degree of p1, p2 is 1 and of p3 is 2. So, 3 ∤ i(K). +Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) < +v3(b − a − 1) + 1. Then φ−1 provides one prime ideal of residual degree 1 and φ1-Newton +polygon of f(x) has three edges of positive slope. The first edge is the line segment joining + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +13 +the points (4, 0) and (5, 1), the second edge join (5, 1) with (6, v3(a + 7)) and the third +edge joins (6, v3(a + 7)) and (7, v3(b − a − 1)). The residual polynomial attached to the +each edge is linear. Hence J = p3 +1p2p3p4, where the residual degree of pi = 1, for each +i = 1, 2, 3, 4 is 1. Thus 3 | i(K). +Let a ≡ 2 mod 9, b3 ≡ −1 mod 3 and v3(b + a + 1) = 2, then for each δ ∈ {1, −1}, +φδ-Newton polygon of f(x) has a single edge of positive slope and the residual polynomial +attached to this edge is linear. Thus J = p3 +1p3 +2, where residual degree of p1 and p2 is 1. +Hence 3 ∤ i(K). +Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) = 2 and v3(b + a + 1) ≥ 3. Then φ1 +provides one prime ideal of degree 1. The φ−1-Newton polygon of f(x) has two edges of +positive slope. The first edge is the line segment joining the points (4, 0) and (6, 2). +The second edge is the line segment joining the points (6, 2) and (7, v3(b − a − 1)). The +residual polynomial attached to each edge is linear. Therefore J = p3 +1p2 +2p3, where for each +i = 1, 2, 3 the residual degree of pi is 1. Hence 3 | i(K). +Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) > +v3(a + b + 1) + 1. Then φ1 provides one prime ideal of residual degree 1 and φ−1-Newton +polygon of f(x) has two edges of positive slope. The first edge is the line segment joinig +the points (4, 0) and (5, 1) and the second edge join (5, 1) with (7, v3(b − a − 1)). The +residual polynomial attached to the first edge is linear and to second edge is (Y −¯1)(Y +¯1) +or Y 2 +¯1 (according as (b+a+1)3 ≡ −1 mod 3 or (b+a+1)3 ≡ 1 mod 3). Thus either +J = p3 +1p2 +2p3, where the residual degree of pi = 1, for each i = 1, 2, 3 or J = p3 +1p2 +2p3, where +the residual degree of p1, p2 is 1 and of p3 is 2. +Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) = +v3(a + b + 1) + 1. Then φ1 provides one prime ideal of residual degree 1 and φ−1 has two +edges of positive slope. The residual polynomial attached to the first edge is linear and +to the second edge is Y 2 − Y − 1 or Y 2 + Y − 1. Thus J = p3 +1p2 +2p3, where the residual +degree of p1, p2 is 1 and of p3 is 2. So, 3 ∤ i(K). +Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) < +v3(a + b + 1) + 1. Then φ−1 provides one prime ideal of residual degree 1 and φ1-Newton +polygon of f(x) has three edges of positive slope. The first edge is the line segment joining +the points (4, 0) and (5, 1), the second edge join (5, 1) with (6, v3(a + 7)) and the third +edge joins (6, v3(a + 7)) and (7, v3(b + a + 1)). The residual polynomial attached to the +each edge is linear. Hence J = p3 +1p2p3p4, where the residual degree of pi = 1, for each +i = 1, 2, 3, 4 is 1. Thus 3 | i(K). +Let a ≡ 5 mod 9 and v3(b) = 1, then v3(a + 7) = 1. In this, either φ1 provides one +prime ideal of residual degree 1 and φ−1 provides two prime ideal of residual degree 1 each +or φ1 provides two prime ideal of residual degree 1 each and φ−1 provides one prime ideal +of residual degree 1 (according as b3 ≡ −1 mod 3 or b3 ≡ 1 mod 3). Thus J = p3 +1p2 +2p3, +where the residual degree of pi, for each i = 1, 2, 3 is 1. Thus 3 | i(K). +Let v3(a + 1) > 1 and v3(b) = 1, then v3(a + 7) = 1 and v3(b − δ(a + 1)) = 1. Thus for +each δ, φδ provides one prime ideal of residual degree 1. So J = p3 +1p3 +2. +Let v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, 1), (1, −1)} mod 3, then + +14 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +v3(b − a − 1) > 1 and v3(b + a + 1) = 1. So the φ−1-Newton polygon of f(x) has two +edges of positive slope and residual polynomial attached to each edge is linear. And φ1 +provides one prime ideal of degree 1. Therefore J = p2 +1p2p3 +3. Thus 3 | i(K). +Let v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, −1), (1, 1)} mod 3, then +v3(b + a + 1) > 1 and v3(b − a − 1) = 1. So the φ1-Newton polygon of f(x) has two +edges of positive slope and residual polynomial attached to each edge is linear. The φ−1 +provides one prime ideal of degree 1. Therefore J = p2 +1p2p3 +3. Thus 3 | i(K). +Case B3: a ≡ 1 mod 3 and b ≡ 0 mod 3. In this case f(x) ≡ x(x2 + 1)3 mod 3. +In this case x-provides one prime ideal of residual degree 1. +If we denote R((p) the +residual degree of the prime ideal p, then for 3OK we have the following possibilities. +Case +Factorization of 3OK +Residual degree +(1) +3OK = p1p3 +2 +R(p1) = 1 and R(p1) = 2 +(2) +3OK = p1p2p3 +R(p1) = 1, R(p1) = 2 and R(p1) = 4 +(3) +3OK = p1p2 +R(p1) = 1 and R(p2) = 6 +(4) +3OK = p1p2 +2p3 +R(p1) = 1, R(p1) = 2 and R(p1) = 2 +(5) +3OK = p1p2p3p4 +R(p1) = 1 and R(pi) = 2, i = 2, 3, 4 +Table +Clearly in each case, there can be atmost 3 prime ideals of residual degree 2 each. But +there are 5 monic irreducible polynomial of degree 2 over F3. Thus in view of Lemma 2.1 +3 ∤ i(K). This completes the proof. +□ +Proof of Theorem 1.7. In this case 5 | i(K) if and only if a7 ≡ 2b6 mod 5. +i.e. (a, b) ∈ {(0, 0), (2, 2), (2, −2), (3, 1), (3, −1)}. +Let (a, b) = (0, 0), then f(x) ≡ x7 mod 7. If 7v5(a) > 6v5(b), then x-Newton polygon +of f(x) has a single edge of positive slope. The residual polynomial attached to this edge +is linear. Therefore 5OK = p7. If 7v5(a) < 6v5(b) and v5(a) ∈ {1, 5}, then x-Newton +polygon of f(x) has two edges of positive slope. The residual polynomial attached to +each edge is linear. +Therefore 5OK = p6 +1p2. +If 7v5(a) < 6v5(b) and v5(a) ∈ {2, 4}, +then x-Newton polygon of f(x) has two edges of positive slope. The residual polynomial +attached to first edge is (Y + ¯1)(Y − ¯1) or (Y + ¯3)(Y − ¯2) and to the second edge is +linear. Therefore p3 +1p3 +2p3. If 7v5(a) < 6v5(b) and v5(a) ∈ {2, 4}, then x-Newton polygon +of f(x) has two edges of positive slope. The residual polynomial attached to first edge is +(Y + ¯1)(Y 2 − Y + ¯1) or (Y − ¯1)(Y 2 + Y + ¯1) and to the second edge is linear. Therefore +5OK = p3 +1p3 +2p3. Thus 5 ∤ i(K). +Let (a, b) = (2, 2), then f(x) ≡ (x + 2)2(x − 1)(x4 + 2x3 + 4x2 + 2x + 2) mod 5, so +5OK = p1p2p3 or 5OK = p1p2p3. Thus 5 ∤ i(K). +Let (a, b) = (2, −2), then f(x) ≡ (x + 1)(x + 3)2(x4 + 3x2 + 4x2 + 3x + 2) mod 5, so + +ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b +15 +5OK = p1p2p3 or 5OK = p2 +1p2p3 or 5OK = p1p2p3p4. Thus 5 ∤ i(K). +Let (a, b) = (2, −2), then f(x) ≡ (x + 1)(x + 3)2(x4 + 3x3 + 4x2 + 3x + 2) mod 5, so +5OK = p1p2p3 or p1p2 +2p3 5OK = p1p2p3p4. Thus 5 ∤ i(K). +Let (a, b) = (3, 1), then f(x) ≡ (x + 4)(x + 3)2(x4 + 4x3 + x2 + x + 2) mod 5, so +5OK = p1p2p3 or p1p2 +2p3 5OK = p1p2p3p4. Thus 5 ∤ i(K). +Let (a, b) = (3, −1), then f(x) ≡ (x + 1)(x + 2)2(x4 + x2 + x2 + 4x + 2) mod 5, so +5OK = p1p2p3 or p1p2 +2p3 5OK = p1p2p3p4. Thus 5 ∤ i(K). +Next we show that 7 ∤ i(K). Suppose there exists distinct non-zero prime ideals p1, · · · , +pr of OK such that 7OK = pe1 +1 · · · per +r , where ei ≥ 1, then by the Fundamental Equality, +e1f1 + · · · + erfr = 7 where fi is the residual degree of pi for i = 1, 2, · · · , r. Since ei ≥ 1 +for all i = 1, 2, · · · , r, therefore there can be at most 7 prime ideals lying above 7, but for +every positive integer h the number of monic irreducible polynomials of degree h in F7[x] +is greater than or equal to 7. So by Lemma 2.1, 7 ∤ i(K). This completes the proof. +□ +References +[1] S. Ahmad, T. Nakahara, A. Hameed, On certain pure sextic fields related to a problem of Hasse, +Int. J. Algebra Compu., 26(3) (2016), 577-583. +[2] S. Ahmad, T. Nakahara, S. M. Husnine, Power integral basis for certain pure sextic fields, Int. J. +Number Theory, 10(8) (2014), 2257-2265. +[3] S. D. Cohen, A. Movahhedi, A. 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Kumar, Non-monogenity of certain octic number fields defined +by trinomials, Colloquium Mathematicum, (2022), DOI: 10.4064/cm8799-3-2022. +[11] A. Jakhar, S. Kaur, A note on non-monogenity of number fields arising from sextic trinomials, +(2022), DOI:10.2989/16073606.2022.2043948. +[12] S. K. Khanduja, A Textbook of Algebraic Number Theory, Unitext series 135, Springer(2022) +ISBN:978-981-16-9149-2 +[13] S. K. Khanduja, S. Kumar, On prolongations of valuations via Newton polygons and liftings of +polynomials, J. Pure Appl. Algebra, 216 (2012), 2648-2656. +[14] R. MacKenzie, J. Scheuneman, A number field without a relative integral basis, Amer. Math. +Monthly, 78 (1971), 882-823. + +16 +ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR +[15] W. Narkiewicz, Elementary and Analytic Theory of Algebraic numbers, (Third edition), Springer +Monographs in Mathematics, Springer-Verlag, Berlin, 2004. +[16] J. Neukirch, Algebraic Number Theory, Berlin-Heidelberg, Springer-Verlag, 1999. +[17] H. Smith, The monogenity of radical extension, Acta Arith., 198(3), 313-327. +[18] A. Soullami, M. Sahmoudi, On sextic integral bases using relative quadratic extension, Bol. Soc. +Paran. Mat., 38(4) (2020), 175-180. +[19] J. Montes and E. Nart, On a theorem of Ore, J. Algebra 146(2) (1992), 318–334. +[20] O. Ore, Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann., 99 (1928), +84–117. +[21] J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, +Trans. Amer. Math. Soc. 364 (1) (2012) 361–416. +(Anuj Jakhar) Department of Mathematics, Indian Institute of Technology (IIT) Madras +(Sumandeep Kaur) Department of Mathematics, Panjab University Chandigarh +(Surender Kumar) Department of Mathematics, Indian Institute of Technology (IIT) +Bhilai + diff --git a/N9AyT4oBgHgl3EQfgvi_/content/tmp_files/load_file.txt b/N9AyT4oBgHgl3EQfgvi_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6dced364698a088e58898ce3da18652a05966036 --- /dev/null +++ b/N9AyT4oBgHgl3EQfgvi_/content/tmp_files/load_file.txt @@ -0,0 +1,686 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf,len=685 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='00365v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='NT] 1 Jan 2023 ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let f(x) = x7 + ax + b be an irreducible polynomial having integer coeffi- cients and K = Q(θ) be an algebraic number field generated by a root θ of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In the present paper, for every rational prime p, our objective is to determine the necessary and sufficient conditions involving only a, b so that p is a divisor of the index of the field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In particular, we provide sufficient conditions on a and b, for which K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In a special case, we show that if either 8 divides both a ± 1, b or 32 divides both a + 4, b, then K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' We illustrate our results through examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Introduction and statements of results Let K = Q(θ) be an algebraic number field with θ in the ring OK of algebraic integers of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let f(x) ∈ Z[x] be the minimal polynomial of θ having degree n over the field Q of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' It is a basic result in algebraic number theory that OK is a free abelian group of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' An algebraic number field K is said to be monogenic if there exists some α ∈ OK such that {1, α, · · · , αn−1} is an integral basis of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case OK = Z[α], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=', [OK : Z[α]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If K does not have any such α, then the field K is said to be non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' It is well-known that every quadratic and cyclotomic field is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' It is important to know that whether a number field is monogenic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' It was Dedekind, who gave the first non-monogenic number field K = Q(ξ), where ξ is a root of the polynomial x3 − x2 − 2x − 8 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' [15, page 64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The problem of testing the monogenity of number fields and constructing power integral bases have been intensively studied (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' [1], [2], [6], [9], [10], [11], [14], [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In 1984, Funakura [5] gave necessary and sufficient conditions on those integers m for which the quartic field Q(m1/4) is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Ahmad, Nakahara and Husnine in [1], [2] proved that for a square free integer m not congruent to ±1 mod 9, a pure field Q(m1/6) having degree six over Q is monogenic when m ≡ 2 or 3 mod 4 and it is non-monogenic when m ≡ 1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 11R04, 11R21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Monogenity, Theorem of Ore, prime ideal factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first author is thankful to SERB grant SRG/2021/000393 and IIT Madras for NFIG RF/22- 23/1035/MA/NFIG/009034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The second author is grateful to the Council of Scientific and Industrial Research, New Delhi for providing financial support in the form of Senior Research Fellowship through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 09/135(0878)/2019-EMR-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The third author is grateful to the University Grants Commi- sion, New Delhi for providing financial support in the form of Junior Research Fellowship through Ref No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1129/(CSIR-NET JUNE 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 1 2 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR In 2017, Ga´al and Remete [6] studied monogenity of algebraic number fields of the type Q(m1/n) where 3 ≤ n ≤ 9 and m is square free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In [9], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Jakhar and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Kumar provides some sufficient conditions for which the number field K defiend by the sextic trinomial x6 + ax + b ∈ Z[x], is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In [7], Ga´al studied monogenity of number fields defined by some sextic irreducible trinomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Recall that for an algebraic number field L = Q(ξ) with ξ an algebraic integer satis- fying a monic irreducible polynomial g(x) over Q, the discriminant D of g(x) and the discriminant dL of L are related by the formula D = (ind ξ)2 · dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1) Throughout this paper, ind θ will denote the index of the subgroup Z[θ] in OK and i(K) will stand for the index of the field K defined by i(K) = gcd{ind α | K = Q(α) and α ∈ OK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' A prime divisor of i(K) is called a common index divisor of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Note that i(K) = 1, for every monogenic number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' But there exist non-monogenic number fields having i(K) = 1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=', K = Q( 3√ 175) is non-monogenic with i(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In what follows, let K = Q(θ) be an algebraic number field with θ a root of an irre- ducible trinomial f(x) = x7 + ax + b ∈ Z[x], then for every rational prime p, we provide necessary and sufficient conditions on a, b, so that p is a common index divisor of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In particular, under these conditions K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Our method is based on the the- ory of Newton polygons and theorem of Ore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In 1878, (see [ [4], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4]) Dedekind gave a criterion to determine whether p divides the index [OK : Z[θ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' When p divides the index [OK : Z[θ]], then a method of Ore 1928, can be used in order to evaluate the prime ideal factorization of pOK (see [19], [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If Ore’s method doesn’t work, then an algorithm developed by Guardia, Montes, and Nart [21], based on higher order Newton polygons can be used to determine the prime ideal factorization of pOK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' For a prime p and a non-zero m belonging to the ring Zp of p-adic integers, vp(m) will denote the highest power of p dividing m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' For a non-zero integer l, let lp denote l pvp(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a rational prime p is such that p6 divides a and p7 divides b, then θ/p is a root of the polynomial x7 + (a/p6)x + (b/p7) having integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So we may assume that for each prime p either vp(a) ≤ 5 or vp(b) ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2) Also, D will stand for the discriminant of f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' One can check that D = −77b6 − 66a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='3) With the above notations and assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2), we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field with θ a root of an irreducible polynomial f(x) = x7 + ax + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If u = v2(D)−6 2 and ρ = 2u + 7b 6a, then 2 | i(K) if and only if one of the following hold: ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 3 (1) a ≡ 3 mod 4 and b ≡ 0 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (2) a ≡ 3 mod 8 and b ≡ 4 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (3) a ≡ 1 mod 4 and b ≡ 0 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (4) a ≡ 1 mod 4, b ≡ 2 mod 4, v2(D) is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (5) a ≡ 1 mod 4, b ≡ 2 mod 4, v2(D) is even and v2(f(ρ)) = 2u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (6) a ≡ 1 mod 4, b ≡ 2 mod 4, v2(D) is even and v2(f(ρ)) ≥ 2u + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (7) a ≡ 28 mod 32 and b ≡ 0 mod 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (8) a ≡ 48 mod 64 and b ≡ 0 mod 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The next three corollaries follow immediately from the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field, where θ satisfies the ir- reducible polynomial x7 + ax + b ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 8 divides both a ± 1 and b, then K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field generated by a root θ of an irreducible polynomial x7 + ax + b belonging to Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If both a + 4 and b are divisible by 32, then K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field generated by a root θ of an irreducible polynomial x7 + ax + b belonging to Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then K is non-monogenic, if both a + 16 and b are divisible by 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field with θ a root of an irreducible polynomial f(x) = x7 + ax + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then 3 | i(K) if and only if one of the following hold: (1) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ 1 mod 3, v3(a + 7) = 2 and v3(b − a − 1) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (2) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3, 2v3(a + 7) > v3(b − a − 1) + 1 and (b − a − 1)3 ≡ −1 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (3) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) < v3(b − a − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (4) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ −1 mod 3, v3(a + 7) = 2 and v3(b + a + 1) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (5) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3, 2v3(a + 7) > v3(a + b + 1) + 1 and (b + a + 1)3 ≡ −1 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (6) a ≡ 2 mod 9, v3(b) = 1, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) = v3(a + b + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (7) a ≡ 5 mod 9 and v3(b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (8) v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, 1), (1, −1)} mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (9) v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, −1), (1, 1)} mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The following corollary follow immediately from the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) where θ satisfies the irreducible polynomial x7 + ax + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then K is non-monogenic, if one of the following hold: (1) a ≡ 5 mod 9 and b ≡ 3 mod 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 4 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR (2) a ≡ 5 mod 9 and b ≡ 6 mod 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field with θ a root of an irreducible polynomial f(x) = x7 + ax + b ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let p ≥ 5 be a rational prime, then p ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Note that when a = 0 and b is a non-zero odd integer, then the index of K is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) be an algebraic number field generated by a root θ of an irreducible polynomial f(x) = x7 + ax+ b ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 15 mod 72 and b ≡ 12 mod 72, then in view of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='7, we have i(K) = 1 We now provide some examples of non-monogenic number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) where θ is a root of f(x) = x7 + 17x+ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Note that f(x) satisfies Eisenstein’s criterion with respect to 17, hence it is irreducible over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, by Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='6 K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K = Q(θ) with θ satisfying f(x) = x7 + 7x + 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' It can be easily seen that f(x) is 7-Eisenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence, in view of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4 K is non-monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Preliminary Results Let K = Q(θ) be an algebraic number field with θ a root of a monic irreducible polynomial f(x) belonging to Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In what follows, OK will stand for the ring of algebraic integers of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' For a rational prime p, let Fp denote the finite field with p elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The following lemma (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' [17, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2]) will play an important role in the proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let K be an algebraic number field and p be a rational prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then p is a common index divisor of K if and only if for some positive integer h, the number of distinct prime ideals of OK lying above p having residual degree h is greater than the number of monic irreducible polynomials of degree h in Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' We shall first introduce the notion of Gauss valuation, φ-Newton polygon and Newton polygon of second order, where φ(x) belonging to Zp[x] is a monic polynomial with φ(x) irreducible over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Gauss valuation of the field Qp(x) of rational functions in an inde- terminate x which extends the valuation vp of Qp and is defined on Qp[x] by vp,x( a0 + a1x + a2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' + asxs) = min{vp(ai), 1 ≤ i ≤ s}, ai ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let p be a rational prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ(x) ∈ Zp[x] be a monic polynomial which is irreducible modulo p and f(x) ∈ Zp[x] be a monic polynomial not divisible by φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let n � i=0 ai(x)φ(x)i with deg ai(x) < deg φ(x), an(x) ̸= 0 be the φ(x)-expansion of ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 5 f(x) obtained on dividing it by the successive powers of φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let Pi stand for the point in the plane having coordinates (i, vp,x(an−i(x))) when an−i(x) ̸= 0, 0 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let µij denote the slope of the line joining the point Pi with Pj if an−i(x)an−j(x) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let i1 be the largest positive index not exceeding n such that µ0i1 = min{ µ0j | 0 < j ≤ n, an−j(x) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If i1 < n, let i2 be the largest index such that i1 < i2 ≤ n with µi1i2 = min{ µi1j | i1 < j ≤ n, an−j(x) ̸= 0} and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ-Newton polygon of f(x) with respect to p is the polygonal path having segments P0Pi1, Pi1Pi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' , Pik−1Pik with ik = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' These segments are called the edges of the φ-Newton polygon and their slopes form a strictly increasing sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' these slopes are non-negative as f(x) is a monic polynomial with coefficients in Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ(x) ∈ Zp[x] be a monic polynomial which is irreducible modulo a rational prime p having a root α in the algebraic closure �Qp of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let f(x) ∈ Zp[x] be a monic polynomial not divisible by φ(x) with φ(x)-expansion φ(x)n + an−1(x)φ(x)n−1 + · ·+a0(x) such that f(x) is a power of φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Suppose that the φ-Newton polygon of f(x) consists of a single edge, say S, having positive slope denoted by l e with l, e coprime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=', min �vp,x(an−i(x)) i | 1 ≤ i ≤ n � = vp,x(a0(x)) n = l e so that n is divisible by e, say n = et and vp,x(an−ej(x)) ≥ lj for 1 ≤ j ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus the polynomial bj(x) := an−ej(x) plj has coefficients in Zp and hence bj(α) ∈ Zp[α] for 1 ≤ j ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The polynomial T(Y ) in an indeterminate Y defined by T(Y ) = Y t+ t� j=1 bj(α)Y t−j having coefficients in Fp[α] ∼= Fp[x] ⟨φ(x)⟩ is called residual polynomial of f(x) with respect to (φ, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The following definition gives the notion of residual polynomial when f(x) is more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ(x), α be as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let g(x) ∈ Zp[x] be a monic poly- nomial not divisible by φ(x) such that g(x) is a power of φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let λ1 < · · · < λk be the slopes of the edges of the φ-Newton polygon of g(x) and Si denote the edge with slope λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In view of a classical result proved by Ore (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5], [13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1]), we can write g(x) = g1(x) · · · gk(x), where the φ-Newton polygon of gi(x) ∈ Zp[x] has a single edge, say S′ i, which is a translate of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let Ti(Y ) belonging to Fp[α][Y ] denote the residual polynomial of gi(x) with respect to (φ, S′ i) described as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' For convenience, the polynomial Ti(Y ) will be referred to as the residual polyno- mial of g(x) with respect to (φ, Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The polynomial g(x) is said to be p-regular with respect to φ if none of the polynomials Ti(Y ) has a repeated root in the algebraic closure of Fp, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In general, if F(x) belonging to Zp[x] is a monic polynomial and f(x) = φ1(x)e1 · · · φr(x)er is its factorization modulo p into irreducible polynomials with 6 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR each φi(x) belonging to Zp[x] monic and ei > 0, then by Hensel’s Lemma there exist monic polynomials f1(x), · · · , fr(x) belonging to Zp[x] such that f(x) = f1(x) · · · fr(x) and f i(x) = φi(x)ei for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The polynomial f(x) is said to be p-regular (with respect to φ1, · · · , φr) if each fi(x) is p-regular with respect to φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' To determine the number of distinct prime ideals of OK lying above a rational prime p, we will use the Newton polygon of second order and the following theorem which is a weaker version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2 of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let L = Q(ξ) be an algebraic number field with ξ satisfying an irre- ducible polynomial g(x) ∈ Z[x] and p be a rational prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ1(x)e1 · · · φr(x)er be the factorization of g(x) modulo p into powers of distinct irreducible polynomials over Fp with each φi(x) ̸= g(x) belonging to Z[x] monic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Suppose that the φi-Newton polygon of g(x) has ki edges, say Sij having slopes λij = lij eij with gcd (lij, eij) = 1 for 1 ≤ j ≤ ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If Tij(Y ) = sij � s=1 Uijs(Y ) is the factorization of the residual polynomial Tij(Y ) into distinct irreducible factors over Fp with respect to (φi, Sij) for 1 ≤ j ≤ ki, then pOL = r� i=1 ki � j=1 sij � s=1 p eij ijs, where pijs are distinct prime ideals of OL having residual degree deg φi(x) × deg Uijs(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let L = Q(γ) where γ is a root of a monic polynomial g(x) = anxn + · · · + a0 ∈ Z[x], a0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let p be a prime number such that g(x) ≡ xn(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Suppose that the p-Newton polygon of g(x) consists of a single edge with positive slope λ = l e, where gcd(l, e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let the residual polynomial Tg(Y ) ∈ Fp[Y ] of g(x) is a power of monic irreducible polynomial ψ(Y ) over Fp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=', Tg(Y ) = ψ(Y )s in Fp[Y ], where s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case, we construct a key polynomial Φ(x) attached with the slope λ such that the following hold: (i) Φ(x) is congruent to a power of x modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (ii) The p-Newton polygon of Φ(x) of first order is one-sided with slope λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (iii) The residual polynomial of Φ(x) with respect to p is ψ(Y ) in Fp[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (iv) deg Φ(x) = e deg ψ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' As described in [8, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2], the data (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' λ, ψ(Y )) determines a p-adic valuation V of the field Qp(x) which satisfies the following properties: (i) V (x) = l where λ = l e with gcd(l, e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (ii) If p(x) = � 0≤i bixi ∈ Zp[x] is any polynomial, then V (p(x)) = e min 0≤i {vp(bi) + iλ} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1) ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 7 We define the above valuation V to the valuation of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If g(x) = u � i=0 ai(x)Φ(x)i ∈ Zp[x] is a Φ-adic expansion of g(x), then the Newton polygon of g(x) with respect to V (also called V -Newton polygon of g(x) of second order) is the lower convex hull of the set of the points (i, V (au−i(x)Φ(x)i)) of the Euclidean plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let the V -Newton polygon of g(x) of second order has k-sides E1, · · · , Ek with positive slopes λ1, · · · , λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let λt = lt et with gcd(lt, et) = 1 and [at, bt] denote the projection to the horizontal axis of the side of slope λt for 1 ≤ t ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then, there is a natural residual polynomial ψt(Y ) of second order attached to each side Et, whose degree coincides with the degree of the side (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' bt−at et ) [8, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Only those integral points of the V - Newton polygon of g(x) which lie on the side, determine a non-zero coefficient of this second order residual polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' We define g(x) to be ψt-regular when the second order residual polynomial ψt(Y ) attached to the side Et of the V -Newton polygon of g(x) of second order is separable in Fp[Y ] ⟨ψ(Y )⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' We define g(x) to be V -regular if g(x) is ψt-regular for each t, 1 ≤ t ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Further, if each residual polynomial ψt(Y ), t ∈ {1, 2, · · · , k}, is irreducible in Fp[Y ] ⟨ψ(Y )⟩ , then each ψt(Y ) provides a prime ideal having residual degree deg ψ · deg ψt and ramification index e · et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 2 is a common index divisor of K, then 2 | D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 2 | b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Case A1: a ≡ 1 mod 2, b ≡ 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case, f(x) ≡ x(1 + x + x2)2(x + 1)2 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ1(x) = x, φ2(x) = 1 + x + x2 and φ3(x) = x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ1-Newton polygon of f(x) has a single edge joining the points (0, 0) and (7, v2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated with this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore φ provides one prime ideal say p1 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So 2OK = p1P, where P is an ideal of OK (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1) Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='6, we see that φ2 provides prime ideals of residual degree multiple of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore keeping in mind Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1, 2 | i(K) if and only if φ2 and φ3 provides atleast two prime ideals of residual degree t, where t ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φi, for i = 2, 3 expansion of f(x) is (x − 3)φ3 2 + (3x + 5)φ2 2 − (4x + 2)φ2 + b + x(a + 1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2) φ7 3 − 7φ6 3 + 21φ5 3 − 35φ4 3 + 35φ3 3 − 21φ2 3 + (a + 7)φ3 + (b − a − 1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='3) The φ2-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (1, 0), (2, 1) and (3, min{v2(b), v2(a + 1)}) and φ3-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (1, 0), (2, 0), (3, 0) , (4, 0), (5, 0), (6, v2(a+7)) and (7, v2(b−a−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 3 mod 4 and b ≡ 2 mod 4, then v2(a + 1) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' For each i = 2, 3, φi-Newton polygon of f(x) has a single edge of slope 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus φ2 provides one prime ideal say p2 of residual degree 2 and φ3 provides one prime ideal say p3 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='6, P = p2 2p2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 8 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR Let a ≡ 3 mod 8 and b ≡ 0 mod 8 , then for each i = 2, 3, the φi-Newton polygon of f(x) has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial of f(x) associated to this edge with respect to φ2 is (Y − (x + 1))(Y − 1) over F2[x] ⟨φ2(x)⟩ and with respect to φ3 is Y 2 +Y +¯1 ∈ F2[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore P = p2p3p4, where residual degree of each pi, for i = 2, 3, 4 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 7 mod 8 and b ≡ 0 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then for each i = 2, 3, the φi-Newton polygon of f(x) has a two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial of f(x) associated to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore P = p2p3p4p5, where residual degree of each p2, p3 is 2 and of p4, p5 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 3 mod 8 and b ≡ 4 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then the φ2-Newton polygon of f(x) has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial of f(x) associated to this edge with respect to φ2 is Y 2 + xY + 1 over F2[x] ⟨φ2(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ3-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial of f(x) associated to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So P = p2p3p4, where residual degree of p2 is 4 and of p3, p4 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 7 mod 8 and b ≡ 4 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then the φ2-Newton polygon of f(x) has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial of f(x) associated to this edge with respect to φ2 is Y 2+xY +x over F2[x] ⟨φ2(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ3-Newton polygon of f(x) has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial of f(x) associated to the edge is Y 2 + Y + ¯1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So P = p2p3, where residual degree of p2 and p3 is 4 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 1 mod 4 and b ≡ 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ2-Newton polygon of f(x) has a single edge of slope 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus φ2 provide one prime ideal say p2 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1), we have 2OK = p1p2 2I, where I is an ideal of OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='4) It is clear that 2 | i(K) if and only if φ3 provides either two prime ideals of residual degree 2 each or atleast one prime ideal of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 1 mod 4 and b ≡ 0 mod 4, then φ3 provides one prime ideal say p3 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus I = p2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' One can observe that if a ≡ 1 mod 4, b ≡ 2 mod 4, v2(b − a − 1) < 2v2(a + 7) and v2(b−a−1) is even, then the φ3-Newton polygon has a single edge of positive slope whose residual polynomial is not square-free, therefore we find some rational µ such that f(x) is x − µ regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 1 mod 4 and b ≡ 2 mod 4 and take µ = −7b 6a , then v2(µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ3(x) = x − µ, then the φ3 expansion of f(x) is φ7 3 + 7µφ6 3 + 21µ2φ5 3 + 35µ3φ4 3 + 35µ4φ3 3 + 21µ5φ2 3 + f ′(µ)φ3 + f(µ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5) A simple calculation shows that f(µ) = −Db 67a7 and f ′(µ) = D 66a6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Clearly v2(f(µ)) = v2(f ′(µ)) = v2(D) − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' It is easy to verify that v2(D) − 6 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v2(D) is odd, then φ3-Newton polygon of f(x) has one edge of positive slope joining the points (5, 0) and (7, v2(D) − 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So φ3 provides one prime ideal say p3 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore I = p2 3 and so 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 9 If v2(D) is even, then f(x) is not x − µ regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So we choose another rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Here we have a ≡ 1 mod 4, b ≡ 2 mod 4 and v2(D) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Take u = v2(D)−6 2 and define ρ = 2u − µ, then v2(ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φ3(x) = x − ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ3 expansion of f(x) is φ7 3 + 7ρφ6 3 + 21ρ2φ5 3 + 35ρ3φ4 3 + 35ρ4φ3 3 + 21ρ5φ2 3 + f ′(ρ)φ3 + f(ρ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='6) One can verify that v2(f(ρ)) ≥ 2u + 1 and v2(f ′(ρ)) = u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v2(f(ρ)) = 2u + 1, then φ3 provides one prime ideal say p3 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore I = p2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v2(f(ρ)) = 2u+2, then φ3 provides one prime ideal say p3 of residual degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore I = p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v2(f(ρ)) ≥ 2u + 3, then φ3 provides two prime ideal say p3 and p4 of residual degree 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore I = p3p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So 2 | i(K) Case A2: a ≡ 0 mod 2, b ≡ 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Here f(x) ≡ x7 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The x-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (6, v2(a)) and (7, v2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let 7v2(a) > 6v2(b), then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2), v2(b) < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The x-Newton polygon of f(x) has a single edge of positive slope v2(b) 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 2OK = p7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' For the remaining case, let 7v2(a) < 6v2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2), we have 1 ≤ v2(a) ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The x-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge say S1 is line segment joining the points (0, 0) and (6, v2(a)) with slope v2(a) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The second edge say S2 is the line segment joining the points (6, v2(a)) and (7, v2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated to the edge S2 is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore S2 provides one prime ideal say q of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2OK = Rq, where R is an ideal of OK (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='7) If v2(a) ∈ {1, 5}, then S1 provides one prime ideal say p1 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So R = p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v2(a) = 3, then the residual polynomial associated to S1 is (Y + ¯1)(Y 2 + Y + ¯1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore R = p2 1p3 2, where residual degree of p1 and p2 is 1 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v2(a) = 2, then λ = 1 3 and the residual polynomial associated to S1 is (Y + ¯1)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' residual polynomial is not square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let ψ(Y ) = Y + ¯1, h = 1 and e = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Since e > 1, for the prime ideal factorization of 2OK, we shall use higher order Newton polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Take Φ(x) = x3 + 2, then Φ(x) is the key polynomial attached to the data (x, λ, ψ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' As in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1), we can define valuation V of second order such that V (Φ) = 3, V (x) = 1 and V (2) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ(x) expansion of f(x) is f(x) = xΦ2(x) − 4xΦ(x) + (a + 4)x + b (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='8) The Φ-Newton polygon of f(x) of second order is the lower convex hull of the points (0, 7), (1, 10) and (2, v), where v = min{3v2(b), 3v2(a + 4) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Since v2(a) = 2, we have v2(b) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 4 mod 16 and b ≡ 0 mod 16, then v = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ-Newton polygon of f(x) of second order has a single edge and the residual polynomial attached to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 10 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR So, R = p6 1, where residual degree of p1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 12 mod 16 and b ≡ 16 mod 32, then v = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ-Newton polygon of f(x) of second order has a single edge whose residual polynomial is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, R = p6 1, where residual degree of p1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K) If a ≡ 12 mod 32 and b ≡ 0 mod 32, Then v = 13, then Φ-Newton polygon of f(x) of second order has a single edge and residual polynomial attached to this edge is Y 2+Y +¯1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, R = p3 2,where residual degree of p2 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 28 mod 32 and b ≡ 0 mod 32, then v > 13, then Φ-Newton polygon of f(x) of second order has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, Φ provides two prime ideals say p1 and p2 of residual degree 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore R = p3 1p3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' hence 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Note that if v2(b) = 3, then the Φ-Newton polygon of f(x) of second order has a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to this edge is not square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So we shall use another key polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let v2(b) = 3 and take Φ(x) = x3 + 2x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then Φ(x) is key polynomial attached to (x, 1 3, ψ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let V be same as given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ(x) expansion of f(x) is f(x) = xΦ2(x) + (4 − 4x − 4x2)Φ(x) + (b − 8 + (a − 4)x + 8x2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='9) The Φ-Newton polygon of f(x) of second order being the lower convex hull of the points (0, 7), (1, 9) and (2, w), where w = V (b − 8 + (a − 4)x + 8x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 12 mod 16 and b ≡ 8 mod 16, then w = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ provide one prime ideal say p1 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, R = p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 4 mod 16 and b ≡ 8 mod 16, then w = 11, then Φ provides one prime ideal say p1 of residual degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, R = p3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let v2(a) = 4, then v2(b) ≥ 5, λ = 2 3, h = 2 and e = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Take Φ(x) = x3 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The valuation V of second order defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1) is such that V (Φ) = 6, V (x) = 2 and V (2) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ(x) expansion of f(x) is xΦ2(x) − 8xΦ(x) + (a + 16)x + b (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='10) Let w′ = min{3v2(b), 3v2(a + 16) + 2}), then w′ ≥ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ-Newton polygon of f(x) of second order is the lower convex hull of the points (0, 14), (1, 17) and (2, w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 16 mod 32 and b ≡ 32 mod 64, then w′ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ provide one prime ideal say p1 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, R = p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 16 mod 64 and b ≡ 0 mod 128, then w′ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, the Φ-Newton polygon of f(x) of second order has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore R = p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 48 mod 64 and b ≡ 0 mod 128, then w′ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, the Φ-Newton polygon of f(x) of second order has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to this edge is Y 2 + Y + ¯1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore R = p3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 16 mod 64 and b ≡ 64 mod 128, then w′ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence R = p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 2 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If a ≡ 48 mod 64 and b ≡ 64 mod 128, then w′ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, Take Φ(x) = x3 + 4x+ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 11 Φ expansion of f(x) is xΦ2(x) − (8x + 4x2)Φ(x) + b − 64 + x(a − 48) + 32x2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='11) The Φ-Newton polygon of f(x) of second order is the lower convex hull of the points (0, 14), (1, 16) and (2, 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus Φ provide two prime ideals say p1 and p2 of residual degree 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So R = p3 1p3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 2 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 3 | i(K), then by using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='3), we have 3 | b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Case B1: Let a ≡ 0 mod 3 and b ≡ 0 mod 3, then f(x) ≡ x7 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The x-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (6, v3(a)) and (7, v3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 7v3(a) > 6v3(b), then x provides one prime ideal of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So 3OK = p6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let 7v3(a) < 6v3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then the x-Newton polygon of f(x) being the lower convex hull of the points (0, 0), (6, v3(a)) and (7, v3(b)) has two edges of positive slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first say S1 is the line segment joining the points (0, 0) and (6, v3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The second edge say S2 is the line segment joining the points (6, v3(a)) and (7, v3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated with the edge S2 is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 3OK = Lq, for some ideal L of OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2) and 7v3(a) > 6v3(b), we have v3(b) ∈ {1, 2, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(a) ∈ {1, 5}, then S1 provides one prime ideal say p1 of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So L = p6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(a) ∈ {2, 4}, then the residual polynomial associated with S1 is (Y − ¯1)(Y + ¯1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So L = p3 1p3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let v2(a) = 3, then the residual polynimial associated with S1 is (Y + δ)2, where δ ∈ {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let ψ(Y ) = Y + δ, λ = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Take φ(x) = x2 + 3δ, h = 1 and e = 2, then Φ(x) is key polynomial attached to the data (x, λ, ψ(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The valuation V on Q2[x], given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1) is such that V (Φ) = 2, V (x) = 1 and V (3) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The Φ(x) expansion of f(x) is xΦ3(x) − 9δΦ2(x) + 27xΦ(x) + b + x(a − 27δ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='12) The Φ(x)-Newton polygon of second order is the lower convex hull of the points (0, 7), (1, 9), (2, 9) and (3, min{2v3(b), 2v3(a − 27δ) + 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' As v3(a) = 3, so by using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='2), we have v3(b) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(b) = 4, then Φ-Newton polygon of f(x) has a single edge of slope 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore L = p6 1, where degree of p1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(b) > 4 and v3(a − 27δ) = 4, then Φ-Newton polygon of f(x) has a single edge of slope 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore L = p6 1, where degree of p1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(b) = 5 and v3(a−27δ) > 4, then Φ-Newton polygon of f(x) has a single edge joining the points (0, 7) and (3, 10) with a lattice point (2, 9) on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated to this edge is Y 3 ± Y ± ¯1, which is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore L = p2 1, where degree of p1 is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(b) > 5 and v3(a−27δ) > 4, then Φ-Newton polygon of f(x) has two edges of positive slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial associated to the first edge is Y 2+¯1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore L = p2 1p2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence we conclude that when a ≡ 0 mod 3 and b ≡ 0 mod 3, 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' 12 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR Case B2: a ≡ −1 mod 3 and b ≡ 0 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case f(x) ≡ x(x − 1)3(x + 1)3 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Clearly x-Newton polygon of f(x) provides one prime ideal say q of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 3OK = qJ, where J is an ideal of OK (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='13) Keepind in mind Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1, 3 | i(K) if and only if x−1 and x+ 1 provides atleast three prime ideals of residual degree 1 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let φδ(x) = (x + δ), where δ ∈ {−1, 1}, then the φδ(x) expansion of f(x) is φ7 δ(x)−7δφ6 δ(x)+21φ5 δ(x)−35δφ4 δ(x)+35δφ3 δ(x)−21δφ2 δ(x)+(a+7)φδ(x)+(b−δ(a+1)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='14) The φδ(x)-Newton polygon of f(x) is the lower convex hull of the points (0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 1), (6, v3(a + 7)) and (7, v3(b − δ(a + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If v3(a + 1) = 1 and v3(b) > 1, then for each δ ∈ {−1, 1}, φδ provides one prime ideal of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore J = p3 1p3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Note that when v3(a + 1) = 1 and v3(b) = 1, then either a ≡ 2 mod 9 or a ≡ 5 mod 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ 1 mod 3 and v3(b − a − 1) = 2, then for each δ ∈ {1, −1}, φδ-Newton polygon of f(x) has a single edge of positive slope and the residual polynomial attached to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus J = p3 1p3 2, where residual degree of p1 and p2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) = 2 and v3(b − a − 1) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ−1 provides one prime ideal of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ1-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge is the line segment joining the points (4, 0) and (6, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The second edge is the line segment joining the points (6, 2) and (7, v3(b − a − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore J = p3 1p2 2p3, where for each i = 1, 2, 3 the residual degree of pi is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) > v3(b − a − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ−1 provides one prime ideal of residual degree 1 and φ1-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge is the line segment joinig the points (4, 0) and (5, 1) and the second edge join (5, 1) with (7, v3(b − a − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to the first edge is linear and to second edge is (Y −¯1)(Y +¯1) or Y 2 + ¯1 (according as (b − a − 1)3 ≡ −1 mod 3 or (b − a − 1)3 ≡ 1 mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus either J = p3 1p2 2p3, where the residual degree of pi = 1, for each i = 1, 2, 3 or J = p3 1p2 2p3, where the residual degree of p1, p2 is 1 and of p3 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) = v3(b − a − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ−1 provides one prime ideal of residual degree 1 and φ1 has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to the first edge is linear and to the second edge is Y 2 − Y − 1 or Y 2 + Y − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus J = p3 1p2 2p3, where the residual degree of p1, p2 is 1 and of p3 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ 1 mod 3, v3(a + 7) ≥ 3, v3(b − a − 1) ≥ 3 and 2v3(a + 7) < v3(b − a − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ−1 provides one prime ideal of residual degree 1 and φ1-Newton polygon of f(x) has three edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge is the line segment joining ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 13 the points (4, 0) and (5, 1), the second edge join (5, 1) with (6, v3(a + 7)) and the third edge joins (6, v3(a + 7)) and (7, v3(b − a − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to the each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence J = p3 1p2p3p4, where the residual degree of pi = 1, for each i = 1, 2, 3, 4 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ −1 mod 3 and v3(b + a + 1) = 2, then for each δ ∈ {1, −1}, φδ-Newton polygon of f(x) has a single edge of positive slope and the residual polynomial attached to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus J = p3 1p3 2, where residual degree of p1 and p2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) = 2 and v3(b + a + 1) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ1 provides one prime ideal of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ−1-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge is the line segment joining the points (4, 0) and (6, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The second edge is the line segment joining the points (6, 2) and (7, v3(b − a − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore J = p3 1p2 2p3, where for each i = 1, 2, 3 the residual degree of pi is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) > v3(a + b + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ1 provides one prime ideal of residual degree 1 and φ−1-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge is the line segment joinig the points (4, 0) and (5, 1) and the second edge join (5, 1) with (7, v3(b − a − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to the first edge is linear and to second edge is (Y −¯1)(Y +¯1) or Y 2 +¯1 (according as (b+a+1)3 ≡ −1 mod 3 or (b+a+1)3 ≡ 1 mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus either J = p3 1p2 2p3, where the residual degree of pi = 1, for each i = 1, 2, 3 or J = p3 1p2 2p3, where the residual degree of p1, p2 is 1 and of p3 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) = v3(a + b + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ1 provides one prime ideal of residual degree 1 and φ−1 has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to the first edge is linear and to the second edge is Y 2 − Y − 1 or Y 2 + Y − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus J = p3 1p2 2p3, where the residual degree of p1, p2 is 1 and of p3 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So, 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 2 mod 9, b3 ≡ −1 mod 3, v3(a + 7) ≥ 3, v3(b + a + 1) ≥ 3 and 2v3(a + 7) < v3(a + b + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Then φ−1 provides one prime ideal of residual degree 1 and φ1-Newton polygon of f(x) has three edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The first edge is the line segment joining the points (4, 0) and (5, 1), the second edge join (5, 1) with (6, v3(a + 7)) and the third edge joins (6, v3(a + 7)) and (7, v3(b + a + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to the each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Hence J = p3 1p2p3p4, where the residual degree of pi = 1, for each i = 1, 2, 3, 4 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let a ≡ 5 mod 9 and v3(b) = 1, then v3(a + 7) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this, either φ1 provides one prime ideal of residual degree 1 and φ−1 provides two prime ideal of residual degree 1 each or φ1 provides two prime ideal of residual degree 1 each and φ−1 provides one prime ideal of residual degree 1 (according as b3 ≡ −1 mod 3 or b3 ≡ 1 mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus J = p3 1p2 2p3, where the residual degree of pi, for each i = 1, 2, 3 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let v3(a + 1) > 1 and v3(b) = 1, then v3(a + 7) = 1 and v3(b − δ(a + 1)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus for each δ, φδ provides one prime ideal of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So J = p3 1p3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, 1), (1, −1)} mod 3, then 14 ANUJ JAKHAR, SUMANDEEP KAUR, AND SURENDER KUMAR v3(b − a − 1) > 1 and v3(b + a + 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So the φ−1-Newton polygon of f(x) has two edges of positive slope and residual polynomial attached to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' And φ1 provides one prime ideal of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore J = p2 1p2p3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let v3(a + 1) > 1, v3(b) > 1 and ((a + 1)3, b3) ∈ {(−1, −1), (1, 1)} mod 3, then v3(b + a + 1) > 1 and v3(b − a − 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So the φ1-Newton polygon of f(x) has two edges of positive slope and residual polynomial attached to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The φ−1 provides one prime ideal of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore J = p2 1p2p3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 3 | i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Case B3: a ≡ 1 mod 3 and b ≡ 0 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case f(x) ≡ x(x2 + 1)3 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case x-provides one prime ideal of residual degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If we denote R((p) the residual degree of the prime ideal p, then for 3OK we have the following possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Case Factorization of 3OK Residual degree (1) 3OK = p1p3 2 R(p1) = 1 and R(p1) = 2 (2) 3OK = p1p2p3 R(p1) = 1, R(p1) = 2 and R(p1) = 4 (3) 3OK = p1p2 R(p1) = 1 and R(p2) = 6 (4) 3OK = p1p2 2p3 R(p1) = 1, R(p1) = 2 and R(p1) = 2 (5) 3OK = p1p2p3p4 R(p1) = 1 and R(pi) = 2, i = 2, 3, 4 Table Clearly in each case, there can be atmost 3 prime ideals of residual degree 2 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' But there are 5 monic irreducible polynomial of degree 2 over F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus in view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1 3 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' In this case 5 | i(K) if and only if a7 ≡ 2b6 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' (a, b) ∈ {(0, 0), (2, 2), (2, −2), (3, 1), (3, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let (a, b) = (0, 0), then f(x) ≡ x7 mod 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 7v5(a) > 6v5(b), then x-Newton polygon of f(x) has a single edge of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to this edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 5OK = p7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 7v5(a) < 6v5(b) and v5(a) ∈ {1, 5}, then x-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to each edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 5OK = p6 1p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 7v5(a) < 6v5(b) and v5(a) ∈ {2, 4}, then x-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to first edge is (Y + ¯1)(Y − ¯1) or (Y + ¯3)(Y − ¯2) and to the second edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore p3 1p3 2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' If 7v5(a) < 6v5(b) and v5(a) ∈ {2, 4}, then x-Newton polygon of f(x) has two edges of positive slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' The residual polynomial attached to first edge is (Y + ¯1)(Y 2 − Y + ¯1) or (Y − ¯1)(Y 2 + Y + ¯1) and to the second edge is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Therefore 5OK = p3 1p3 2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 5 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let (a, b) = (2, 2), then f(x) ≡ (x + 2)2(x − 1)(x4 + 2x3 + 4x2 + 2x + 2) mod 5, so 5OK = p1p2p3 or 5OK = p1p2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 5 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let (a, b) = (2, −2), then f(x) ≡ (x + 1)(x + 3)2(x4 + 3x2 + 4x2 + 3x + 2) mod 5, so ON COMMON INDEX DIVISOR OF THE NUMBER FIELDS DEFINED BY x7 + ax + b 15 5OK = p1p2p3 or 5OK = p2 1p2p3 or 5OK = p1p2p3p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 5 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let (a, b) = (2, −2), then f(x) ≡ (x + 1)(x + 3)2(x4 + 3x3 + 4x2 + 3x + 2) mod 5, so 5OK = p1p2p3 or p1p2 2p3 5OK = p1p2p3p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 5 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let (a, b) = (3, 1), then f(x) ≡ (x + 4)(x + 3)2(x4 + 4x3 + x2 + x + 2) mod 5, so 5OK = p1p2p3 or p1p2 2p3 5OK = p1p2p3p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 5 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Let (a, b) = (3, −1), then f(x) ≡ (x + 1)(x + 2)2(x4 + x2 + x2 + 4x + 2) mod 5, so 5OK = p1p2p3 or p1p2 2p3 5OK = p1p2p3p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Thus 5 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Next we show that 7 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Suppose there exists distinct non-zero prime ideals p1, · · · , pr of OK such that 7OK = pe1 1 · · · per r , where ei ≥ 1, then by the Fundamental Equality, e1f1 + · · · + erfr = 7 where fi is the residual degree of pi for i = 1, 2, · · · , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' Since ei ≥ 1 for all i = 1, 2, · · · , r, therefore there can be at most 7 prime ideals lying above 7, but for every positive integer h the number of monic irreducible polynomials of degree h in F7[x] is greater than or equal to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' So by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content='1, 7 ∤ i(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} +page_content=' □ References [1] S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQfgvi_/content/2301.00365v1.pdf'} diff --git a/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf b/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5b2198059aaa7df7f114b1d09e639c254b46bed9 --- /dev/null +++ b/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4d8a6295f5c49daa4eb766d7fbf8473a09f1bc4af9206d9aae32d77c2adb150a +size 2659472 diff --git a/NtFJT4oBgHgl3EQf0C3f/content/tmp_files/2301.11646v1.pdf.txt b/NtFJT4oBgHgl3EQf0C3f/content/tmp_files/2301.11646v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f13b4f26c730300a8020f05a43fdb1ba6d6f7a51 --- /dev/null +++ b/NtFJT4oBgHgl3EQf0C3f/content/tmp_files/2301.11646v1.pdf.txt @@ -0,0 +1,1577 @@ +Real-Space Imaging of the Tailored Plasmons in Twisted Bilayer Graphene + +F. Hu1,2*, Suprem R. Das2,3,4,5*, Y. Luan1,2*, T.-F. Chung6,7, Y. P. Chen6,7,8,9, Z. Fei1,2† + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA +3Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011, USA +4Department of Industrial and Manufacturing Systems Engineering, Kansas State University, +Manhattan, KS 66506, USA +5Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS +66506, USA +6Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA +7Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA +8School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana +47907, USA +9Purdue Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA + +*These authors contributed equally to this work. + +†Corresponding author: Z.F. (zfei@iastate.edu) + +Abstract + +We report a systematic plasmonic study of twisted bilayer graphene (TBLG) – two +graphene layers stacked with a twist angle. Through real-space nanoimaging of TBLG single +crystals with a wide distribution of twist angles, we find that TBLG supports confined infrared +plasmons that are sensitively dependent on the twist angle. At small twist angles, TBLG has a +plasmon wavelength comparable to that of single-layer graphene (SLG). At larger twist angles, the +plasmon wavelength of TBLG increases significantly with apparently lower damping. Further +analysis and modeling indicate that the observed twist-angle-dependence of TBLG plasmons in +the Dirac linear regime is mainly due to the Fermi-velocity renormalization, a direct consequence +of interlayer electronic coupling. Our work unveils the tailored plasmonic characteristics of TBLG +and deepens our understanding of the intriguing nano-optical physics in novel van der Waals (vdW) +coupled two-dimensional (2D) materials. + +Main text + +Graphene Dirac plasmons [1-6], which are collective oscillations of Dirac fermions in +graphene, have been widely investigated in recent years by using both the electron energy loss +spectroscopy [7-9] and optical imaging/spectroscopy [10-21] techniques. These quasiparticles +demonstrate many superior characteristics including high confinement, long lifetime, strong field +enhancement, broad spectral range, electrical tunability and a broad spectral range from terahertz +to infrared [1-21]. So far, plasmons in single layer graphene (SLG) have been extensively studied +and are generally well understood. One convenient way to create new plasmonic materials with +novel physics and properties is by stacking graphene with graphene and other 2D materials into +vdW materials or heterostructures. Indeed, the 2D nature of graphene makes it extremely sensitive +to interlayer coupling that could dramatically modify the properties of Dirac fermions and their +plasmonic excitations. For example, earlier studies about Bernal-stacked BLG [20,22] and + +graphene/hBN heterostructures [23,24] have demonstrated many unique plasmonic characteristics +compared to those of SLG. +In this Letter, we report a systematic nano-infrared imaging study of plasmons in TBLG +[Fig. 1(a)], which is formed when two misorientated graphene layers are stacked together by vdW +forces. Depending on the twist angle () between the two graphene layers, moiré patterns with +different periodicities could form [Fig. 1(b)]. Due to the interlayer coupling and modulation of +Dirac fermions by moiré superlattice potential, the electronic structure of TBLG shows distinct +features compared to SLG and Bernal-stacked BLG, and it varies systematically with . For +example, TBLG with a sizable features two separated Dirac cones [Fig. 1(c)] in the momentum +space [25-34]. Moreover, the Fermi velocity ( +tBLG +F +v +) close to the charge neutrality point is +renormalized compared to that of SLG ( +SLG +Fv +), namely +tBLG +F +v + drops systematically below +SLG +Fv + as + decreases [25-30]. Therefore, TBLG is a unique system where the Fermi velocity of Dirac +fermions could become an adjustable variable in experimental studies. The unique electronic +properties of TBLG have led to observations of many interesting optical phenomena through far- +field spectroscopic experiments [35-38]. So far, plasmonic responses of TBLG have not been +explored experimentally despite the potential rich physics according to theoretical predictions +[39,40]. + +Here we utilize a scattering-type scanning near-field optical microscope (s-SNOM) to +perform nano-infrared imaging studies of TBLG plasmons. The s-SNOM apparatus is built based +on an atomic force microscope (AFM). As illustrated in Fig. 1(a), the infrared light (solid arrow) +from a continuous-wave infrared laser is focused at the apex of a metalized AFM tip. The laser- +illuminated tip acts as both a launcher and a detector of surface plasmons [13-23]. The back- +scattered light (dashed arrow) off the tip-sample system contains essential information about +plasmons underneath the tip. The s-SNOM collects simultaneously the topography, near-field +scattering amplitude (s) and phase (). By analyzing both the s and  data images, we can +determine the key plasmonic parameters of TBLG. Our samples were grown by the chemical vapor +deposition (CVD) method on copper foils [41-43] and then transferred to the standard SiO2/Si +substrates (Supplemental Material [44]). As shown in Fig. 1(d), both SLG and TBLG are single- +crystal grains with a hexagonal shape and the TBLG grains are typically located at the center of +relatively larger SLG grains. Occasionally, we also see hexagon-like shapes with slightly curved +edges (Fig. S5) [45,46], but in all cases, these SLG or TBLG single-crystals demonstrate a six-fold +rotational symmetry (Supplemental Material [44]). According to the previous studies [45,47], the +six-fold flake symmetry correlates strictly and accurately with the lattice orientation, so it is +convenient to determine the twist angle with a relatively good accuracy (1) by comparing the +orientations of the TBLG and SLG grains. + +Representative s-SNOM imaging data are shown in Fig. 2, where we plot both the +normalized amplitude [Fig. 2(b)] and phase [Fig. 2(c)] signals of a typical sample region +containing two TBLG grains. The data images were taken at an excitation laser energy of E = 0.11 +eV that is away from the strong optical phonon resonance of SiO2 centered at about 0.14 eV [13]. +Therefore, the near-field responses of graphene at our excitation energy are mainly due to +plasmons [14]. Figure 2(a) sketches the sample configuration, where we can conveniently +determine the twist angles of TBLG from the orientations of the hexagonal grains. For example, +the TBLG sample labeled as ‘TBLG1’ has a twist angle of about 26 relative to SLG and the one +labeled as ‘TBLG2’ has a twist angle close to 1. Their different twist angles result in distinct near- +field responses. As shown in Figs. 2(b) and 2(c), TBLG1 has significant higher near-field + +amplitude compared to SLG but shows no clear phase contrast with respect to the latter. On the +contrary, the amplitude of TBLG2 is almost the same as that of SLG and its phase is slightly +weaker. Such dramatic differences in the near-field responses are clear indications of the strong  +dependence of TBLG. More near-field data images are given in Figs. S1 and S2, where additional +TBLG samples with various twist angles are shown. In all the samples we measured (partly shown +in Figs. 2, S1 and S2), the near-field amplitude of TBLG is comparable to SLG for  ≤ 3, and +gradually increases from an intermediate signal (  5) to a maximum value ( > 7). The phase +signal of TBLG, on the other hand, is roughly the same as SLG for  > 7, but slightly declines as + approaches 0. The above  dependence is more clearly seen in Figs. 4(c) and 4(d), where we +summarized the extracted amplitude and phase signal data points (squares) from tens of TBLG +samples that we measured. + +The unique near-field responses discussed above are directly linked to the plasmons in +TBLG. Indeed, we found direct evidence of plasmons in the high-resolution imaging data (Figs. 3 +and S3) taken over five small sample regions (marked with dashed squares) in Fig. S1. These +regions (labeled with ‘P1’  ‘P5’) are chosen to be at the edge of SLG or the boundaries between +SLG and TBLG. The amplitude images are shown in Figs. 3(a) – 3(e), where we observe bright +fringe(s) close to the SLG edge and the SLG/TBLG boundaries. This can be seen more clearly in +the line profiles [grey solid curves in Fig. 3(f) – 3(j)] taken perpendicular to the edges or boundaries +in the amplitude images (along blue dashed lines). Here in these line profiles, the peak features +correspond to the bright fringes in the images. + +According to previous studies [14-24], the bright fringes registered by the s-SNOM are +generated due to the constructive interference between tip-launched and edge- or boundary- +reflected plasmons. The plasmonic origin of the observed fringes is further confirmed by the +spectroscopic imaging data (Fig. S4), where we observed a systematic evolution of the bright +fringes with laser energy, consistent with the dispersion nature of plasmons. There are two main +observations from these plasmonic fringes data (Fig. 3). First, fringes are clear and strong close to +the SLG/TBLG3 (≈27) and SLG/TBLG4 (≈12) boundaries. As  decreases, the fringes +become weaker and fewer at the SLG/TBLG5 (≈5) boundary and then barely seen at the +SLG/TBLG6 boundary (≈3). Second, in the case of SLG/TBLG3 [Fig. 3(b)] and SLG/TBLG4 +[Fig. 3(c)] boundaries, we can easily identify two to three fringes. Nevertheless, at the edge of +SLG [Fig. 3(a)], we can only see one bright fringe. Note that the edge of SLG is a nearly-perfect +plasmon reflector. The plasmon reflection at the SLG/TBLG boundaries, on the other hand, is in +principle weaker. Therefore, we can tell directly from the fringes data that our SLG sample has a +relatively higher plasmon damping compared to TBLG with relatively large . Figure S3 plot the +near-field phase images and the corresponding line profiles, where plasmonic interference fringes +are also seen. The amplitude and phase imaging data are consistent and complementary to each +other. They are all considered in our modeling as discussed in detail below. + +To extract quantitative information about plasmons in SLG and TBLG, we performed +numerical modeling of both the plasmonic fringes profiles (Figs. 3 and S3) and the -dependent +near-field amplitude and phase signals [Figs. 4(c) and 4(d)] by using the so-called spheroid model. +In this model, the s-SNOM tip is approximated as a highly-elongated conducting spheroid (Fig. +S7) and we evaluate the complex scattering signal by computing the total radiating dipole of the +coupled tip-sample system (Supplemental Material [44]). We wish to emphasize that our model +has been proven to effective in describing s-SNOM responses of graphene with quantitative +accuracy [14,16,23]. The main modeling parameter of the sample is the optical conductivity + +(i) that is directly linked to the complex plasmon wavevector (qp = q1 + iq2) under the +long-wavelength approximation: + +0(1 +) +p +s +q +i +E + + + + + +. (1) +Here + is the reduced plank constant, 0 is the vacuum permittivity, and s is the relative +permittivity of SiO2. For convenience, our analysis and discussions are based on the following two +parameters: the plasmon wavelength (p = 2/q1) and damping rate (p = q2/q1). Based on Eq. 1, +we know that the plasmon wavelength (p) is roughly proportional to , and the damping rate (p) +scales linearly with . + +We first fit the plasmonic fringe profiles of SLG [Figs. 3(f) and S3(f)]. Through the fitting, +we extract the plasmon wavelength ( +SLG +p + +) and damping rate ( +SLG +p + +) of SLG at E = 0.11 eV to be +about 279 nm and 0.2, respectively. Based on Eq. (1), we can establish a simple relation between +SLG +p + + and the carrier density (n) under the Drude approximation: +2 +SLG +SLG +2 +0 +2 +| +| +(1 +) +F +p +s +e +v +n +E + + +  + + +. (2) +Here +SLG +6 +10 m/s +F +v + + is the Fermi velocity of SLG. Equation (2) allows us to estimate the carrier +density of SLG to be n  1.2 × 1013 cm-2, which is a typical value for highly-hole-doped CVD +samples on SiO2/Si substrates at ambient conditions [16]. The relatively high doping is mainly due +to the impurities on the surface of SiO2 as well as the water and oxygen molecules in the air [48]. +Considering that all the samples studied here share the same substrate and atmospheric conditions, +they are expected to share roughly an equal density of external dopants and therefore a similar +carrier density [21,22]. +Based on the extracted parameters of SLG, we then determine both the plasmon +wavelength ( +tBLG +p + +) and damping rate ( +tBLG +p + +) of TBLG by fitting the fringe profiles at the +SLG/TBLG boundaries (Figs. 3 and S3). Through fitting, we estimate that ( +tBLG +p + +, +tBLG +p + +) of +TBLG3 ( ≈ 27), TBLG4 ( ≈ 12), TBLG5 ( ≈ 5) and TBLG6 ( ≈ 3) to be (393 nm, 0.10), +(387 nm, 0.11), (340 nm, 0.16) and (278 nm, 0.28), respectively. These numbers are plotted in +Figs. 4(a) and 4(b) as data points. Note that the first two numbers of +tBLG +p + + can be read out directly +from the fringe profiles of TBLG3 and TBLG4 by doubling the fringe period [arrows in Figs. 3(g) +and 3(h)]. Nevertheless, precise modeling of the complex fringe profiles is required to extract both +tBLG +p + + and +tBLG +p + +, and to analyze data from TBLG samples without strong fringes [Figs. 3(d) and +3(e)]. In the latter case, the modeling fits mainly the s and  signals of TBLG in contrast to SLG. +In Figs. 4(c) and 4(d), we show the modeling curve of s and  contrast signals of TBLG versus +SLG at a wide distribution of twist angles (red curves), which match well the trend of the +experimental data points with twist angles above 3 (marked with dashed lines). At twist angles +below 3, the experimental data points clearly deviate from the modeling curve, which will be +discussed in the following paragraphs. The smooth +tBLG( ) +p + + and +tBLG( ) +p + + parameters [red curves +in Figs. 4(a) and 4(b)] used to model the s and  contrast signals are fully consistent with the +discrete data points obtained from fringe profile fitting. + +Now we wish to discuss the origin of the -dependence of +tBLG +p + + and +tBLG +p + +. We first pay +attention to twist angles above 3, where TBLG is in the Dirac regime [Fig. 1(c)] [25-29]. Here + +we assume that carriers are equally distributed among the two graphene layers, which is reasonable +considering no external gating. The general results won’t change much even with slightly unequal +carrier distribution among the two graphene layers (Fig. S6 and Supplemental Material [44]). +Under the equal carrier distribution assumption, +tBLG +p + + can be written as +2 +tBLG +tBLG +2 +0 +2 +2 +| +| +( ) +( ), +(1 +) +p +F +s +e +n v +E + + + + +  + + + (3) +where the Fermi velocity of TBLG ( +tBLG +F +v +) is proven to be sensitively dependent on  due to the +Fermi velocity renormalization. The amount of Fermi velocity renormalization is determined by +the interlayer coupling energy (t) of TBLG [inset of Fig. 4(a)] as described by the following +equation: [25] +tBLG +SLG +2 +SLG +( ) +[1 9( +) ] +F +F +F +t +v +v +v +K +  + + + . (4) +Here, +(8 / 3 )sin( / 2) +K +a + + + + + is the momentum separation of the two Dirac cones [Fig. 1(c)], and +a = 0.246 nm is the lattice constant. Equation 4 indicates that t is the one single parameter that +controls +tBLG( ) +F +v + and hence +tBLG( ) +p + + . Here we set t to be 0.1 eV, which is roughly consistent with +previous studies [29,34]. With such a t setting, we calculated +tBLG( ) +p + + based on Eqs. (3) and (4), +which is shown as the red curve in Fig. 4(a). Other choices of t will lead to either faster or slower +decreasing of +tBLG +F +v + and hence +tBLG +p + + as  drops [inset of Fig. 4(a)]. + +The origin for the higher +tBLG +p + + at smaller  in the Dirac regime ( ≥ 3) is likely due to +the stronger charge scattering rates [49,50]. According to previous literature [51], the charge +scattering rates () due to either long-range Coulomb scattering or short-range defect scattering +are inverse proportional to the Fermi velocity. Therefore, as  decreases,  rises and thus +tBLG +p + + +increases. Note that interband transitions are forbidden due to the Pauli blocking for  ≥ 3, where +threshold energy for interband transitions ( +tBLG +2 +F +E +) is estimated to be over 0.2 eV, far above our +laser energy (0.11 eV). + +Finally, we wish to discuss briefly TBLG samples with twist angles below 3, where the +Dirac approximation begins to fail [26-29]. In this regime, we find that the amplitude signal of the +TBLG samples deviates from the projected trend of the modeling curves and stay close to that of +SLG [Fig. 4(c)]. With quantitative modeling (Fig. S8), we estimate that the +tBLG +p + + at small twist +angles ( ≤ 2) is in the range from 278 to 314 nm. According to previous theoretical studies [26- +28], the lowest-energy bands of TBLG with small twist angles become flat or nearly-flat close to +the charge neutrality point, which could lead to an extremely small +tBLG +p + + (Eq. 3). The finite +tBLG +p + + +of TBLG ( ≤ 2) observed here suggests that the Fermi surface of our highly-doped samples is +away from these relatively flat bands. The phase signals [Fig. 4(d)] of TBLG ( ≤ 2) appear to be +slightly smaller than that of SLG and large-twist-angle TBLG, indicating even higher plasmon +damping rates: +tBLG +p + +( ≤ 2) = 0.2 - 0.4 (Fig. S8). The higher damping is most likely due to +interband transitions, which are enabled in TBLG ( ≤ 2) at our excitation energy (0.11 eV) due +to the small energy separations between the lowest-energy bands. More detailed discussions about +TBLG with  ≤ 2 are given in the Supplemental Material [44]. Future studies with more + +comprehensive experiments of small-twist-angle TBLG and more precise determinations of twist +angles are needed to explore further TBLG plasmons in the non-Dirac regime. + +By combining the state-of-the-art s-SNOM technique with rigorous numerical modeling, +we performed a systematic nano-infrared imaging study of TBLG single crystals with various twist +angles. In the Dirac linear regime, we found that TBLG support infrared plasmons with parameters +that vary systematically with the twist angle between the two graphene planes. The underlining +physics behind the observed twist angle dependence is the Fermi velocity renormalization, which +is originated from the interlayer electronic coupling. Our study establishes TBLG as a unique +platform where the Fermi velocity, the fundamentally important parameter of Dirac fermions, has +become an adjustable variable in nano-optical and plasmonic studies of Dirac materials. + + +F.H., Y.L. and Z.F. acknowledge startup support from Iowa State University and the +royalty funds from the U.S. Department of Energy, Ames Laboratory. The nano-optical setup is +partially supported by W. M. Keck Foundation. S.R.D. acknowledges the U.S. Department of +Energy, Ames Laboratory. The synthesis of TBLG samples at Purdue University is partially +supported by NSF CMMI (grant 1538360) and NSF EFMA (Grant No. 1641101). + +References +[1] S. Das Sarma and E. H. Hwang, Phys. Rev. Lett. 102, 206412 (2009). +[2] M. Jablan, H. Buljan, and M. Soljačić, Phys. Rev. B 80, 245435 (2009). +[3] A. Vakil and N. Engheta, Science 332, 1291 (2011). +[4] A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat. Photon. 6, 749 (2012). +[5] D. N. Basov, M. M. Fogler, and F. 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S. Novoselov, and A. K. Geim, Rev. Mod. +Phys. 81, 109-162 (2009). + +Figure Captions +FIG. 1. (a) Illustration of the nano-infrared imaging experiment of a TBLG/SLG single crystal. +The solid and dashed arrows mark the directions of the incident laser beam and back-scattered +photons, respectively. (b) Sketch of the crystal structure of TBLG revealing moiré periodic pattern. +The double-sided arrow marks the moiré period. (c) Calculated band structure of TBLG with a +twist angle of 5 with the continuum model [25]. Here the momentum unit K equals to the +separation between the two Dirac points (K1 and K2) in the momentum space. (d) Optical image +of representative TBLG/SLG single-crystal samples. The scale bar represents 5 m. + +FIG. 2. (a) Sketch of the sample geometry indicating two adjacent TBLG grains with different +twist angles with respect to SLG. (b) and (c) The near-field images plotting scattering amplitude +(s) and phase (), respectively. In both images, the amplitude or phase signal is normalized to that +of SLG. The laser energy is set to be at E = 0.11 eV. The scale bars represent 3 m. + +FIG. 3. (a)-(e) High-resolution near-field amplitude images of the five small regions (‘P1’ – ‘P5’) +marked in Fig. S1 (squares), respectively. The white dashed lines in the images mark the SLG edge +and the TBLG/SLG boundaries. The scale bars represent 400 nm. (f)-(j), Experimental (grey solid) + +and modeled (red dashed) amplitude profiles taken along the blue dashed lines in the +corresponding near-field images above. The blue arrows in (g) and (h) mark the size of +tBLG +p + +that +is twice the fringe period. The vertical dashed lines mark the boundaries between SLG and TBLG. + +FIG. 4. (a) The +tBLG +p + + data points extracted by modeling the fringe profiles in Fig. 3. Inset plots +the calculated +tBLG( ) +Fv + normalized to +SLG +Fv + considering different t. (b) The +tBLG +p + + data points +extracted by modeling the fringe profiles in Fig. 3. The red curves in (a) and (b) are used for +calculations of the amplitude and phase signals in (c) and (d). The blue arrows in (a) and (b) mark +the values of +SLG +p + + and +SLG +p + +, respectively. (c) The -dependent near-field amplitude from both +experiment (squares) and modeling (red curve). (d) The -dependent near-field phase from both +experiment (squares) and modeling (red curve). Both the amplitude (c) and phase (d) of TBLG are +normalized to those of SLG. The vertical dashed lines in (c) and (d) mark  = 3. + + +Figure 1 +K1 +K2 +θ +(b) +(d) +(c) +(a) +Energy (eV) +TBLG +SLG +TBLG +SLG +�K +kx (�K) +ky (�K) +-1 +-0.5 +0 +1 +0.5 +-2 +-1 +0 +1 +2 +1 +0 +-1 +SiO2 + + +Figure 2 +(a) +θ � +o +26 +(b) +TBLG1 +SLG +2 +SiO +(c) +� (norm.) + +s (norm.) +2 +0 +TBLG2 +SLG +2 +0 +θ � +o +1 + +� +� +� + + + +Figure 3 +(a) +(f) +(b) +(c) +(d) +(e) +(g) +(h) +(i) +(j) +SLG +SiO2 +P1 (SLG) +P2 (TBLG3, � ~ 27o) +P3 (TBLG4, � ~ 12o) +P4 (TBLG5, � ~ 5o) +P5 (TBLG6, � ~ 3o) +s (norm.) +2 +0 + + +-0.5 + +0.5 + +0 + +x (�m) +-0.5 + +0.5 + +0 + +x (�m) + +-0.5 + +0.5 + +0 + +x (�m) + +-0.5 + +0.5 + +0 + +x (�m) + +-0.5 + +0.5 + +0 + +x (�m) +s (norm.) + +0 + +1 +2 + +0 + +1 +2 + +0 + +1 +2 + +0 + +1 +2 + +0 + +1 +2 +SLG +SiO2 +SLG +TBLG3 +SLG +TBLG4 +TBLG5 +SLG +SLG +TBLG6 +TBLG3 +TBLG4 +TBLG5 +SLG +SLG +SLG +SLG +TBLG6 + +Figure 4 + +sTBLG sSLG +θ (degree) +(c) +ψTBLG ψSLG +θ (degree) +λp (nm) +θ (degree) +γp +θ (degree) +(d) +(a) +(b) +0 +10 +20 +30 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +10 +20 +30 +0.0 +0.1 +0.2 +0.3 +0 +10 +20 +30 +0.0 +0.5 +1.0 +0 +10 +20 +30 +0 +100 +200 +300 +400 +0 +10 +20 +0 +1 +t=0.15 eV +t=0.1 eV +t=0.05 eV +vF (norm.) +θ (degree) +/ +/ + +Supplemental Material +Real-Space Imaging of the Tailored Plasmons in Twisted Bilayer Graphene + +F. Hu1,2*, Suprem R. Das2,3,4,5*, Y. Luan1,2*, T.-F. Chung6,7, Y. P. Chen6,7,8,9, Z. Fei1,2† + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, Iowa 50011, USA +3Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011, USA +4Department of Industrial and Manufacturing Systems Engineering, Kansas State University, +Manhattan, KS 66506, USA +5Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS +66506, USA +6Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA +7Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA +8School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana +47907, USA +9Purdue Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA + +*These authors contributed equally to this work. + +† Email: (Z.F.) zfei@iastate.edu + + +List of contents + +1. Nano-optical imaging setup + +2. Sample fabrication procedures + +3. Additional s-SNOM imaging data + +4. Flake shape of single-crystal SLG/TBLG samples + +5. Unintentional doping of the samples + +6. Analysis of unequal carrier distribution + +7. Quantitative modeling of the s-SNOM signals + +8. Analysis and discussions of small-twist-angle TBLG + + + + + + + +1. Nano-infrared imaging setup + + +For nano-infrared imaging experiments in the current work, we used a scattering-type +scanning near-field optical microscope (s-SNOM) apparatus (www.neaspec.com) that is built +based on an atomic force microscope (AFM). The AFM is operated in the tapping mode with a +tapping frequency of about 270 kHz and a tapping amplitude of about 50 nm. The AFM tips used +in the current work are platinum iridium coated silicon tips with a radius of curvature of about 25 +nm at the tip apex (www.nanoworld.com). For signal detection, we used a mercury cadmium +telluride photodiode (www.kolmartech.com). A pseudo-heterodyne interferometer is implemented +in our s-SNOM to extract both the near-field amplitude (s) and phase (of the complex near- +field signal. To suppress the background signal, we demodulated the near-field signal at the 3rd +harmonics of the tapping frequency of the AFM tip. For optical excitation, we used a continuous- +wave CO2 laser (www.accesslaser.com). The photon energy of the laser can be discretely tunable +from 0.11 to 0.13 eV. Our nano-infrared imaging experiments were all performed at ambient +conditions. + +2. Sample fabrication procedures + +The growth of the hexagon-shaped TBLG/SLG single crystals was achieved by using the +atmospheric-pressure chemical vapor deposition (CVD) method. A mixture of methane carrier gas +was atomically cracked at high temperature (1050 °C) with argon/hydrogen gas before controllably +getting deposited onto a pre-cleaned copper foil. After the growth process, the methane flow was +stopped, and the sample was cooled down to room temperature in the furnace with argon/hydrogen +gas flow uninterrupted. For s-SNOM studies, our samples on copper foil were transferred onto the +standard SiO2/Si wafers used a wet-chemical etching and transfer method. In short, a 100-nm-thick +polymethyl methacrylate (PMMA) protection layer was applied to one side of the copper foil and +the TBLG on the other side of the foil was plasma etched using an oxygen plasma. The copper +was then etched overnight in an etchant solution. After multiple washing of the floating +TBLG/PMMA stack with deionized water and a mild aqueous acid, the stack was transferred onto +the SiO2/Si wafers. The PMMA layer was stripped off TBLG in about 4 hours after baking/drying +on a hotplate at 90 °C. Finally, the sample was electronically cleaned at 300 °C for 2 hours. More +detailed introductions about the growth procedures are given in the earlier study (Ref. [41] in the +main text). + +3. Additional s-SNOM imaging data + +In Figs. S1-S4, we provide additional near-field imaging data about TBLG. Figure S1 plots +four typical TBLG crystals (‘TBLG3’ – ‘TBLG6’) that we used for high-resolution imaging. The +high-resolution imaging data taken at five specific locations (‘P1’ to ‘P5’ marked in Fig. S1) are +plotted in Fig. 3 (amplitude images and profiles) of the main text and Fig. S3 (phase images and +profiles). Figure S2 plots near-field amplitude and phase images at four different locations, where +a total of eight SLG/TBLG crystals (‘TBLG7’ – ‘TBLG14’) are included. Figure S4 plots the +excitation-energy-dependent s-SNOM imaging data at the boundary between ‘SLG’ and ‘TBLG3’. +One can see that the plasmonic interference fringes at the SLG/TBLG3 boundary show a +systematic variation with photon energy. Through modeling of the fringe profiles, we extracted +the plasmon wavelength of TBLG (SLG) to be about 393 nm (279 nm), 370 nm (262 nm) and 350 +nm (248 nm) for excitation energies of 0.110, 0.116 and 0.121 eV, respectively. The energy- +dependent plasmon wavelengths are consistent with the dispersion nature of plasmons. + + + +FIG. S1. (a)-(d) Sketches of the sample geometries of four additional TBLG single crystals +(‘TBLG3’ – ‘TBLG6’) with different twist angles. (e)-(h) The near-field amplitude images of the +samples sketched in (a)-(d), respectively. (i)-(l) The near-field phase images of the samples +sketched in (a)-(d), respectively. The excitation laser energy is set to be at E = 0.11 eV. In all the +images, the amplitude or phase signal is normalized to that of SLG. The squares mark the five +positions (‘P1’ ─ ‘P5’) where we collected high-resolution imaging data as shown in Fig. 3 of the +main text and Fig. S3 of the Supplemental Material. The scale bars represent 3 m. + +(a) +(b) +(c) +(d) +TBLG4 +TBLG5 +TBLG3 +TBLG6 +日~3° +日~270 +SLG +SLG +SLG +SLG +SiO, +SiO2 +SiO2 +SiO2 +(e) +(f) +(g) +(h) +P1 +NI +P5 +s (norm.) +P4 +P3 +0 +(i) +(0) +(k) +(0) +P1 +2 +P5 +y (norm.) +P4 +0 +FIG. S2. (a)-(d) Sketches of the sample geometry of four samples regions that include a total of +eight additional TBLG single crystals (‘TBLG7’ – ‘TBLG14’). (e)-(h) Near-field amplitude +images of sample regions sketched in (a)-(d), respectively. (i)-(l) The near-field phase images of +sample regions sketched in (a)-(d), respectively. In all the images, the amplitude or phase signal is +normalized to that of SLG. The scale bars represent 3 m. + + + + + + +(a) +TBLG7 +b +C +d) +TBLG11 +9~0 +6~18° +9~00 +TBLG9 +TBLG13 +SLG +SLG +SiO2 +SLG +SLG +SiO2 +Sio2 +TBLG8 +TBLG14 +TBLG10 +TBLG12 +~70 +6~28° +SLG +SLG +SiO2 +SLG +SLG +(e) +(f) +(g) +(h) +2 +s (norm.) +0 +0) +(k) +2 +w(norm.) +0 +FIG S3. (a)-(e) Near-field phase images of the five small regions (‘P1’ – ‘P5’) marked in Fig. S1 +(squares). The white dashed lines in the images mark the edges of SLG and the boundaries between +TBLG and SLG. The scale bars in all the images represent 400 nm. (f)-(j) Experimental (grey solid) +and modeled (red dashed) phase profiles taken perpendicular to the SLG edge and the TBLG/SLG +boundaries. In all the near-field images, the phase signal is normalized to that of SLG. The +experimental phase profiles were taken along the blue dotted lines in the corresponding near-field +images. The vertical dashed lines mark the boundaries between SLG and TBLG. + + +FIG. S4. Near-field amplitude images of the SLG/TBLG3 boundary (see Fig. 3 in the main text) +at various excitation laser energies: E = 0.110 eV (a), 0.116 eV (b) and 0.121 eV (c). Here we plot +the near-field amplitude normalized to that of SLG. The scale bars represent 400 nm. + +4. Flake shape of single-crystal SLG/TBLG samples + +Most of our single-crystal TBLG/SLG samples appear in a hexagonal shape with straight +edges. Occasionally, we also see hexagon-like shapes with slightly-curved edges. For example, in +Fig. S5a (or Fig. 3b), we found the edge of TBLG3 is not exactly straight. A large-area image is +shown in Fig. S5b, where we mark carefully the boundaries of SLG and TBLG3 grains with green +and blue curves. These curves are replotted in Fig. S5c, where one can see that the edges of the +TBLG3 grain are not straight. Instead, it has a hexagonlike shape with slight negative curvature at +the edges. This is, in fact, one of the typical shapes of single-crystal graphene flakes. As reported +by an earlier study (Ref. [45] in the main text), various types of six-fold symmetric shapes of +graphene flakes (from flower-like to hexagon-like) could occur by varying systematically the CVD +growth conditions, among which both the perfect hexagonal shape and slight-curved hexagonal +shape are included. The lattice orientation of graphene is consistent with the six-fold symmetry of +the flake, as verified by diffraction experiments in Refs. [45] and [47] of the main text. An earlier +study also shows slightly-curved hexagon shape in twisted bilayer and trilayer graphene flakes + +E= 0.110 eV +E= 0.116 eV +(a) +(b) +(c) +E= 0.121 eV +2 +s(norm.) +0P1 (SLG) +P2 (TBLG3, 0 ~ 27°) +P3 (TBLG4, 0 ~ 12°) +P4 (TBLG5, 0 ~ 5°) +P5 (TBLG6, 0 ~ 3°) +(a) +(q) +(c) +(d) +(e) +SiO2 +2 +SLG +TBLG5 +SLG +TBLG4 +(wvou) +TBLG3 +SLG +SLG +TBLG6 +SLG +0 +(f) +(wvou) +- +1 +SiQ2 +SLG +TBLG3! +SLG +SLG +TBLG4 +TBLG5! +SLG +SLG +TBLG6 +0 +0 +0 +二 +0 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +x (μm) +x (μm) +x (μm) +x (μm) +x (μum)(Ref. [46] in the main text). To sum up, the TBLG3 sample with slight-curved edges is also typical +for CVD-grown single-crystal samples and we followed the symmetry of the entire flake when +determining the twist angle. + + + +FIG. S5. (a) High-resolution near-field amplitude image of the TBLG3/SLG boundary, a replot of +Fig. 3b in the main text. (b) The large-area near-field amplitude of the TBLG3/SLG sample, a +replot of Fig. S1e. (c) Sketch of the geometry of the TBLG3/SLG sample. + +5. Unintentional doping of the samples + +Our CVD-grown samples are all highly hole-doped at ambient conditions due to the high +density of impurities on the amorphous SiO2 surface as well as water and oxygen molecules in the +air (Ref. [48] in the main text). Considering that all the data presented in the paper were from +samples produced from the same batch and transferred onto the same wafer, they share nearly +identical chemical and environmental conditions. Therefore, we are safe to assume that all the +TBLG and SLG samples share roughly an equal density of external dopants and therefore a similar +carrier density. Indeed, previous studies have confirmed that adjacent SLG and Bernal-stacked or +randomly-stacked BLG share roughly the same carrier density (Refs. [21,22] in the main text). By +fitting the plasmonic fringe profile of SLG, we can accurately determine the plasmon wavelength +of SLG be about 279 nm, based on which we estimate the carrier density of SLG and TBLG is n + 1.2 × 1013 cm-2. + +6. Analysis of unequal carrier distribution + +In the current paper, we assume equal carrier distribution among the top and bottom layers +of TBLG when analyzing the data. This is a reasonable assumption considering the following facts. +First, there is no gating or dedicated chemical doping to introduce doping asymmetry, so strong +asymmetry is unlikely to occur due to the natural doping by the air or substrate. Second, even with +small doping asymmetry, the results won’t change much from the current analysis based on equal +carrier distribution. Detailed analysis and discussions about doping asymmetry are given below. + +Introducing unequal carrier distribution among the two graphene layers (or equivalently, +adding a built-in electric field between the two graphene layers) can indeed affect the optical +conductivity and plasmon wavelength of TBLG, and hence the near-field scattering signals. Here +we only consider the Dirac linear regime (twist angle  ≥ 3), where the plasmon wavelength of +TBLG obeys the following relationship: +2 +2 +2 +( +) +tBLG +tBLG +top +bottom +top +bottom +tBLG +top +bottom +p +F +F +F +E +E +v +n +n + + + + + + + + + + + +, (1) +where the superscripts “top” and “bottom” denote the top and bottom graphene layers, respectively. +The total carrier density +total +top +bottom +n +n +n + + +. For convenience, we define the parameter +( +)/ +top +bottom +total +x +n +n +n + + + to characterize the doping asymmetry, which is in the range [-1, 1]. We do + +(a) +(b) +(c) +SLG +TBLG3 +TBLG3 +TBLG3 +SLG +SiO2 +SLG +SiO2not consider x < -1 or x > 1, which means opposite charges distributed among the two graphene +layers – an unlikely situation without dual gating setup. With the parameter x, Eq. 1 can be re- +written as + + + +( 1 +1 +) +/ 2 +tBLG +tBLG +total +p +Fv +x +x +n + + + + + +. (2) +Accordingly, we can obtain the formula of SLG assuming the same total carrier density: +SLG +SLG +total +p +Fv +n + + +, so the ratio between +tBLG +p + + and +SLG +p + + can be written as +/ +( +/ +)( 1 +1 +) / +2 +tBLG +SLG +tBLG +SLG +p +p +F +F +v +v +x +x + + + + + + +. (3) + +In the Dirac regime of TBLG ( ≥ 3), +/ +tBLG +SLG +F +F +v +v + + + is the Fermi velocity renormalization +factor that is a constant at every given twist angle, so +/ ( +) +( 1 +1 +) / +2 +tBLG +SLG +p +p +x +x + + + + + + + is +solely dependent on x. In Fig. S6, we plot the x-dependence curve of +tBLG +p + + normalized to +SLG +p + +, +namely +/ ( +) +tBLG +SLG +p +p + + +. Here +/ ( +) +tBLG +SLG +p +p + + + reaches the maximum +2 when x = 0 (namely equal +carrier distribution among graphene layers). Introducing doping asymmetry (x ≠ 0) will reduce the +plasmon wavelength of TBLG. The +/ ( +) +tBLG +SLG +p +p + + + reaches the minimum 1 when x = -1 or 1 +(carriers are fully distributed in the top or bottom graphene layer). From Fig. S6, we also notice +that the variation of +/ ( +) +tBLG +SLG +p +p + + + is very small when x is in a reasonable range. For example, for +|x| ≤ 0.5, the variation of +/ ( +) +tBLG +SLG +p +p + + + is only within 3% (see Fig. S6). In the case of our highly- +doped TBLG samples (ntotal  1.2 × 1013 cm-2), |x| = 0.5 means that the one layer has a carrier +density of 0.3 × 1013 cm-2 corresponding to a Fermi energy of 0.2 eV, and the other layer has 0.9 +× 1013 cm-2 corresponding to a Fermi energy of 0.35 eV. This is, in fact, a huge doping asymmetry +that in principle cannot be created without external gating. + + +FIG. S6. The asymmetric doping dependence of the +/ ( +) +TBLG +SLG +p +p + + +. The parameter +( +)/ +top +bottom +total +x +n +n +n + + + characterizes the doping asymmetry. + + +1.5 +1.4 +< 3% +- +- +DTS +- +- +1.3 +- +- +- +- +- +TBLG +1.2 +- +- +2 +1.1 +- +1 +1.0 +- +1 +- +1 +1 +- +-1.0 +-0.5 +0.0 +0.5 +1.0 +x7. Quantitative modeling of the s-SNOM signals + +To model the near-field amplitude and phase signals as well as the plasmon fringe profiles, +we model our AFM tip as a metallic spheroid (Refs. [14,16,23] in the main text): the length of the +spheroid is 2L and the radius of curvature at the tip ends is a (Fig. S7). Here, a is set to be about +25 nm according to the manufacturer and L is set to be 500 nm. Note that L is not a very sensitive +parameter in the modeling when L >> a. The complex near-field signal (before demodulation) +scales with the total radiating dipole p of the spheroid. Therefore, to compute the near-field signal, +we calculate p at different z coordinates of the lower end of the AFM tip. By calculating p at +different z, we can perform ‘demodulation’ to get the nth harmonics of the near-field signal (n = 3 +in this work). To calculate the line profiles perpendicular to the fringes, we also considered the in- +plane coordinate (x) of the tip. Calculating p at different x and z coordinates allows us to plot the +modeling profiles of near-field amplitude and phase. The modeling parameters of the sample (SLG +or TBLG) are (1, 2), or more conveniently (p, p). As introduced in the main text, p is roughly +proportional to , and p scales linearly with . + + +FIG. S7. Sketch of the spheroid model that we used to model the plasmonic responses of SLG and +TBLG. + +8. Analysis and discussions of small-twist-angle TBLG + +In the main text, our analysis and discussions are mainly focused on TBLG samples with +relatively large twist angles ( ≥ 3), where Dirac dispersion approximation stays true. Here, we +wish to add more discussions about plasmonic responses of TBLG samples with small twist angles +( ≤ 2) based on our experimental data. + +(1) Near-field amplitude and phase of TBLG ( ≤ 2). +For convenience, we first summarize in Fig. S8 the near-field signal (s, ) data points of +all measured TBLG samples with small twist angles ( ≤ 2, red dots). From Fig. S8, one can see +that the amplitude ratio between TBLG ( ≤ 2) and SLG ( +tBLG +SLG +/ +s +s +) is 1.04  0.08 and the phase +ratio between TBLG ( ≤ 2) and SLG ( +tBLG +SLG +/ + + +) is 0.92  0.07. In other words, the amplitude + +Spheroid tip +pT +2L +Z4 +x +Sample +Ztip +[2p(x), /p(x)] +SiO2 +Sisignal of TBLG ( ≤ 2) is close to that of SLG with small variations, and the phase signal of +TBLG ( ≤ 2) is slightly weaker than that of SLG with small variations. + + +(2) Plasmon wavelength and damping rate of TBLG ( ≤ 2). +As discussed in the main text, the near-field signals (s and ) are directly linked to the +plasmonic parameters (p and p) of SLG and TBLG. Based on the s and  experimental data +points (Fig. S8), we estimate with numerical modeling that the plasmon wavelength (p) of TBLG +( ≤ 2) is in the range of 278 to 314 nm and the plasmon damping rate (p) of TBLG ( ≤ 2) is +in the range of 0.2 to 0.4. In Fig. S8, we also mark SLG (green star) and large-twist-angle TBLG +(  30, blue shaded region). Compared to SLG, small-twist-angle TBLG ( ≤ 2) has higher p +and the roughly equal p (or slightly higher, but less than 13% variation). From Fig. S8, one can +also see that the smaller scattering phase of TBLG ( ≤ 2) compared to SLG is mainly due to the +overall higher p. The lack of amplitude contrast between TBLG ( ≤ 2) and SLG is mainly due +to their roughly equal p. + + +FIG. S8. Statistics analysis of scattering amplitude versus scattering phase signals of TBLG, both +of which are normalized to those of SLG. Red dots summarize experimental data points of TBLG +with  ≤ 2. Black dots are theoretical calculations assuming various (p, p) settings. Black curves +connecting the black dots are drew to guide the eye. Green star and blue shaded region mark SLG +and large-angle TBLG (  30) respectively. + + +(3) Further discussions of TBLG ( ≤ 2). +Based on the analysis and modeling shown above, we can conclude that all the small-twist- +angle TBLG samples ( ≤ 2) we measured have a roughly equal p and an overall higher p +compared to those of SLG. Now we wish to discuss these interesting findings. The roughly equal +p of TBLG ( ≤ 2) compared to SLG is particularly surprising at first glance considering that +TBLG could become flat or nearly-flat bands as  approaches 0. A relatively flat band generally + +1.2 +- +- +- +·simulation +1.1 +SLG +0.1 +1.0 +0.2 +TBLG(0→30°) +86 +SLG +0.9 +0.3 +0.4 +0.8 +Yp= 0.5 +0.7 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +STBLG / SsLGmeans an extremely-small plasmon wavelength if the Fermi surface lies on the band. Nevertheless, +according to previous first-principle calculated band structures (Ref. [26] in the main text), the +bands of TBLG at small twist angles becomes relatively flat only close to the charge neutrality +point. The Fermi energy of our highly hole-doped TBLG samples is in principle far away from the +charge neutrality point, where the band at the Fermi surface stay dispersive even for small-twist +angle TBLG. For example, estimation based on Fig. 2 in Ref. [26] of the main text indicates that +the Fermi surface of TBLG with  = 2 is about -0.03 eV, where the band stays dispersive. +Estimations based on Fig. 3 in Ref. [26] of the main text suggest that highest-energy valence band +of TBLG (where the bands are relatively flat) with  = 1.6, 1.3 and 1.0 is completely empty +and the Fermi surface moves to lower valence bands, where the bands stay dispersive. +The overall higher p of TBLG ( ≤ 2) compared to SLG is most likely due to Landau +damping caused by interband transitions. While interband transitions at our excitation laser energy +(0.11 eV) are forbidden in SLG or TBLG with relatively large . For  ≤ 2, interband transitions +are enabled due to the small energy separations between lowest-energy bands. For example, +according to the calculated band structures in Fig. 2 and Fig. 3 in Ref [26] of the main text, +interband transitions at our excitation laser energy (0.11 eV) are forbidden in TBLG with  = 2.9, +but are fully enabled for  = 2.0, 1.9, 1.6, 1.3 and 1.0. + + + + + + + + + diff --git a/NtFJT4oBgHgl3EQf0C3f/content/tmp_files/load_file.txt b/NtFJT4oBgHgl3EQf0C3f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..94800dbae60830355f022705e2efb49d41021503 --- /dev/null +++ b/NtFJT4oBgHgl3EQf0C3f/content/tmp_files/load_file.txt @@ -0,0 +1,949 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf,len=948 +page_content='Real-Space Imaging of the Tailored Plasmons in Twisted Bilayer Graphene F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Hu1,2*, Suprem R.' metadata={'source': 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Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Iowa State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Ames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Iowa 50011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 4Department of Industrial and Manufacturing Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Kansas State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Manhattan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' KS 66506,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 5Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Kansas State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Manhattan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' KS 66506,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 6Birck Nanotechnology Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 7Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 8School of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 9Purdue Quantum Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' †Corresponding author: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='edu) Abstract We report a systematic plasmonic study of twisted bilayer graphene (TBLG) – two graphene layers stacked with a twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Through real-space nanoimaging of TBLG single crystals with a wide distribution of twist angles, we find that TBLG supports confined infrared plasmons that are sensitively dependent on the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' At small twist angles, TBLG has a plasmon wavelength comparable to that of single-layer graphene (SLG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' At larger twist angles, the plasmon wavelength of TBLG increases significantly with apparently lower damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Further analysis and modeling indicate that the observed twist-angle-dependence of TBLG plasmons in the Dirac linear regime is mainly due to the Fermi-velocity renormalization, a direct consequence of interlayer electronic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Our work unveils the tailored plasmonic characteristics of TBLG and deepens our understanding of the intriguing nano-optical physics in novel van der Waals (vdW) coupled two-dimensional (2D) materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Main text Graphene Dirac plasmons [1-6], which are collective oscillations of Dirac fermions in graphene, have been widely investigated in recent years by using both the electron energy loss spectroscopy [7-9] and optical imaging/spectroscopy [10-21] techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' These quasiparticles demonstrate many superior characteristics including high confinement, long lifetime, strong field enhancement, broad spectral range, electrical tunability and a broad spectral range from terahertz to infrared [1-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' So far, plasmons in single layer graphene (SLG) have been extensively studied and are generally well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' One convenient way to create new plasmonic materials with novel physics and properties is by stacking graphene with graphene and other 2D materials into vdW materials or heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indeed, the 2D nature of graphene makes it extremely sensitive to interlayer coupling that could dramatically modify the properties of Dirac fermions and their plasmonic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, earlier studies about Bernal-stacked BLG [20,22] and graphene/hBN heterostructures [23,24] have demonstrated many unique plasmonic characteristics compared to those of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In this Letter, we report a systematic nano-infrared imaging study of plasmons in TBLG [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(a)], which is formed when two misorientated graphene layers are stacked together by vdW forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Depending on the twist angle (\uf071) between the two graphene layers, moiré patterns with different periodicities could form [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Due to the interlayer coupling and modulation of Dirac fermions by moiré superlattice potential, the electronic structure of TBLG shows distinct features compared to SLG and Bernal-stacked BLG, and it varies systematically with \uf071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, TBLG with a sizable \uf071\uf020features two separated Dirac cones [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(c)] in the momentum space [25-34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Moreover, the Fermi velocity ( tBLG F v ) close to the charge neutrality point is renormalized compared to that of SLG ( SLG Fv ), namely tBLG F v drops systematically below SLG Fv as \uf071 decreases [25-30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Therefore, TBLG is a unique system where the Fermi velocity of Dirac fermions could become an adjustable variable in experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The unique electronic properties of TBLG have led to observations of many interesting optical phenomena through far- field spectroscopic experiments [35-38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' So far, plasmonic responses of TBLG have not been explored experimentally despite the potential rich physics according to theoretical predictions [39,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here we utilize a scattering-type scanning near-field optical microscope (s-SNOM) to perform nano-infrared imaging studies of TBLG plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The s-SNOM apparatus is built based on an atomic force microscope (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(a), the infrared light (solid arrow) from a continuous-wave infrared laser is focused at the apex of a metalized AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The laser- illuminated tip acts as both a launcher and a detector of surface plasmons [13-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The back- scattered light (dashed arrow) off the tip-sample system contains essential information about plasmons underneath the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The s-SNOM collects simultaneously the topography, near-field scattering amplitude (s) and phase (\uf079).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' By analyzing both the s and \uf079 data images, we can determine the key plasmonic parameters of TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Our samples were grown by the chemical vapor deposition (CVD) method on copper foils [41-43] and then transferred to the standard SiO2/Si substrates (Supplemental Material [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(d), both SLG and TBLG are single- crystal grains with a hexagonal shape and the TBLG grains are typically located at the center of relatively larger SLG grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Occasionally, we also see hexagon-like shapes with slightly curved edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S5) [45,46], but in all cases, these SLG or TBLG single-crystals demonstrate a six-fold rotational symmetry (Supplemental Material [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' According to the previous studies [45,47], the six-fold flake symmetry correlates strictly and accurately with the lattice orientation, so it is convenient to determine the twist angle with a relatively good accuracy (\uf0b11\uf0b0) by comparing the orientations of the TBLG and SLG grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Representative s-SNOM imaging data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2, where we plot both the normalized amplitude [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2(b)] and phase [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2(c)] signals of a typical sample region containing two TBLG grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The data images were taken at an excitation laser energy of E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV that is away from the strong optical phonon resonance of SiO2 centered at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='14 eV [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Therefore, the near-field responses of graphene at our excitation energy are mainly due to plasmons [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure 2(a) sketches the sample configuration, where we can conveniently determine the twist angles of TBLG from the orientations of the hexagonal grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, the TBLG sample labeled as ‘TBLG1’ has a twist angle of about 26\uf0b0 relative to SLG and the one labeled as ‘TBLG2’ has a twist angle close to 1\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Their different twist angles result in distinct near- field responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2(b) and 2(c), TBLG1 has significant higher near-field amplitude compared to SLG but shows no clear phase contrast with respect to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' On the contrary, the amplitude of TBLG2 is almost the same as that of SLG and its phase is slightly weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Such dramatic differences in the near-field responses are clear indications of the strong \uf071 dependence of TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' More near-field data images are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1 and S2, where additional TBLG samples with various twist angles are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In all the samples we measured (partly shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2, S1 and S2), the near-field amplitude of TBLG is comparable to SLG for \uf071 ≤ 3\uf0b0, and gradually increases from an intermediate signal (\uf071 \uf0bb 5\uf0b0) to a maximum value (\uf071 > 7\uf0b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The phase signal of TBLG, on the other hand, is roughly the same as SLG for \uf071 > 7\uf0b0, but slightly declines as \uf071 approaches 0\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The above \uf071 dependence is more clearly seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(c) and 4(d), where we summarized the extracted amplitude and phase signal data points (squares) from tens of TBLG samples that we measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The unique near-field responses discussed above are directly linked to the plasmons in TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indeed, we found direct evidence of plasmons in the high-resolution imaging data (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 and S3) taken over five small sample regions (marked with dashed squares) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' These regions (labeled with ‘P1’ \uf02d ‘P5’) are chosen to be at the edge of SLG or the boundaries between SLG and TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The amplitude images are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(a) – 3(e), where we observe bright fringe(s) close to the SLG edge and the SLG/TBLG boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' This can be seen more clearly in the line profiles [grey solid curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(f) – 3(j)] taken perpendicular to the edges or boundaries in the amplitude images (along blue dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here in these line profiles, the peak features correspond to the bright fringes in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' According to previous studies [14-24], the bright fringes registered by the s-SNOM are generated due to the constructive interference between tip-launched and edge- or boundary- reflected plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The plasmonic origin of the observed fringes is further confirmed by the spectroscopic imaging data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S4), where we observed a systematic evolution of the bright fringes with laser energy, consistent with the dispersion nature of plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' There are two main observations from these plasmonic fringes data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' First, fringes are clear and strong close to the SLG/TBLG3 (\uf071\uf020≈\uf02027\uf0b0) and SLG/TBLG4 (\uf071\uf020≈\uf02012\uf0b0) boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As \uf071\uf020 decreases, the fringes become weaker and fewer at the SLG/TBLG5 (\uf071\uf020≈\uf0205\uf0b0) boundary and then barely seen at the SLG/TBLG6 boundary (\uf071\uf020≈\uf0203\uf0b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Second, in the case of SLG/TBLG3 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(b)] and SLG/TBLG4 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(c)] boundaries, we can easily identify two to three fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Nevertheless, at the edge of SLG [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(a)], we can only see one bright fringe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Note that the edge of SLG is a nearly-perfect plasmon reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The plasmon reflection at the SLG/TBLG boundaries, on the other hand, is in principle weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Therefore, we can tell directly from the fringes data that our SLG sample has a relatively higher plasmon damping compared to TBLG with relatively large \uf071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure S3 plot the near-field phase images and the corresponding line profiles, where plasmonic interference fringes are also seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The amplitude and phase imaging data are consistent and complementary to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' They are all considered in our modeling as discussed in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' To extract quantitative information about plasmons in SLG and TBLG, we performed numerical modeling of both the plasmonic fringes profiles (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 and S3) and the \uf071-dependent near-field amplitude and phase signals [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(c) and 4(d)] by using the so-called spheroid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In this model, the s-SNOM tip is approximated as a highly-elongated conducting spheroid (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S7) and we evaluate the complex scattering signal by computing the total radiating dipole of the coupled tip-sample system (Supplemental Material [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' We wish to emphasize that our model has been proven to effective in describing s-SNOM responses of graphene with quantitative accuracy [14,16,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The main modeling parameter of the sample is the optical conductivity (\uf073\uf020\uf03d\uf020\uf073\uf031\uf020\uf02b\uf020i\uf073\uf032) that is directly linked to the complex plasmon wavevector (qp = q1 + iq2) under the long-wavelength approximation: 0(1 ) p s q i E \uf065 \uf065 \uf073 \uf0bb \uf02b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (1) Here is the reduced plank constant, \uf0650 is the vacuum permittivity, and \uf065s is the relative permittivity of SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For convenience, our analysis and discussions are based on the following two parameters: the plasmon wavelength (\uf06cp = 2\uf070/q1) and damping rate (\uf067p = q2/q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1, we know that the plasmon wavelength (\uf06cp) is roughly proportional to \uf073\uf032, and the damping rate (\uf067p) scales linearly with \uf073\uf031\uf02f\uf073\uf032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' We first fit the plasmonic fringe profiles of SLG [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(f) and S3(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Through the fitting, we extract the plasmon wavelength ( SLG p \uf06c ) and damping rate ( SLG p \uf067 ) of SLG at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV to be about 279 nm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (1), we can establish a simple relation between SLG p \uf06c and the carrier density (n) under the Drude approximation: 2 SLG SLG 2 0 2 | | (1 ) F p s e v n E \uf070 \uf06c \uf065 \uf065 \uf0bb \uf02b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (2) Here SLG 6 10 m/s F v \uf0bb is the Fermi velocity of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Equation (2) allows us to estimate the carrier density of SLG to be n \uf0bb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 × 1013 cm-2, which is a typical value for highly-hole-doped CVD samples on SiO2/Si substrates at ambient conditions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The relatively high doping is mainly due to the impurities on the surface of SiO2 as well as the water and oxygen molecules in the air [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Considering that all the samples studied here share the same substrate and atmospheric conditions, they are expected to share roughly an equal density of external dopants and therefore a similar carrier density [21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Based on the extracted parameters of SLG, we then determine both the plasmon wavelength ( tBLG p \uf06c ) and damping rate ( tBLG p \uf067 ) of TBLG by fitting the fringe profiles at the SLG/TBLG boundaries (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Through fitting, we estimate that ( tBLG p \uf06c , tBLG p \uf067 ) of TBLG3 (\uf071 ≈ 27\uf0b0), TBLG4 (\uf071 ≈ 12\uf0b0), TBLG5 (\uf071 ≈ 5\uf0b0) and TBLG6 (\uf071 ≈ 3\uf0b0) to be (393 nm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='10), (387 nm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11), (340 nm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='16) and (278 nm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='28), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' These numbers are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(a) and 4(b) as data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Note that the first two numbers of tBLG p \uf06c can be read out directly from the fringe profiles of TBLG3 and TBLG4 by doubling the fringe period [arrows in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(g) and 3(h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Nevertheless, precise modeling of the complex fringe profiles is required to extract both tBLG p \uf06c and tBLG p \uf067 , and to analyze data from TBLG samples without strong fringes [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3(d) and 3(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In the latter case, the modeling fits mainly the s and \uf079 signals of TBLG in contrast to SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(c) and 4(d), we show the modeling curve of s and \uf079 contrast signals of TBLG versus SLG at a wide distribution of twist angles (red curves), which match well the trend of the experimental data points with twist angles above 3\uf0b0 (marked with dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' At twist angles below 3\uf0b0, the experimental data points clearly deviate from the modeling curve, which will be discussed in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The smooth tBLG( ) p \uf06c \uf071 and tBLG( ) p \uf067 \uf071 parameters [red curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(a) and 4(b)] used to model the s and \uf079 contrast signals are fully consistent with the discrete data points obtained from fringe profile fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Now we wish to discuss the origin of the \uf071-dependence of tBLG p \uf06c and tBLG p \uf067 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' We first pay attention to twist angles above 3\uf0b0, where TBLG is in the Dirac regime [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(c)] [25-29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here we assume that carriers are equally distributed among the two graphene layers, which is reasonable considering no external gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The general results won’t change much even with slightly unequal carrier distribution among the two graphene layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S6 and Supplemental Material [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Under the equal carrier distribution assumption, tBLG p \uf06c can be written as 2 tBLG tBLG 2 0 2 2 | | ( ) ( ), (1 ) p F s e n v E \uf070 \uf06c \uf071 \uf071 \uf065 \uf065 \uf0bb \uf02b (3) where the Fermi velocity of TBLG ( tBLG F v ) is proven to be sensitively dependent on \uf071 due to the Fermi velocity renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The amount of Fermi velocity renormalization is determined by the interlayer coupling energy (t) of TBLG [inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(a)] as described by the following equation: [25] tBLG SLG 2 SLG ( ) [1 9( ) ] F F F t v v v K \uf071 \uf03d \uf02d \uf044 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (4) Here, (8 / 3 )sin( / 2) K a \uf070 \uf071 \uf044 \uf03d is the momentum separation of the two Dirac cones [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1(c)], and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='246 nm is the lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Equation 4 indicates that t is the one single parameter that controls tBLG( ) F v \uf071 and hence tBLG( ) p \uf06c \uf071 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here we set t to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='1 eV, which is roughly consistent with previous studies [29,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' With such a t setting, we calculated tBLG( ) p \uf06c \uf071 based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (3) and (4), which is shown as the red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Other choices of t will lead to either faster or slower decreasing of tBLG F v and hence tBLG p \uf06c as \uf071 drops [inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The origin for the higher tBLG p \uf067 at smaller \uf071 in the Dirac regime (\uf071 ≥ 3\uf0b0) is likely due to the stronger charge scattering rates [49,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' According to previous literature [51], the charge scattering rates (\uf047) due to either long-range Coulomb scattering or short-range defect scattering are inverse proportional to the Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Therefore, as \uf071 decreases, \uf047 rises and thus tBLG p \uf067 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Note that interband transitions are forbidden due to the Pauli blocking for \uf071 ≥ 3\uf0b0, where threshold energy for interband transitions ( tBLG 2 F E ) is estimated to be over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 eV, far above our laser energy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Finally, we wish to discuss briefly TBLG samples with twist angles below 3\uf0b0, where the Dirac approximation begins to fail [26-29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In this regime, we find that the amplitude signal of the TBLG samples deviates from the projected trend of the modeling curves and stay close to that of SLG [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' With quantitative modeling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8), we estimate that the tBLG p \uf06c at small twist angles (\uf071 ≤ 2\uf0b0) is in the range from 278 to 314 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' According to previous theoretical studies [26- 28], the lowest-energy bands of TBLG with small twist angles become flat or nearly-flat close to the charge neutrality point, which could lead to an extremely small tBLG p \uf06c (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The finite tBLG p \uf06c of TBLG (\uf071 ≤ 2\uf0b0) observed here suggests that the Fermi surface of our highly-doped samples is away from these relatively flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The phase signals [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4(d)] of TBLG (\uf071 ≤ 2\uf0b0) appear to be slightly smaller than that of SLG and large-twist-angle TBLG, indicating even higher plasmon damping rates: tBLG p \uf067 (\uf071 ≤ 2\uf0b0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The higher damping is most likely due to interband transitions, which are enabled in TBLG (\uf071 ≤ 2\uf0b0) at our excitation energy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV) due to the small energy separations between the lowest-energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' More detailed discussions about TBLG with \uf071 ≤ 2\uf0b0 are given in the Supplemental Material [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Future studies with more comprehensive experiments of small-twist-angle TBLG and more precise determinations of twist angles are needed to explore further TBLG plasmons in the non-Dirac regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' By combining the state-of-the-art s-SNOM technique with rigorous numerical modeling, we performed a systematic nano-infrared imaging study of TBLG single crystals with various twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In the Dirac linear regime, we found that TBLG support infrared plasmons with parameters that vary systematically with the twist angle between the two graphene planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The underlining physics behind the observed twist angle dependence is the Fermi velocity renormalization, which is originated from the interlayer electronic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Our study establishes TBLG as a unique platform where the Fermi velocity, the fundamentally important parameter of Dirac fermions, has become an adjustable variable in nano-optical and plasmonic studies of Dirac materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' acknowledge startup support from Iowa State University and the royalty funds from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Department of Energy, Ames Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The nano-optical setup is partially supported by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' acknowledges the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Department of Energy, Ames Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The synthesis of TBLG samples at Purdue University is partially supported by NSF CMMI (grant 1538360) and NSF EFMA (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1641101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Das Sarma and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Hwang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' B 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 121405(R) (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Principi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' B 90, 165408 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Castro Neto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Guinea, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Peres, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Novoselov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Geim, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 81, 109-162 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure Captions FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a) Illustration of the nano-infrared imaging experiment of a TBLG/SLG single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The solid and dashed arrows mark the directions of the incident laser beam and back-scattered photons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (b) Sketch of the crystal structure of TBLG revealing moiré periodic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The double-sided arrow marks the moiré period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (c) Calculated band structure of TBLG with a twist angle of 5\uf0b0 with the continuum model [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here the momentum unit \uf044K equals to the separation between the two Dirac points (K1 and K2) in the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (d) Optical image of representative TBLG/SLG single-crystal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bar represents 5 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a) Sketch of the sample geometry indicating two adjacent TBLG grains with different twist angles with respect to SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (b) and (c) The near-field images plotting scattering amplitude (s) and phase (\uf079), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In both images, the amplitude or phase signal is normalized to that of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The laser energy is set to be at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bars represent 3 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a)-(e) High-resolution near-field amplitude images of the five small regions (‘P1’ – ‘P5’) marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1 (squares), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The white dashed lines in the images mark the SLG edge and the TBLG/SLG boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bars represent 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (f)-(j), Experimental (grey solid) and modeled (red dashed) amplitude profiles taken along the blue dashed lines in the corresponding near-field images above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The blue arrows in (g) and (h) mark the size of tBLG p \uf06c that is twice the fringe period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The vertical dashed lines mark the boundaries between SLG and TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a) The tBLG p \uf06c data points extracted by modeling the fringe profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Inset plots the calculated tBLG( ) Fv \uf071 normalized to SLG Fv considering different t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (b) The tBLG p \uf067 data points extracted by modeling the fringe profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The red curves in (a) and (b) are used for calculations of the amplitude and phase signals in (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The blue arrows in (a) and (b) mark the values of SLG p \uf06c and SLG p \uf067 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (c) The \uf071-dependent near-field amplitude from both experiment (squares) and modeling (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (d) The \uf071-dependent near-field phase from both experiment (squares) and modeling (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Both the amplitude (c) and phase (d) of TBLG are normalized to those of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The vertical dashed lines in (c) and (d) mark \uf071 = 3\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure 1 K1 K2 θ (b) (d) (c) (a) Energy (eV) TBLG SLG TBLG SLG �K kx (�K) ky (�K) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 2 1 0 1 2 1 0 1 SiO2 Figure 2 (a) θ � o 26 (b) TBLG1 SLG 2 SiO (c) � (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') 2 0 TBLG2 SLG 2 0 θ � o 1 � � � Figure 3 (a) (f) (b) (c) (d) (e) (g) (h) (i) (j) SLG SiO2 P1 (SLG) P2 (TBLG3, � ~ 27o) P3 (TBLG4, � ~ 12o) P4 (TBLG5, � ~ 5o) P5 (TBLG6, � ~ 3o) s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 x (�m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 x (�m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 x (�m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 x (�m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 x (�m) s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 SLG SiO2 SLG TBLG3 SLG TBLG4 TBLG5 SLG SLG TBLG6 TBLG3 TBLG4 TBLG5 SLG SLG SLG SLG TBLG6 Figure 4 sTBLG sSLG θ (degree) (c) ψTBLG ψSLG θ (degree) λp (nm) θ (degree) γp θ (degree) (d) (a) (b) 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 0 10 20 30 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='15 eV t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='1 eV t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='05 eV vF (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') θ (degree) / / Supplemental Material Real-Space Imaging of the Tailored Plasmons in Twisted Bilayer Graphene F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Hu1,2*, Suprem R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Das2,3,4,5*, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Luan1,2*, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Chung6,7, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Chen6,7,8,9, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Fei1,2† 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Ames Laboratory, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Department of Energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Iowa State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Ames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Iowa 50011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 3Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Iowa State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Ames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Iowa 50011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 4Department of Industrial and Manufacturing Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Kansas State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Manhattan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' KS 66506,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 5Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Kansas State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Manhattan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' KS 66506,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 6Birck Nanotechnology Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 7Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 8School of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA 9Purdue Quantum Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Purdue University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' West Lafayette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indiana 47907,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' USA These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' † Email: (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='edu List of contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Nano optical imaging setup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Sample fabrication procedures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Additional s SNOM imaging data 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Flake shape of single-crystal SLG/TBLG samples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Unintentional doping of the samples 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Analysis of unequal carrier distribution 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Quantitative modeling of the s-SNOM signals 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Analysis and discussions of small-twist-angle TBLG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Nano infrared imaging setup For nano-infrared imaging experiments in the current work, we used a scattering-type scanning near-field optical microscope (s-SNOM) apparatus (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='neaspec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='com) that is built based on an atomic force microscope (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The AFM is operated in the tapping mode with a tapping frequency of about 270 kHz and a tapping amplitude of about 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The AFM tips used in the current work are platinum iridium coated silicon tips with a radius of curvature of about 25 nm at the tip apex (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='nanoworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For signal detection, we used a mercury cadmium telluride photodiode (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='kolmartech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' A pseudo-heterodyne interferometer is implemented in our s-SNOM to extract both the near-field amplitude (s) and phase (\uf079\uf029\uf020of the complex near- field signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' To suppress the background signal, we demodulated the near-field signal at the 3rd harmonics of the tapping frequency of the AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For optical excitation, we used a continuous- wave CO2 laser (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='accesslaser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The photon energy of the laser can be discretely tunable from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='13 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Our nano-infrared imaging experiments were all performed at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Sample fabrication procedures The growth of the hexagon-shaped TBLG/SLG single crystals was achieved by using the atmospheric-pressure chemical vapor deposition (CVD) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' A mixture of methane carrier gas was atomically cracked at high temperature (1050 °C) with argon/hydrogen gas before controllably getting deposited onto a pre-cleaned copper foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' After the growth process, the methane flow was stopped, and the sample was cooled down to room temperature in the furnace with argon/hydrogen gas flow uninterrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For s-SNOM studies, our samples on copper foil were transferred onto the standard SiO2/Si wafers used a wet-chemical etching and transfer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In short, a 100-nm-thick polymethyl methacrylate (PMMA) protection layer was applied to one side of the copper foil and the TBLG on the other side of the foil was plasma etched using an oxygen plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The copper was then etched overnight in an etchant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' After multiple washing of the floating TBLG/PMMA stack with deionized water and a mild aqueous acid, the stack was transferred onto the SiO2/Si wafers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The PMMA layer was stripped off TBLG in about 4 hours after baking/drying on a hotplate at 90 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Finally, the sample was electronically cleaned at 300 °C for 2 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' More detailed introductions about the growth procedures are given in the earlier study (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [41] in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Additional s SNOM imaging data In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1-S4, we provide additional near-field imaging data about TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure S1 plots four typical TBLG crystals (‘TBLG3’ – ‘TBLG6’) that we used for high-resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The high-resolution imaging data taken at five specific locations (‘P1’ to ‘P5’ marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 (amplitude images and profiles) of the main text and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S3 (phase images and profiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure S2 plots near-field amplitude and phase images at four different locations, where a total of eight SLG/TBLG crystals (‘TBLG7’ – ‘TBLG14’) are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Figure S4 plots the excitation-energy-dependent s-SNOM imaging data at the boundary between ‘SLG’ and ‘TBLG3’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' One can see that the plasmonic interference fringes at the SLG/TBLG3 boundary show a systematic variation with photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Through modeling of the fringe profiles, we extracted the plasmon wavelength of TBLG (SLG) to be about 393 nm (279 nm), 370 nm (262 nm) and 350 nm (248 nm) for excitation energies of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='110, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='116 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='121 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The energy- dependent plasmon wavelengths are consistent with the dispersion nature of plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a)-(d) Sketches of the sample geometries of four additional TBLG single crystals (‘TBLG3’ – ‘TBLG6’) with different twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (e)-(h) The near-field amplitude images of the samples sketched in (a)-(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (i)-(l) The near-field phase images of the samples sketched in (a)-(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The excitation laser energy is set to be at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In all the images, the amplitude or phase signal is normalized to that of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The squares mark the five positions (‘P1’ ─ ‘P5’) where we collected high-resolution imaging data as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 of the main text and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S3 of the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bars represent 3 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a) (b) (c) (d) TBLG4 TBLG5 TBLG3 TBLG6 日~3° 日~270 SLG SLG SLG SLG SiO, SiO2 SiO2 SiO2 (e) (f) (g) (h) P1 NI P5 s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') P4 P3 0 (i) (0) (k) (0) P1 2 P5 y (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') P4 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a)-(d) Sketches of the sample geometry of four samples regions that include a total of eight additional TBLG single crystals (‘TBLG7’ – ‘TBLG14’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (e)-(h) Near-field amplitude images of sample regions sketched in (a)-(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (i)-(l) The near-field phase images of sample regions sketched in (a)-(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In all the images, the amplitude or phase signal is normalized to that of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bars represent 3 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a) TBLG7 b C d) TBLG11 9~0 6~18° 9~00 TBLG9 TBLG13 SLG SLG SiO2 SLG SLG SiO2 Sio2 TBLG8 TBLG14 TBLG10 TBLG12 ~70 6~28° SLG SLG SiO2 SLG SLG (e) (f) (g) (h) 2 s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') 0 0) (k) 2 w(norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') 0 FIG S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a)-(e) Near-field phase images of the five small regions (‘P1’ – ‘P5’) marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1 (squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The white dashed lines in the images mark the edges of SLG and the boundaries between TBLG and SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bars in all the images represent 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (f)-(j) Experimental (grey solid) and modeled (red dashed) phase profiles taken perpendicular to the SLG edge and the TBLG/SLG boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In all the near-field images, the phase signal is normalized to that of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The experimental phase profiles were taken along the blue dotted lines in the corresponding near-field images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The vertical dashed lines mark the boundaries between SLG and TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Near-field amplitude images of the SLG/TBLG3 boundary (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 in the main text) at various excitation laser energies: E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='110 eV (a), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='116 eV (b) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='121 eV (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here we plot the near-field amplitude normalized to that of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The scale bars represent 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Flake shape of single-crystal SLG/TBLG samples Most of our single-crystal TBLG/SLG samples appear in a hexagonal shape with straight edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Occasionally, we also see hexagon-like shapes with slightly-curved edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S5a (or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3b), we found the edge of TBLG3 is not exactly straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' A large-area image is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S5b, where we mark carefully the boundaries of SLG and TBLG3 grains with green and blue curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' These curves are replotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S5c, where one can see that the edges of the TBLG3 grain are not straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Instead, it has a hexagonlike shape with slight negative curvature at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' This is, in fact, one of the typical shapes of single-crystal graphene flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As reported by an earlier study (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [45] in the main text), various types of six-fold symmetric shapes of graphene flakes (from flower-like to hexagon-like) could occur by varying systematically the CVD growth conditions, among which both the perfect hexagonal shape and slight-curved hexagonal shape are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The lattice orientation of graphene is consistent with the six-fold symmetry of the flake, as verified by diffraction experiments in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [45] and [47] of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' An earlier study also shows slightly-curved hexagon shape in twisted bilayer and trilayer graphene flakes E= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='110 eV E= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='116 eV (a) (b) (c) E= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='121 eV 2 s(norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=') 0P1 (SLG) P2 (TBLG3, 0 ~ 27°) P3 (TBLG4, 0 ~ 12°) P4 (TBLG5, 0 ~ 5°) P5 (TBLG6, 0 ~ 3°) (a) (q) (c) (d) (e) SiO2 2 SLG TBLG5 SLG TBLG4 (wvou) TBLG3 SLG SLG TBLG6 SLG 0 (f) (wvou) - 1 SiQ2 SLG TBLG3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' SLG SLG TBLG4 TBLG5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' SLG SLG TBLG6 0 0 0 二 0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 x (μm) x (μm) x (μm) x (μm) x (μum)(Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [46] in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' To sum up, the TBLG3 sample with slight-curved edges is also typical for CVD-grown single-crystal samples and we followed the symmetry of the entire flake when determining the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (a) High-resolution near-field amplitude image of the TBLG3/SLG boundary, a replot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3b in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (b) The large-area near-field amplitude of the TBLG3/SLG sample, a replot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (c) Sketch of the geometry of the TBLG3/SLG sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Unintentional doping of the samples Our CVD-grown samples are all highly hole-doped at ambient conditions due to the high density of impurities on the amorphous SiO2 surface as well as water and oxygen molecules in the air (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [48] in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Considering that all the data presented in the paper were from samples produced from the same batch and transferred onto the same wafer, they share nearly identical chemical and environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Therefore, we are safe to assume that all the TBLG and SLG samples share roughly an equal density of external dopants and therefore a similar carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Indeed, previous studies have confirmed that adjacent SLG and Bernal-stacked or randomly-stacked BLG share roughly the same carrier density (Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [21,22] in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' By fitting the plasmonic fringe profile of SLG, we can accurately determine the plasmon wavelength of SLG be about 279 nm, based on which we estimate the carrier density of SLG and TBLG is n \uf0bb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 × 1013 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Analysis of unequal carrier distribution In the current paper, we assume equal carrier distribution among the top and bottom layers of TBLG when analyzing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' This is a reasonable assumption considering the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' First, there is no gating or dedicated chemical doping to introduce doping asymmetry, so strong asymmetry is unlikely to occur due to the natural doping by the air or substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Second, even with small doping asymmetry, the results won’t change much from the current analysis based on equal carrier distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Detailed analysis and discussions about doping asymmetry are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Introducing unequal carrier distribution among the two graphene layers (or equivalently, adding a built-in electric field between the two graphene layers) can indeed affect the optical conductivity and plasmon wavelength of TBLG, and hence the near-field scattering signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here we only consider the Dirac linear regime (twist angle \uf071 ≥ 3\uf0b0), where the plasmon wavelength of TBLG obeys the following relationship: 2 2 2 ( ) tBLG tBLG top bottom top bottom tBLG top bottom p F F F E E v n n \uf06c \uf073 \uf073 \uf073 \uf0b5 \uf03d \uf02b \uf0b5 \uf02b \uf0b5 \uf02b , (1) where the superscripts “top” and “bottom” denote the top and bottom graphene layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The total carrier density total top bottom n n n \uf03d \uf02b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For convenience, we define the parameter ( )/ top bottom total x n n n \uf03d \uf02d to characterize the doping asymmetry, which is in the range [-1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' We do (a) (b) (c) SLG TBLG3 TBLG3 TBLG3 SLG SiO2 SLG SiO2not consider x < -1 or x > 1, which means opposite charges distributed among the two graphene layers – an unlikely situation without dual gating setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' With the parameter x, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1 can be re- written as ( 1 1 ) / 2 tBLG tBLG total p Fv x x n \uf06c \uf0b5 \uf02b \uf02b \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (2) Accordingly, we can obtain the formula of SLG assuming the same total carrier density: SLG SLG total p Fv n \uf06c \uf0b5 , so the ratio between tBLG p \uf06c and SLG p \uf06c can be written as / ( / )( 1 1 ) / 2 tBLG SLG tBLG SLG p p F F v v x x \uf06c \uf06c \uf03d \uf02b \uf02b \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (3) In the Dirac regime of TBLG (\uf071 ≥ 3\uf0b0), / tBLG SLG F F v v \uf068 \uf03d is the Fermi velocity renormalization factor that is a constant at every given twist angle, so / ( ) ( 1 1 ) / 2 tBLG SLG p p x x \uf06c \uf068\uf06c \uf03d \uf02b \uf02b \uf02d is solely dependent on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S6, we plot the x-dependence curve of tBLG p \uf06c normalized to SLG p \uf068\uf06c , namely / ( ) tBLG SLG p p \uf06c \uf068\uf06c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here / ( ) tBLG SLG p p \uf06c \uf068\uf06c reaches the maximum 2 when x = 0 (namely equal carrier distribution among graphene layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Introducing doping asymmetry (x ≠ 0) will reduce the plasmon wavelength of TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The / ( ) tBLG SLG p p \uf06c \uf068\uf06c reaches the minimum 1 when x = -1 or 1 (carriers are fully distributed in the top or bottom graphene layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S6, we also notice that the variation of / ( ) tBLG SLG p p \uf06c \uf068\uf06c is very small when x is in a reasonable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, for |x| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5, the variation of / ( ) tBLG SLG p p \uf06c \uf068\uf06c is only within 3% (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In the case of our highly- doped TBLG samples (ntotal \uf0bb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 × 1013 cm-2), |x| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 means that the one layer has a carrier density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='3 × 1013 cm-2 corresponding to a Fermi energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 eV, and the other layer has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='9 × 1013 cm-2 corresponding to a Fermi energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='35 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' This is, in fact, a huge doping asymmetry that in principle cannot be created without external gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The asymmetric doping dependence of the / ( ) TBLG SLG p p \uf06c \uf068\uf06c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The parameter ( )/ top bottom total x n n n \uf03d \uf02d characterizes the doping asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='4 < 3% - - DTS - - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='3 - - - - - TBLG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 - - 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='1 - 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 - 1 - 1 1 - -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 x7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Quantitative modeling of the s-SNOM signals To model the near-field amplitude and phase signals as well as the plasmon fringe profiles, we model our AFM tip as a metallic spheroid (Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [14,16,23] in the main text): the length of the spheroid is 2L and the radius of curvature at the tip ends is a (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here, a is set to be about 25 nm according to the manufacturer and L is set to be 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Note that L is not a very sensitive parameter in the modeling when L >> a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The complex near-field signal (before demodulation) scales with the total radiating dipole p of the spheroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Therefore, to compute the near-field signal, we calculate p at different z coordinates of the lower end of the AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' By calculating p at different z, we can perform ‘demodulation’ to get the nth harmonics of the near-field signal (n = 3 in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' To calculate the line profiles perpendicular to the fringes, we also considered the in- plane coordinate (x) of the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Calculating p at different x and z coordinates allows us to plot the modeling profiles of near-field amplitude and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The modeling parameters of the sample (SLG or TBLG) are (\uf0731, \uf0732), or more conveniently (\uf06cp, \uf067p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As introduced in the main text, \uf06cp is roughly proportional to \uf073\uf032, and \uf067p scales linearly with \uf073\uf031\uf02f\uf073\uf032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Sketch of the spheroid model that we used to model the plasmonic responses of SLG and TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Analysis and discussions of small-twist-angle TBLG In the main text, our analysis and discussions are mainly focused on TBLG samples with relatively large twist angles (\uf071 ≥ 3\uf0b0), where Dirac dispersion approximation stays true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Here, we wish to add more discussions about plasmonic responses of TBLG samples with small twist angles (\uf071 ≤ 2\uf0b0) based on our experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (1) Near-field amplitude and phase of TBLG (\uf071 ≤ 2\uf0b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For convenience, we first summarize in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8 the near-field signal (s, \uf079) data points of all measured TBLG samples with small twist angles (\uf071 ≤ 2\uf0b0, red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8, one can see that the amplitude ratio between TBLG (\uf071 ≤ 2\uf0b0) and SLG ( tBLG SLG / s s ) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='04 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='08 and the phase ratio between TBLG (\uf071 ≤ 2\uf0b0) and SLG ( tBLG SLG / \uf079 \uf079 ) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='92 \uf0b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In other words, the amplitude Spheroid tip pT 2L Z4 x Sample Ztip [2p(x), /p(x)] SiO2 Sisignal of TBLG (\uf071 ≤ 2\uf0b0) is close to that of SLG with small variations, and the phase signal of TBLG (\uf071 ≤ 2\uf0b0) is slightly weaker than that of SLG with small variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (2) Plasmon wavelength and damping rate of TBLG (\uf071 ≤ 2\uf0b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' As discussed in the main text, the near-field signals (s and \uf079) are directly linked to the plasmonic parameters (\uf06cp and \uf067p) of SLG and TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Based on the s and \uf079 experimental data points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8), we estimate with numerical modeling that the plasmon wavelength (\uf06cp) of TBLG (\uf071 ≤ 2\uf0b0) is in the range of 278 to 314 nm and the plasmon damping rate (\uf067p) of TBLG (\uf071 ≤ 2\uf0b0) is in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8, we also mark SLG (green star) and large-twist-angle TBLG (\uf071 \uf0e0 30\uf0b0, blue shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Compared to SLG, small-twist-angle TBLG (\uf071 ≤ 2\uf0b0) has higher \uf067p and the roughly equal \uf06cp (or slightly higher, but less than 13% variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8, one can also see that the smaller scattering phase of TBLG (\uf071 ≤ 2\uf0b0) compared to SLG is mainly due to the overall higher \uf067p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The lack of amplitude contrast between TBLG (\uf071 ≤ 2\uf0b0) and SLG is mainly due to their roughly equal \uf06cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Statistics analysis of scattering amplitude versus scattering phase signals of TBLG, both of which are normalized to those of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Red dots summarize experimental data points of TBLG with \uf071 ≤ 2\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Black dots are theoretical calculations assuming various (\uf06cp, \uf067p) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Black curves connecting the black dots are drew to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Green star and blue shaded region mark SLG and large-angle TBLG (\uf071 \uf0e0 30\uf0b0) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' (3) Further discussions of TBLG (\uf071 ≤ 2\uf0b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Based on the analysis and modeling shown above, we can conclude that all the small-twist- angle TBLG samples (\uf071 ≤ 2\uf0b0) we measured have a roughly equal \uf06cp and an overall higher \uf067p compared to those of SLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Now we wish to discuss these interesting findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The roughly equal \uf06cp of TBLG (\uf071 ≤ 2\uf0b0) compared to SLG is particularly surprising at first glance considering that TBLG could become flat or nearly-flat bands as \uf071 approaches 0\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' A relatively flat band generally 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 - - - ·simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='1 SLG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 TBLG(0→30°) 86 SLG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='8 Yp= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='8 STBLG / SsLGmeans an extremely-small plasmon wavelength if the Fermi surface lies on the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Nevertheless, according to previous first-principle calculated band structures (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [26] in the main text), the bands of TBLG at small twist angles becomes relatively flat only close to the charge neutrality point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The Fermi energy of our highly hole-doped TBLG samples is in principle far away from the charge neutrality point, where the band at the Fermi surface stay dispersive even for small-twist angle TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, estimation based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [26] of the main text indicates that the Fermi surface of TBLG with \uf071 = 2\uf0b0 is about -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='03 eV, where the band stays dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' Estimations based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' [26] of the main text suggest that highest-energy valence band of TBLG (where the bands are relatively flat) with \uf071 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='6\uf0b0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='3\uf0b0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0\uf0b0 is completely empty and the Fermi surface moves to lower valence bands, where the bands stay dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' The overall higher \uf067p of TBLG (\uf071 ≤ 2\uf0b0) compared to SLG is most likely due to Landau damping caused by interband transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' While interband transitions at our excitation laser energy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV) are forbidden in SLG or TBLG with relatively large \uf071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For \uf071 ≤ 2\uf0b0, interband transitions are enabled due to the small energy separations between lowest-energy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' For example, according to the calculated band structures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content=' 3 in Ref [26] of the main text, interband transitions at our excitation laser energy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='11 eV) are forbidden in TBLG with \uf071 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='9\uf0b0, but are fully enabled for \uf071 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0\uf0b0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='9\uf0b0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='6\uf0b0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='3\uf0b0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} +page_content='0\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFJT4oBgHgl3EQf0C3f/content/2301.11646v1.pdf'} diff --git a/ONE3T4oBgHgl3EQfxAvE/vector_store/index.pkl b/ONE3T4oBgHgl3EQfxAvE/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..03743463fc1cea0f412d7d7196ede456c2b4e611 --- /dev/null +++ b/ONE3T4oBgHgl3EQfxAvE/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:769a0dfa1c31bdc7b5288060a4a512aced38fd34aac40c11f390433eb648679c +size 661135 diff --git a/OtE3T4oBgHgl3EQfCAnu/vector_store/index.faiss b/OtE3T4oBgHgl3EQfCAnu/vector_store/index.faiss new file mode 100644 index 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Science Park 904, 1098 XH Amsterdam, The Nether- +lands +cInstitute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyen- +daalseweg 135, Nijmegen, the Netherlands +E-mail: pbraat@nikhef.nl, mpostma@nikhef.nl +Abstract: Pions of a dark sector gauge group can be strongly interacting massive particle +(SIMP) dark matter, produced by the freeze-out of 3 → 2 interactions, with naturally large +self-interactions. We study if adding a dark photon to the set-up can do it all: i) main- +tain thermalization with the visible sector, ii) resonantly enhance the 3 → 2 interactions, +thus allowing for a perturbative pion description, and iii) provide a velocity dependent self- +interaction that can affect small scale structure formation. We find that this is marginally +excluded, as the required kinetic mixing is too small to maintain thermal equilibrium with +the SM. Dropping the small scale structure requirement iii), a viable setup is reproduced for +dark charges of αd = 0.01 − 1 and a dark pion mass mπ ≥ 30 MeV. Late time annihilations +are non-negligible making the SIMP dark pion a bit WIMPy. +arXiv:2301.04513v1 [hep-ph] 11 Jan 2023 + +Contents +1 +Introduction +1 +2 +Lagrangian +2 +3 +Self-interactions +4 +3.1 +Bullet cluster bound +5 +3.2 +Resonant self-interactions +5 +4 +Relic density +6 +4.1 +SIMP freeze-out +7 +4.2 +Annihilation scenario +8 +5 +Kinetic mixing +9 +5.1 +Thermal equilibrium between the dark and visible sector +9 +5.2 +Annihilations subdominant during freeze-out +10 +5.3 +CMB bound and other bounds on kinetic mixing +10 +6 +SIMP scenarios +11 +6.1 +Self-interacting resonant SIMP DM +13 +6.2 +SIMP DM +14 +7 +Conclusion +16 +A Cross sections +17 +A.1 Scattering/annihilation cross section +17 +A.2 Thermally averaged cross section +18 +A.3 S-channel resonance and narrow width approximation +19 +B Pion self-interactions +19 +B.1 +Amplitude +19 +B.2 +Cross section +20 +B.3 +Thermally averaged cross section +21 +C Dark photon decay rate +22 +C.1 Dark photon decay rate into dark pions ΓV →ππ. +22 +C.2 Dark photon decay rate in SM particles. +23 +D Dark pion annilation and scattering +23 +D.1 Dark pion annihiliation into SM fermions +23 +D.2 Pion-electron scattering +25 +– i – + +E Cross section for 3 → 2 dark pion interactions +26 +E.1 +Pion 5pnt interaction from the WZW-term +26 +E.2 +Dark photon interactions from WZW term +28 +E.2.1 +Amplitude +29 +E.3 +Resonance contribution from mV ≈ 2mπ +29 +1 +Introduction +Despite ongoing experimental and theoretical efforts, the nature of DM remains elusive. An +attractive possibility is that DM is a thermal relic, and its abundance is determined by +freeze-out from the thermal plasma. Although most attention has been on weakly interact- +ing particles (WIMPs), their parameter space is increasingly constrained [1–3], and other +explanations have come to prominence. One such alternative is strongly interactive massive +particles (SIMPs) [4–6]. In the SIMP scenario, dark matter freeze-out occurs in a dark sector +via 3 → 2 interactions, which are typically stronger than in the WIMP scenario. As a result, +SIMPs have a lower mass (typically MeV-GeV scale), to which direct detection experiments +are less sensitive [7]. To avoid overheating the dark sector during freeze-out, a portal coupling +maintains thermal equilibrium with the visible Standard Model sector. +Cosmological observations question the collisionless dark matter (CDM) paradigm. For +example, observations of dark matter halo density profiles do not match the expected NFW- +profile [8, 9] of CDM [10–15]. This discrepancey is known as the cusp vs. core problem, +see e.g. [16] for a recent review. Although the inclusion of baryonic effects in the numerical +simulations may resolve the discrepancy [17–21], it is also possible that the resolution lies +in the dark matter properties. Indeed, dark matter with strong self-interactions naturally +alleviate the problem by transferring heat from the inner to the outer parts of the halo, thus +smoothening the density profile [22, 23]. The required interactions are scale dependent – a +factor 10 difference between galaxies and galaxy clusters – pointing to a velocity-dependent +self-interaction [23]. +The archetypical SIMPs are the pseudo Nambu-Goldstone bosons – the dark pions – +of a condensed dark Yang-Mills theory, with the Wess-Zumino-Witten term providing the +five-point interactions [24–26]. +The dark pions have naturally large self-interactions, and +may address the small scale problems of CDM as well [27], except that the interactions are +not velocity dependent. Moreover, satisfying both the relic density and the self-interaction +constraints (or more conservatively, the upper bound on the self-interactions from the Bullet +cluster observations), is only possible for non-perturbatively large pion couplings, invalidating +the chiral perturbation theory approach [4]. Both of these issues can be overcome with extra +vector bosons in the model [28], and in this paper we will consider adding a massive dark +photon. +For finetuned dark photon mass, almost twice the dark pion mass, the photon +mediated self-interactions can be on resonance, thus giving rise to a velocity-dependent effect +– 1 – + +[29]. In addition, the WZW-interactions may likewise be resonantly enhanced, bringing the +freeze-out interactions back in the resonant regime. +In addition, the dark photon can maintain equilibrium with the SM through kinetic +mixing [30, 31] with the SM photon [5, 6]. In this case dark pion annihilations to SM final +states should be included in the relic density calculations as well. The annihilation rate grows +at late times, as the dark pions lose kinetic energy and the annihilation cross section gets more +and more resonantly enhanced. As a result, even after freeze-out of the (resonantly enhanced) +WZW interactions, the annihilations can still be important and affect the final relic density. +Late time annihilations in photons and electrons are bounded by nucleosynthesis and cosmic +microwave background observations. +In this paper we will study if one particle – the dark photon – can do it all, solve the +small scale structure problems of CDM, enhance the freeze-out interactions such that the relic +density is obtained for non-perturbative pion couplings, and maintain thermal equilibrium +with the SM – this is dubbed the resonant self-interacting dark matter (RSIDM) scenario. +We also include collider bounds on dark photon kinetic mixing and millicharged particles, as +well as cosmological constraints from nucleosynthesis and the cosmic microwave background. +We find that RSIDM cannot affect small scale structure formation while maintaining thermal +equilibrium with the SM – although this possibility is only marginally excluded and more +precise calculations may be needed to make definite statements. +We will also study how +parameter space opens up if the requirement that dark matter affects small scale structure +formation is dropped. +This paper is structured as follows. Section 2 introduces the dark sector, with the dark +pions and dark photon. This is followed by a discussion of the dark matter self-interactions +and Bullet cluster bound in section 3; analytical estimates for the freeze-out temperature and +final relic density, including both WZW interactions and annihilations, in section 4; and the +constraints on the kinetic mixing parameter in section 5. In section 6 we then discuss the +parameter space for which the the relic density can be obtained in a perturbative set-up, the +self-interactions can be resonantly enhanced to address the small scale structure problems +of CDM, and the dark photon can keep the dark and visible sector in thermal equilibrium +during freeze-out. +In addition to the analytical estimates we will also provide numerical +results. We end with concluding remarks in section 7. For completeness, we have also added +the computation of the various (thermally averaged) cross sections in the appendix. Our +results for the WZW and pion self-interactions agree with the literature; new is the photon +mediated resonant contributions to the various cross section. +2 +Lagrangian +Strongly interacting massive particles (SIMPs) freeze out via 3 → 2 dark matter interactions +[7, 32]. The large required number changing interactions can naturally be obtained in a dark +sector with a non-abelian symmetry, with dark pions playing the role of the DM [4]. In this +paper we study the phenomenology of this set-up if we add a dark photon [5, 6]. The dark +– 2 – + +photon can provide a portal between the dark and SM sectors, and – for tuned masses – can +resonantly enhance the dark pion interactions. +We thus consider a dark sector with an SU(Nc) × U(1) gauge symmetry, with Nf dark +quarks in the fundamental representation of the gauge group. The quark mass matrix is +assumed diagonal M = mq1, allowing for a Wess-Zumino-Witten (WZW) term in the action +for Nc ≥ 3 colors [24–26]. A dimension-four kinetic mixing operator connects the dark U(1) +group with the SM hypercharge [30, 31], and provides a portal between the dark and visible +sectors. +At a scale Λ the non-abelian gauge group condenses, and the approximate flavor symmetry +of the light left- and right-handed quarks is broken down to the diagonal subgroup. The +(N2 +f − 1) dark pions are the pseudo-Goldstone bosons of the this symmetry breaking. They +can be naturally lighter than the scale Λ, which sets the mass of the baryons in the theory. +Depending on their couplings to other sectors, including the SM, the dark pions can be stable +on cosmological timescales and thus are a good dark matter candidate. +At energies below the condensation scale the effective action is +S = +� +d4x +�f2 +π +4 Tr(DµU)†DµU + ζf3 +π +2 Tr(MU + h.c.) − 1 +4V 2 +µν − 1 +2mV V 2 +µ − ϵVµJµ +SM +� ++ ΓWZW +(2.1) +with ζ = O(1). The first two terms are the leading order operators of the chiral effective +Lagrangian describing the dark pion dynamics. Here U = e2iπ/fπ, π = πaT a with T a gen- +erators of SU(Nf), and fπ ∼ √NcΛ/(4π) the pion decay constant. The covariant derivative +is DµU = ∂µU + igd[Q, U]Vµ, with Vµ the dark photon field and gd the dark U(1) gauge +coupling. We choose the charge matrix [6, 33] +Q = Diag(1, −1, 1, −1, ...), +(2.2) +with Nf entries. With this charge assignment Tr(Q2T a) = 0 and the mixed anomaly vanishes, +avoiding decay of the neutral dark pion into to two (dark) photons [34, 35]. +The next two terms are the dark photon kinetic term, where we defined the gauge field +strength Vµν = ∂µVν−∂νVµ, and the St¨uckelberg mass term for the dark photon. To avoid pion +decay the dark photon mass should exceed twice the pion mass; in the limit that the photon +mass is close to that threshold, the photon-pion interactions can be resonantly enhanced. We +parameterize1 +mV = mπ(2 + δm) > 2mπ. +(2.3) +The last term in eq. (2.1) between the square brackes couples the dark photon to the SM +vector current Jµ +SM = � qf ¯fγµf, and qf the electric charge of the SM fermion f. This term +arises as a consequence of kinetic mixing between the dark U(1) group and SM hypercharge; +after redefining the fields to make the kinetic terms canonical, and diagonalizing the mass +1Here mπ is the mass of the charged – with unit charge – dark pions, which interact with the dark photon. +The mass of the charged dark pions receive loop corrections, and is slightly larger than the mass of the neutral +pions. +– 3 – + +matrix, the result is the coupling in eq. (2.1) [36]. Here we used that ϵ ≪ 1 and have dropped +the O(ϵ2) terms, and we have neglected the coupling to the Z-boson, valid if the dark photon +mass is small compared to the electroweak scale. +Finally, the last term is the WZW action, which is present if the 5th homotopy group of +the coset space π5(G/H) is non-trivial; this is the case for Nf ≥ 3 mass degenerate flavors. +Expanding the action in pion fields, the Lagrangian is the sum of the Lagrangian for +chiral perturbation theory (χPT), the dark photon Lagrangian, and the terms from the WZW +action: L = LχPT + LV + LWZW. The relevant dark pion and and dark photon interactions +are: +LχPT = Tr(∂π)2 − m2 +πTr(π2) + +1 +3f2π +Tr +� +(2π∂π)(π∂π) − 2(ππ)(∂π∂π) + m2 +ππ4� ++ 2igdV µTr ((∂µπ)[Q, π]) , +LV = −1 +4V 2 +µν − 1 +2mV V 2 +µ − ϵVµ +� +f +qf ¯fγµf +LWZW = +2Nc +15π2f5π +ϵµνρσTr [π∂µπ∂νπ∂ρπ∂σπ] − i Ncgd +3π2f3π +ϵµνρσVµTr (Q∂µπ∂ρπ∂σπ) , ++ Ncg2 +d +4π2fπ +ϵµνρσ(∂µVν)VρTr +� +Q2∂σπ +� +. +(2.4) +The first two terms in the chiral lagrangian are the kinetic and mass term for the pion fields, +with mass m2 +π = 2ζfπmq. Chiral perturbation theory is perturbative for +ξ ≡ mπ +fπ +≲ 4π. +(2.5) +The 3rd term gives the 4pnt pion interactions, and the last term the pion-dark photon coupling +(V 2π interaction). The first two terms in the dark photon Lagrangian are the kinetic and +mass term for the dark photon field, and the last term the coupling of the dark photon to +the SM fermions from kinetic mixing. Finally, the WZW lagrangian contains the 5pnt pion +interaction, and additional dark photon-pion couplings with an odd number of pions (V 3π and +2V π interactions). We have only included the most relevant, lowest dimensional operators. +3 +Self-interactions +The dark pion can scatter via a 4-point contact interaction appearing in the χPT Lagrangian, +and via the exchange of a dark photon. The cross sections for these contributions are cal- +culated in the non-relativistic limit in appendix B. The photon exchange contribution is +subdominant, unless enhanced by an s-channel resonance which can appear for fine-tuned +dark photon masses eq. (2.3). The cross section is can then be approximated by a sum of +the velocity independent contact interaction and velocity dependent resonance contribution +σSI ≈ σ4pnt +SI ++ σres +SI . +– 4 – + +For SIMP dark matter the self-interactions are naturally large, and the Bullet clus- +ter observations [37–39] put a strong constraint on the cross section. In the resonant self- +interacting dark matter (RSIDM) scenario [29, 40], with a judicial choice of parameters, the +self-interactions can affect structure formation on small scales in a velocity dependent way, +and thus may address the putative problems of collisionless dark matter [22, 41–43]. +3.1 +Bullet cluster bound +The s-wave part of the dark pion self-interaction cross section is eq. (B.12) +σ4pnt +SI +mπ += 3κSIξ4 +64πm3π += 2.2 × 105 cm2 +g +�MeV +mπ +�3 3ξ4κSI +64π +≤ aint +cm2 +g , +(3.1) +with κSI = (N4 +f − 2 +3N2 +f + 2)/(N2 +f (N2 +f − 1)) = 1 + O(N−2 +f ) and MeV−3 ≈ 2.2 × 105 cm2/g. +The cross section is not very sensitive to the number of quark flavors Nf. The Bullet cluster +observation puts an absolute upper bound on the self-interaction cross section, given by +aint ≈ 1. In the RSIDM scenario, where the resonant interactions from dark photon exchange +become important at small scales, the data is fit by a smaller s-wave contribution and aint ≈ +0.11 (and the r.h.s. of eq. (3.1) becomes an equality) [29]. The bound on the cross section +can be translated in a condition on the dark pion mass +mπ ≥ 14.9 MeV +�ξ4κSI +aint +�1/3 +. +(3.2) +3.2 +Resonant self-interactions +The velocity dependent contribution to the self-interactions from nearly on-shell dark photon +exchange can be written in Breit-Wigner form eq. (B.15) +σres +SI = +4πS +mπE(v) +Γd(v)2/4 +(E(v) − E(vR))2 + Γ(v)2/4, +(3.3) +with S = 3Sf/N 2 +π the ratio of multiplicities of the resonance dark photon (3 polarizations) +and DM particles (Nπ = N2 +f − 1) times a symmetry factor Sf = 2 that takes into account +that there are two identical particles in the final state of both the self-interaction process and +in dark photon decay. E(v) = mπv2/4 is the kinetic energy in terms of the relative velocity +v, and E(vR) = mπδm the resonant kinetic energy. Further, Γd(v) is the running decay rate +of the dark photon into dark sector pions, and Γ = Γd + Γv the total running decay rate, +which includes the decay into the visible SM sector fermions and pions via kinetic mixing. +The velocity dependence of the decay widths can be parameterized as Γi = mV γivni with +i = d, v; explicitly (see appendix C) +Γd(v) = mV +�C4αd +24Sf +� +v3, +Γv(v) = mV +� αϵ2 +3πSf +� +v0, +(3.4) +with αd = g2 +d/(4π) and α = e2/(4π) the dark sector and SM fine-structure constants. +– 5 – + +The resonance is peaked for v = vR, with vR = 3.6 × 10−4c from small scale structure +data [29]. This determines δm +δm = +�vR +2 +�2 += 3.2 × 10−8. +(3.5) +The height of the peak is fit by mπ = 4000 MeVS1/3(Bdγd)1/3, with Bd = Γd/Γ the branching +ratio for decay into the dark sector. This determines the dark photon gauge coupling: +αd = 1.3 × 10−4 � +mπ +100 MeV +�3 +N2 +π +BdC4 +(3.6) +The resonant enhancement of the cross section is large, and small αd is required to avoid too +large self-interactions. +For RSIDM both the mass and the dark gauge coupling are given in terms of ξ, which +together with the kinetic mixing parameter ϵ are the only free parameters left. The combined +constraints eq. (3.2) with aint = 0.11 and eqs. (3.5) and (3.6) give +mπ = 31.0 MeV +� +ξ4κSI +�1/3 , +αd = 3.7 × 10−6ξ4 N2 +πκSI +C4Bd +, +δm = 3.2 × 10−8. +(3.7) +4 +Relic density +The dark pions can freeze out via 3 → 2 number changing SIMP interactions and via annihi- +lation into SM fermions. In the SIMP scenario thermal equilibrium with the SM sector should +be maintained through freeze-out, to avoid entropy production and heating up of the dark +sector. We will assume this is the case in this section, and return to the question whether +the dark photon can be responsible for this in the next section. +The SIMP interactions get a contribution form the 5-point coupling in the WZW La- +grangian, and from diagrams with dark photon exchange allowed by the (3π)V coupling in +the WZW Lagrangian; with our choice of dark charges eq. (2.2) the π(2V ) coupling vanishes. +The thermally averaged cross section is calculated in appendix E, and is given in eqs. (E.10) +and (E.27). Introducing the ‘time’ variable +x = mπ +T , +(4.1) +it can be written in the form +⟨σv2⟩3→2 ≈ ⟨σv2⟩5pnt +3→2 + ⟨σv2⟩res +3→2 = α3→2 +x2m5π +(1 + αresx5/2e−δm x). +(4.2) +The first term comes from the 5pnt pointlike interaction. The 2nd term is from dark photon +exchange, which is dominated by an s-channel resonance if the dark photon is nearly on shell +δm ≪ 1, and we have we used the narrow width approximation to evaluate the thermally +– 6 – + +averaged cross section. We have calculated the latter term only for Nf = 3 flavors. The +effective couplings are +α3→2 = +5 +√ +5 +1536π5 +N2 +c κ3→2ξ10 +Nf +, +αres|Nf=3 = 128π5/2 +15 +αd +ξ4 +(4.3) +with κ3→2 = N2 +f (N2 +f −4)/N 2 +π = 1+O(1/Nf). The resonant contribution dominates at freeze- +out x = xf for αd/ξ4 ≳ 3.8 × 10−6xf20 with xf20 = xf/20. For RSIDM eq. (3.7) this is the +case for ξ ≲ O(1). +Dark pions can annihilate into SM electrons (and depending on their mass, into heavier +charged SM particles) via the kinetic mixing portal. Annihilation is also dominated by the +s-channel resonance and the thermally averaged cross section is eq. (D.9) +⟨σv⟩ann = αann +x3/2e−δm x +m2π +, +αann = 32π√πϵ2αBd +N2π +(4.4) +with as before Bd = Γd/Γ the branching ratio for decay into dark sector states. We note +that both the resonant part of the WZW interactions and the resonant annihilations are +independent of the mass splitting δm, except for the exponential factor, which determines +below which temperature x ≳ 1/δm these interactions are ‘turned off’. +The Boltzmann equation for the dark pions reads +˙n + 3Hn = −⟨σv⟩ann(n2 − n2 +eq) − ⟨σv2⟩3→2(n3 − n2neq), +(4.5) +which in terms of the number density fraction Y = n/s becomes +dY +dx = −λ3→2 +x7 (1 + αresx5/2e−δm x)(Y 3 − YeqY 2) − λanne−δm x +√x +(Y 2 − Y 2 +eq). +(4.6) +Here +λ3→2 = s(mπ)2α3→2 +m5πH(mπ) , +λann = s(mπ)αann +m2πH(mπ) +(4.7) +with s(mπ) = (2π2g∗sm3 +π)/45 and H(mπ) = (π√g∗m2 +π)/(3 +√ +10mpl) the entropy density and +Hubble constant at T = mπ. +The relic dark matter density matches observations [44] for +Ωπ,0 = Nπmπs0Y∞ +ρc +⇒ +mπY∞ = 0.4 × 10−6 MeV +(4.8) +with Y∞ = n/s the asymptotic number density fraction, and s0 the entropy density today. +4.1 +SIMP freeze-out +Consider first the case that freeze out of the dark pion is determined by the WZW interactions, +either by the 5pnt contact interaction or the dark photon mediated contribution. This means +annihilation are subdominant at freeze out ⟨σv⟩ann ≲ ⟨σv2⟩3→2 neq at x = xf. +We will +– 7 – + +estimate the bound this condition gives on ϵ in the next section. Note, however, that the +annihilation cross section grows at late times, and even if negligible at freeze out, annihilation +may still affect the final relic density significantly. +To describe freeze out we thus set the annihilation contribution to zero in the Boltzmann +eq. (4.6). At late times the equilibrium distributions can be dropped, which allows to solve +for the asymptotic distribution +lim +x→∞ +dY +dx = −λ3→2 +x7 (1 + αresx5/2)Y 3 +⇒ +Yf ≃ +√ +3x3 +f +√λ3→2 +1 +� +1 + 12 +7 αresx5/2 +f +(4.9) +where we introduced the notation Yf for the asymptotic number density after freeze-out of +the SIMP reactions. The freeze-out temperature can be estimated from n2 +π⟨σv2⟩3→2 ≃ H +which gives +x3e2x�� +x=xf ≃ N2 +πα3→2mπ +(2π)3H(mπ)(1 + αresx5/2) ≡ C3→2(1 + αresx5/2) +(4.10) +where we used that the non-relativistic number density is n = Nπm3 +π(2πx)−3/2e−x. In the +limit that the 5pnt interaction respectively the resonant contribution dominates the cross +section we can estimate the freeze-out temperature using that xne2x = c gives x ≈ ln √c − +n +2 ln(ln √c). +The SIMP interactions are negligible after freeze out, but annihilations may still have +an effect. To estimate this we solve the Boltzmann equation with the boundary condition +Y (xf) = Yf: +dY +dx = −λanne−δm x +√x +Y 2, +⇒ +Y∞ ≈ Yf +√ +δm +√ +δm + √πYfλann +(4.11) +where we assumed that (δm xf) ≪ 1. The annihilation rate increases at late time, and is +most efficient just before the exponential cutoff at x = 1/δm kicks in; this is how the δm- +dependence appears in the estimate for the relic density. It follows that annihilations are +negligible if +√πYfλann +√ +δm +< 1 +⇒ +ϵ < 7.6 × 10−9Nπ +� +Bdδm +� mπ +MeV +� +. +(4.12) +that is, only for very small kinetic mixing. +4.2 +Annihilation scenario +In the opposite limit that 3 → 2 interactions are always subdominant, ⟨σv⟩ann ≥ ⟨σv2⟩3→2 neq +at freeze-out, the relic density is set by annihilation reactions only. We can still use eq. (4.11) +for the relic density, but now Yf(xf) is the number density as annihilations freeze out. The +freeze-out temperature can be estimated from nπ⟨σv⟩ann ≃ H which gives +x−2ex ≃ +Nπαannmπ +(2π)3/2H(mπ) ≡ Cann +⇒ +xf ≃ ln Cann + 2 ln(ln Cann). +(4.13) +– 8 – + +We estimate the freeze out density +Yf ≈ n +s +��� +xf += +x7/2 +f +λann +(4.14) +where we used eq. (4.13). If xf in eq. (4.14) is smaller than that for SIMP reactions eq. (4.10), +it follows freeze-out is dominated by annihilations. An earlier freeze out means a larger density +Yf. Hence we can write the asymptotic number density as +Y∞ ≈ Yf +√ +δm +√ +δm + √πYfλann +, +Yf = max +� +Yf +�� +ann, Yf +�� +3→2 +� +(4.15) +with the freeze out density for 3 → 2 reactions and annihilation given in eqs. (4.9) and (4.10), +and eqs. (4.13) and (4.14) respectively. +5 +Kinetic mixing +Kinetic mixing between the dark and SM photons provides a portal between the dark and +visible sector. In the SIMP scenario, in which the relic density is determined by the freeze- +out of 3 → 2 dark pion number changing interactions, both sectors need to be in thermal +equilibrium during freeze-out to avoid heating up the dark sector. The SIMP scenario further +requires that dark pion annihilation into SM particles is subdominant during freeze-out – +although, as we have seen in the previous subsection, annihilation still may affect the relic +density at late times. In this subsection we determine the constraints on the mixing parameter +ϵ that these two requirements give. The relevant cross sections are computed in appendix D. +We also quickly review the relevant cosmological and collider bounds on the kinetic mixing +parameter. +5.1 +Thermal equilibrium between the dark and visible sector +The dark and visible sector can be kept in thermal equilibrium via pion scattering with SM +electrons and positrons. The non-relativistic cross section for this process is eq. (D.16) +σscat = Ascatϵ2 p2 +m4π +, +Ascat = 2πC4αDα +Nπ +(5.1) +This can be straightforwardly generalized to include muon and SM pion scattering as well,2 +see below eq. (D.16), in case freeze-out occurs at temperatures exceeding the muon and pion +mass. Here p ≈ Ee is the incoming electron momentum, and C4 = 4 (8) for Nf = 3 (4) the +same color factor as appearing in the dark photon decay rate eq. (3.4). The scattering rate +2Scattering off muons was included in the numerical results, but its effect for thermalisation was found to +be negligible. +– 9 – + +can be estimated as Γscat ≈ ⟨neE2 +e⟩(σv)scat/E2 +e [5], and demanding that it exceeds the Hubble +rate Γscat > H at the time of freeze out gives the bound +ϵ > +� +H(Tf)m4π +⟨neE2e⟩Ascat += 4.6 × 10−8x3/2 +f20 +� +mπ +100 MeV +4 +ge +Nπ +C4αd +, +(5.2) +where we used ⟨neE2 +e⟩|x=xf = ge45ζ(5)m5 +π +4π2x5 +f +, H(T) = H(mπ)/x2 +f, and as before xf20 = xf/20. +Further, ge = 4 are the degrees of freedom of the electron/positron pair. To get the numerical +value we used α = 1/137 and gs = 17.25. For RSIDM the bounds eq. (3.7) eliminate the dark +pion mass and gauge coupling dependence. +If the dark photon is to maintain thermal equilibrium with the SM, annihilations cannot +be neglected for the relic density calculation if (comparing eq. 4.12 and eq. 5.2) +mπ ≳ 3.7 × 10−2 MeV +αdδm +10 +C4Nπ +x3 +f20 +(3.7) += 3600 MeV +�8xf20 +Nπ +�3/4 +. +(5.3) +The last equality applies to RSIDM, for which annihilations thus always play a role, depleting +the dark matter abundance at late times. +5.2 +Annihilations subdominant during freeze-out +Annihilations are subdominant at freeze-out if ⟨σv⟩ann < ⟨σv2⟩3→2 nπ at x = xf eq. (4.10), +which translates to +� +α3→2(1 + αresx5/2 +f +)H(mπ)/mπ > αannx7/2 +f +. This gives an upper bound +on the kinetic mixing parameter +ϵ < 4 × 10−8 � +mπ +100 MeV +�1/4 ξ5/2 +x7/4 +f20 +Nπ +√Nc +N1/4 +f +� +1 + 3 × 105 αdx5/2 +f20 +ξ4 +�1/4 +, +(5.4) +where we set κ3→2, Bd to unity. +5.3 +CMB bound and other bounds on kinetic mixing +BBN and CMB observations place bounds on the energy injected in the photon fluid from late +time dark pion annihilations. These constraints can be stringent as the thermally averaged +cross section grows as ⟨σv⟩ ∝ x3/2e−δmx at late times. +For self-interacting resonant DM +the mass splitting δm ∼ 3 × 10−8 is small eq. (3.5), and the exponential suppression of the +cross section only kicks in shortly before or after the CMB is formed, depending on the mass +of the dark pions. +The thermally averaged cross section thus peaks at this time. +CMB +observations are more stringent for s-wave annihilations (as opposed to more stringent BBN +bounds for p-wave annihilations) [45]. +Since our thermally averaged cross sections scales +inversely proportional to the velocity, CMB bounds are stronger than bounds from BBN, and +we thus only consider the former. +The CMB bound is given by pann < 3.3 × 10−31 cm3s−1MeV−1, where pann ≡ f(z) ⟨σv⟩ +mπ +and f(z) = 0.01−1 is a function that quantifies the efficiency of energy injection in the CMB +– 10 – + +[44]. Recasting this to a bound on the mixing parameter ϵ, the constraint is given by (setting +f(z) = 1) +ϵ ≲ 9.4 × 10−14� +mπ +10 MeV +�3/4 +exp +� +1.1 +mπ +10 MeV +� +, +(5.5) +which vanishes above dark pion masses of 150 MeV. This bound was derived for s-wave +scattering. In our model, the annihilation cross section increases for later times until the +exponential cut-off kicks in, so the bound is expected to be stronger at late times. At the +same time, the energy injection into the CMB is maximized at z ∼ 600, where f(z) ≈ 1 +[46, 47]. Although applying the CMB bound naively for our case at the different choices of z +affects the exact constraint on the mixing parameter somewhat, this has no consequences on +parameter space of the SIMP scenarios discussed in the next section, as this is dominated by +the constraintes from thermalisation and beam dump experiments. For simplicity, then, we +imposed the CMB bound at z ∼ 600, and that is what is shown in our figures in section 6. +For larger δm only the BBN bound applies. +Following [45], energy injection during +BBN is most efficient in the range 1/T ∼ 102 − 103 MeV−1. Requiring the annihilations +to be suppressed at this time (δmx ≳ 1), the BBN bound can evaded for mass splittings +δm ≳ 10−3. +Finally, there are bounds from dark photon searches at beam dump or fixed target ex- +periments at electron or proton colliders. In these experiments large number of dark photons +can be produced from Bremsstrahlung or secondary meson decays. The experiments typi- +cally search for highly displaced vertices in the detector [48, 49]. For our region of interest, +10−4 ≲ ϵ ≲ 10−8 and 10 MeV ≲ mγ′ ≲ 1 GeV we consider bounds from NuCal[50, 51], +CHARM[52], and E137 [53]. Given the p-wave nature of our scattering cross section, scatter- +ing at late times is heavily suppressed. Bounds on millicharged particles from direct detection +experiments like XENON are therefore too weak to constrain the kinetic mixing parameters +and are therefore not considered. +6 +SIMP scenarios +In the ‘standard’ SIMP scenario the dark pion relic density results from the freeze out of +the 5pnt 3 → 2 WZW processes. We reproduce this set-up by turning off the dark photon +interactions αd = 0. The Bullet cluster observation puts an upper bound on the pion self- +interactions, and consequently on the pion mass eq. (3.2). The correct relic density is obtained +for ξ ∼ 4π for Nf = Nc = 3, uncomfortably close to the perturbativity bound ξ ≲ 4π [4]. +Increasing the number of colors improves the situation slightly, but a large number of colors +is needed to be within the perturbative regime. +The problem with only the 3 → 2 interactions, and no resonance enhancement, is that +dark matter is over produced. The dark pions contribute a fraction to the DM density eq. (4.8) +R ≡ +Ωπ +ΩDM +≈ +9 × 102x3 +f20 +ξ3Nc +� +Nf +aint +. +(6.1) +– 11 – + +Bullet cluster +perturbativity +WZW +WZW+res +10 +50 +100 +500 1000 +5000 104 +0 +2 +4 +6 +8 +10 +12 +14 +Figure 1: Relic density constraint on ξ for different values of the dark pion mass for the +WZW-term only (red), and including the resonance via the dark photon (blue). The dashed +lines correspond to the estimate of eq. (4.9) (setting αd = 0 for the WZW estimate). The gray +shaded areas are excluded by ξ > 4π, where the χPT description breaks down, and the Bullet +cluster bound on the DM self-interaction σ/m ≤ 1 cm2/g. All lines are for Nf = Nc = 3. +The freeze-out temperature only depends logarithmicly on the model parameters, and ranges +from xf20 = 0.68 − 1.1 for ξ = 1 − 10. The relic density is reproduced for R = 1, which +requires large ξ ≳ 4π. +This is illustrated in fig. 1, which shows the numerical solution to the Boltzmann equation +without (red) and with (blue) adding the 3 → 2 resonance, where αd is chosen from eq. (3.7). +The dashed lines correspond to the estimate of eq. (4.9) with αres = 0 (αres ̸= 0) for the red +(blue) curve. The shaded areas are excluded by the perturbativity cutoff on ξ and the Bullet +cluster bound on the self-interaction. +Without the dark photon resonance, the required value of ξ is at or above the perturbativ- +ity cutoff. Including the resonance, but not considering annihilations, the situation improves +significantly as the increased 3 → 2 interactions reduce the dark matter density. The observed +relic density is obtained for a larger dark pion mass for a fixed ξ. For such a large pion mass, +however, the kinetic mixing parameter is too small to maintain kinetic equilibrium with the +SM eq. (5.3) and either an additional portal interaction is required or annihilations should be +considered. +In the following subsections we discuss two SIMP scenarios that give the correct relic +density, satisfy self-interaction constraints, and the kinetic mixing portal interaction maintains +thermal equilibrium with the SM during freeze-out. First, we focus on the possibility that the +– 12 – + +dark pions are RSIDM. The 3 → 2 freeze-out interactions can be resonantly enhanced for large +enough αd/ξ4 ≳ 10−6. Given the small mass difference δm in eq. (3.5), annihilations become +important at late times, and reduce the relic density further. Second, we consider the more +classical SIMP scenario, and drop the requirement that the self-interactions can affect small +scale structure formation. The self-interactions should still satisfy the upper bound from the +bullet cluster. Both 3 → 2 interactions and annihilations may be resonantly enhanced, by +tuning the dark photon mass, but now have more freedom in the resonant condition δm and +the dark gauge coupling αd to satisfy all constraints. +6.1 +Self-interacting resonant SIMP DM +Consider first the RSIDM scenario that the relic density is produced via the SIMP mechanism, +i.e. via the freeze out of 3 → 2 reactions, and that resonant DM self-interactions can address +the small scale structure problems. The requirements on the self-interactions eq. (3.7) fixes +the parameters mπ, αd, δm in terms of ξ, which in turn is determined from the correct relic +density eqs. (4.8) to (4.10). +For large enough kinetic mixing the relic density will be reduced by (late-time) annihi- +lations, which reduces the required ξ value. Assuming annihilations are subdominant during +freeze-out and Bd ≈ 1, then Y∞ is given by eq. (4.11) and R now becomes +R = +2.7 × 103x3 +f20 +ξ3 +1 +Rres + 3.0 × 1018x3 +f20ϵ2/ξ17/3 , +(6.2) +where Rres ≡ +� +1 + 30x5/2 +f20 accounts for the resonant enhancement of the WZW interactions, +and the ϵ-dependent term for late-time annihilations. Annihilations significantly reduce R +for ϵ > 7.6 × 10−12ξ17/6x−3/2 +f20 +for sufficiently small ξ ≲ O(1), but rapidly shut off for larger ξ. +Solving for the observed relic density R = 1 gives +ϵ = +� +9.3 × 10−16x3 +f20ξ8/3 − 3.4 × 10−19Rresξ17/3 x−3/2 +f20 . +(6.3) +Figure 2 shows the value of ϵ as a function of the dark pion mass mπ for which R = 1 from +numerically solving the Boltzmann equations (red curve), as well as the analytical estimate +eq. (6.3) above (dashed curve). The observed relic density can be obtained with perturbative +couplings, e.g. ξ = 1 for kinetic mixing ϵ = 3×10−8. Note, however, that there is a maximum +value +ϵ ≤ 2.9 × 10−7 +(6.4) +to get the observed relic density, at a dark pion mass mπ ∼ 400 MeV. For larger dark pion +masses, the 3 → 2 interactions alone underproduce DM given the imposed relations on the +self-interaction eq. (3.7), so additional annihilations are not of any help; this explains the sharp +cutoff of the red curve at large pion masses. For such small kinetic mixing annihilations are +subdominant during SIMP freeze-out eq. (5.4), and decay is predominantly into dark sector +pions Bd ≈ 1, validating our assumptions for the freeze out calculation. +– 13 – + +Thermalisation +CMB +Direct detection +10 +50 +100 +500 +1000 +10-9 +10-8 +10-7 +10-6 +10-5 +Figure 2: Kinetic mixing parameter space constraints. The red (numerically) and dashed +(estimate eq. (6.3)) lines show the values of ϵ for which the correct DM relic density is +produced, and for which the dark pions have to required self-interaction. The blue shaded +area excludes those values of ϵ for which the dark photon cannot maintain kinetic equilibrium +with the SM sector. The purple shaded area depicts the CMB constraint on DM annihilations. +The grey shaded area is excluded from beam dump searches from E137, nuCAL and CHARM. +In addition to the relic density constraint fig. 2 also shows the bounds from the CMB, +colliders, and from the requirement of thermal equilibrium between the dark and visible +sector during SIMP freeze out eq. (5.2). It is the latter requirement that rules out most of +the parameter space. Because the annihilations are highly efficient, and only a small portion +of the DM should be depleted, the annihilation cross section should be suppressed by small +values of ϵ. For these small ϵ, the heat transfer to the SM is not fast enough to prevent +the dark bath from heating up. It is thus not possible for the dark pions to be resonant +self-interacting DM, and affect small scale structure formation. +Around mπ ∼ 500 MeV kinetic mixing is marginally too small to maintain thermal +equilibrium with the SM. Additional effects might affect this part of parameter space which +could render the model viable. Alternatively, one could allow for (partial) heating of the dark +bath to study the effect on the dark bath. This is left for future research. +6.2 +SIMP DM +We now consider the SIMP scenario, in which the relic density is determined by 3 → 2 inter- +actions and possibly additional annihilations, but self-interactions are too weak to affect small +scale structure formation. As we have seen eq. (6.1), with just the 5pnt WZW interactions +– 14 – + +and given the Bullet cluster bound, too much DM is produced for perturbative couplings ξ. +The DM density can be reduced by a resonant enhancement of the WZW interactions and +by annihilations. No longer constrained by the small scale structure data, the value of δm +can now be larger. This immediately avoids CMB and BBN constraints as the thermally +averaged annihilation cross section ∝ e−δm x eq. (4.4) is exponentially suppressed in these +eras. Moreover, for larger δm dark photons will predominantly decay to dark pions, thus +evading dark photon searches at beam dump experiments. For concreteness we will fix the +mass splitting to δm = 10−3 throughout this subsection. This choice avoids the cosmological +constraints, while it can still give rise to interesting phenomenology at late times. +In the parameter space region where the dark photon maintains thermal equilibrium with +the SM sector during freeze-out of the WZW-interactions, the kinetic mixing parameter is +always large enough that late time – after freeze-out – annihilations cannot be neglected. For +general mπ, ξ, δm and αd the required mixing parameter that reproduces the correct relic +density is +ϵ = 1.5 × 10−12 Nπ +Yf +�√ +δm +� mπ +MeV +� � +3.8 × 107 +� mπ +MeV +� +Yf − 15 +� +, +(6.5) +with Yf from eq. (4.9). Parameter space opens up significantly compared to the RSIDM +scenario, as all other parameters are free except for the Bullet cluster bound on the dark +matter self-interactions. +Figure 3 shows the numerical result for the kinetic mixing parameter as a function of the +dark pion mass that reproduces the observed relic density. The red, orange and purple curves +correspond to different values of ξ = 2, 5, 10. In all plots the mass splitting is δm = 10−3. The +numerical results are in excellent agreement with the analytical estimate eq. (6.5). Along the +curves three different regions can be identified; i) a part where the self-interactions are larger +than allowed by the Bullet cluster constraint, which is excluded (dotted); ii) a part where the +3 → 2 interactions dominate freeze-out, and annihilations are only important at later times +times (solid); and iii) annihilations are the dominant freeze-out process (dashed). +The blue region in the plots is excluded by the thermalisation requirement eq. (5.2). For +larger αd a smaller kinetic mixing parameter is required to maintain thermal equilibrium with +the SM. +The left top plot shows the result for αd = 0.01. For such small gauge couplings, the +resonance enhancement of the WZW interaction is negligible. DM is over produced unless +annihilations are important. In fact, we see that for the parameter space allowed by the +Bullet cluster constraints, the annihilations are actually so large that they always dominate +freeze out. Hence, in this scenario the dark pions are WIMP rather than SIMP dark matter. +In the right top plot the gauge coupling is increased αd = 0.1, but still resonance effects +on the WZW interactions are small except for large ξ. Indeed, for ξ ∼ 10 the dark pion can +be SIMP DM. Although late time annihilations reduce the relic density somewhat – this is +what generates the slope of the solid curve as a function of mixing parameter ϵ – the effect is +not strong enough to allow for much smaller ξ than in the ‘standard’ scenario. +– 15 – + +10 +50 +100 +500 +1000 +10-8 +10-7 +10-6 +10-5 +10 +50 +100 +500 +1000 +10-8 +10-7 +10-6 +10-5 +10 +50 +100 +500 +1000 +10-8 +10-7 +10-6 +10-5 +Figure 3: The values of ϵ for which the correct relic density is reproduced as a function +of the pion mass, for different values of ξ (coloured lines) and αd. The blue shaded area is +excluded by the thermalisation requirement eq. (5.2). Dotted lines violate the Bullet cluster +constraint on the self-interaction. The solid lines represent the part of parameter space where +the 3 → 2 interactions are the dominant freeze-out interaction. The dashed lines are when +2 → 2 annihilations via the dark photon are important at freeze-out. +Finally, the bottom plot is for seizable couplings αd, and the WZW interactions are sig- +nicficantly enhanced for smaller ξ = 1−5 as well, allowing SIMP dark matter in a perturbative +set-up. The slope of the solid part of the curves shows the impact of late time annihilations +as a function of kinetic mixing. The curves asymptote to a constant value for small mixing +and annihilations are negligible at all times, thus providing a lower bound on the dark pion +mass for a given ξ. +7 +Conclusion +We have studied a dark sector containing dark pions and a dark photon. The dark pions +are stable and can be SIMP dark matter, that is produced by freeze-out of 3 → 2 WZW- +interactions. The dark photon mixes kinetically with the SM sector, and can maintain thermal +– 16 – + +equilibrium during freeze out. For a fine-tuned dark photon mass mV ≈ 2mπ the WZW are +resonantly enhanced, which opens up the possibility that 1) the observed relic density is +produced in the perturbative regime ξ = mπ/fπ ≲ 4π of the effective chiral Lagrangian. +In addition, the pion self-interactions are resonantly enhanced and become velocity depen- +dent, which opens up the possibility that it can address the small scale structure problems +of collisionless dark matter – this scenario is dubbed resonant self-interaction dark matter +RSIDM. +We found that the RSIDM scenario is not possible for all dark pion masses considered. +Because of the highly efficient dark pion annihilations, the value of the mixing parameter +required to reproduce the relic density is too small to maintain kinetic equilibrium with the +SM. For dark pion masses of mπ ∼ 500 MeV the difference is marginal, and more precise +calculations are required to assess the viability of the model in this region. In particular, one +could allow for (partial) heating of the dark bath to study the effect on the dark bath. This +is left for future research. +If we give up the demand that the self-interactions have an effect on small scale structure +formation, and consequently are only constraint by an upper bound from observations of +the Bullet cluster, then parameter space opens up and smaller ξ-values become possible for +sufficiently dark gauge couplings αd ∼ 1. +Acknowledgments +The authors thank Jordy de Vries for very useful discussions. This work was funded by an +NWO-klein2 grant (OCENW.KLEIN.427). +A +Cross sections +We list here the definitions of the (thermally averaged) cross sections. This appendix also +serves to set the notation. +A.1 +Scattering/annihilation cross section +The cross section for scattering with two particles in both the initial and final state, labeled +by α and β respectively, is +σα→β = +1 +4FSβ +� +� � +β=1,2 +� +pβ +� +� (2π)4δ4(Pα − Pβ)| ¯ +Mα→β|2 CM += +� +dΩ +1 +(8π)2sSβ +pout +pin +| ¯ +Mα→β|2 +(A.1) +with +� +p = +� +d3p/(2Ep(2π)3). We use P µ for 4-momenta, and pi for 3-momenta. The second +expression is valid in the center of mass frame (CM), with pin = |pα| (pout = |pβ|) the absolute +value of the three-momentum of either incoming (outgoing) particle. Sβ = N! for N identical +particles in the final state, to avoid overcounting in the phase space integral. | ¯ +M|2 is the +– 17 – + +amplitude averaged over initial and summed over final state particles. The flux factor can be +written as F = E1E2|v1 − v2| = +� +(p1.p2)2 − m2 +1m2 +2 +CM += pin +√s with s = (E1 + E2)2 the center +of mass energy squared. +A.2 +Thermally averaged cross section +The thermally averaged cross sections can be defined in terms of the scattering rates appearing +in the Boltzmann equation for the DM particle [32] +˙n + 3Hn = − +� +α,β +∆αβ(˜γα→β − ˜γβ→α) = −⟨σv⟩ann(n2 − n2 +eq) − ⟨σv2⟩3→2(n3 − n2neq), (A.2) +with n = nDM the number density of dark matter. We have included both DM annihilation +and 3 → 2 interactions. ∆αβ = (Ndm +α +− Ndm +β ) is the difference between the number of DM +particles in the initial (Ndm +α ) and final state (Ndm +β ). The collision terms are +ˆγα→β(fα) = +1 +SαSβ +�� +α +� +α +Nαfα +� � +�� +β +� +β +� +� (2π)4δ4(Pα − Pβ)| ¯ +Mα→β|2 +(A.3) +with fα, Nα the distribution functions and degrees of freedom of the initial states, and +Sα (Sβ) = N! for N identical particles in the initial (final) state. Assuming kinematic equi- +librium for the DM and chemical equilibrium for all other particles gives the relations +ˆγα→β(fi) = +� n +neq +�Ndm +α +γα→β(feq +i ), +γα→β = γβ→α +(A.4) +with γα→β ≡ ˆγα→β(feq +α ) and feq +α = e−Eα/T the Maxwell-Boltzmann distribution. We can then +express the thermally averaged cross sections appearing in the Boltzmann equation eq. (A.2) +in terms of the collision rates as follows +⟨σv⟩ann = 2γann +n2eq +, +⟨σv2⟩3→2 = γ3→2 +n3eq +. +(A.5) +For annihilations the momentum integrations in γann can be partially done, and the final +expression is given in terms of one remaining integral over the center of mass energy [54, 55]. +The thermally averaged cross section is +⟨σv⟩ann = +1 +2Tm2 +1m2 +2K2( m1 +T )K2( m2 +T ) +� ∞ +(m1+m2)2 ds K1( +√s +T )(pinE1E2vmølσ) +(A.6) +with the Møller velocity related to the flux factor as F = (E1E2)vmøl, and the factor +∆ann/Sα = 1 is set to unity. The equilibrium number density is defined as +neq +α = +Nα +(2π)3 +� +d3pαfeq +α = Nαm2 +αT +2π2 +K2(mα +T ) mα≫T += +Nα +�mαT +2π +�3/2 +e−mα/T , +(A.7) +with the last expression valid in the non-relativistic limit. For m1 = m2 ≡ m we can rewrite +this in dimensionless variables +⟨σv⟩ann = +4x∆ +SαK2(x)2 +� ∞ +1 +d˜s +√ +˜s(˜s − 1)K1(2 +√ +˜sx)σ(˜s), +(A.8) +with ˜s = s/(4m2) and x = m/T. +– 18 – + +A.3 +S-channel resonance and narrow width approximation +If the DM interactions are mediated by a massive meditor particle – in our case, the dark +photon – there will be an s-channel resonance for momenta that the mediator is nearly on shell +s ≈ m2 +V . To incorporate this effect, we include the decay rate in the dark photon propagator +Dµν(P 2) = +−igµν +P 2 − m2 +V + iϵ → +−igµν +P 2 − (mV − i 1 +2Γ)2 ≈ +−igµν +P 2 − m2 +V + imV Γ +(A.9) +where we used that Γ2 ≪ m2 +V . Γ = Γd + Γv is the total decay width of the dark photon, +which is the sum of the decay rate into pions and decay rate into SM fermions, i.e. into the +dark and visible sector. In the resonance limit the most enhanced terms in the cross section +will be ∝ Dµν(s)2, which can be evaluated in the narrow width approximation +1 +(s − mV )2 + m2 +V Γ2 ≈ +π +mV Γδ(s − m2 +V ) + O(Γ2/m2 +V ). +(A.10) +B +Pion self-interactions +In this appendix we calculate the pion self-interaction cross section σSI = σ(ππ → ππ), which +has contributions from 4pnt self-interactions and from dark photon exchange. +B.1 +Amplitude +Dark photon mediated self-interaction +The 4pnt pion interaction follows from eq. (2.4) +L ⊃ +1 +3f2π +� +2πa∂πbπc∂πd − 2πaπb∂πc∂πd + m2 +ππaπbπcπd� � +Tr[T aT bT cT d] + perm. +� +. +(B.1) +There are 4! possible orderings of the pions a, b, c.d. Consider first the amplitude for the +{acbd}-term plus the cyclic permutations: +M4pnt +{acbd} = −4Tr[T aT cT bT d] +3f2π +� +(Pc · Pd + Pa · Pb) + 1 +2(Pb · Pd + Pc · Pb + Pa · Pc + Pd · Pa) − m2 +π +� += − 2 +f2π +Tr[T aT cT bT d] +� +s − 2m2 +π +� +, +(B.2) +where we took Pa, Pb as incoming momenta, and Pc, Pd as outgoing (∂πa → −iPa, ∂πc → iPc), +and on the 2nd line we used the Mandelstam variables +s = (Pa + Pb)2, +t = (Pa − Pc)2, +u = (Pa − Pd)2. +(B.3) +The results are similar for the other possible permutations, and the total amplitude is [27] +M4pnt +abcd = M(πaπb → πcπd) = − 2 +f2π +� +Tr[T aT bT cT d] + (b ↔ d) +� � +t − 2m2 +π +� +− 2 +f2π +� +Tr[T aT cT bT d] + (c ↔ d) +� � +s − 2m2 +π +� +− 2 +f2π +� +Tr[T aT cT dT b] + (b ↔ c) +� � +u − 2m2 +π +� +(B.4) +– 19 – + +Dark photon mediated self-interaction +The pion-dark photon interactions that follow +from the covariant derivatives in the chiral Lagangian eq. (2.4) can be written in the form +L ⊃ = −2igdVµ +� +πa(∂πb) − (∂πa)πb� +Tr +� +[T a, T b]Q] +� +. +(B.5) +The (V 2π)-vertex interaction and dark photon propagator (in Lorentz gauge) are then +Aµ +ab = 2igd(Pa − Pb)µTr([T a, T b]Q), +Dµν(P 2) = +−igµν +P 2 − m2 +V + iϵ +(B.6) +with both momenta Pa, Pb incoming. The amplitude for ππ → ππ scattering via dark photon +exchange is +MV +abcd = iAµ +abDµν(s)Aν +cd − iAµ +acDµν(t)Aν +bd − iAµ +adDµν(u)Aν +bc += 4g2 +d +� +(t − u) +s − m2 +V + iϵCabcd + +(s − u) +t − m2 +V + iϵCacbd + +(s − t) +u − m2 +V + iϵCadbc +� +(B.7) +with color factor +Cabcd ≡ Tr([T a, T b]Q)Tr([T c, T d]Q). +(B.8) +Resonant scattering arises for P 2 ≈ m2 +V , and we include the decay width in the propagator +eq. (A.9) to describe this. +B.2 +Cross section +The matrix element squared summed over final and averaged over initial states is | ¯ +M|2 +SI = +1 +N2π +� +abcd |Mabcd|2 with Nπ = (N2 +f − 1) the number of pions. The amplitude is the sum of +the 4pnt interaction eq. (B.4) and the photon exchange contribution eq. (B.7). The latter is +subdominant, except for momenta near the s-channel resonance s ≈ m2 +V ; we can then neglect +interference terms and the non-resonant contributions, and approximate the amplitude +| ¯ +MSI|2 ≈ +1 +N2π +� +abcd +� +|M4pnt +abcd|2 + |Mres +abcd|2� +(B.9) +with Mres = MV |s≈m2 +V the s-channel resonance contribution from the photon exchange +diagram. +4pnt self-interaction +Starting with the 4-pnt contribution, the amplitude squared can be +calculated using the Feyncalc Mathematica program [56–58] +| ¯ +M4pnt +SI +|2 = 6κSI +m4 +π +f4π +− κSI,2 +(st + tu + us) +f4π +, +(B.10) +with +κSI = +(3N4 +f − 2N2 +f + 6) +3N2 +f (N2 +f − 1) += 1 + O(N−1 +f ), +κSI,2 = +N2 +f +(N2 +f − 1) = 1 + O(N−1 +f ). +(B.11) +– 20 – + +The cross section eq. (A.1) for pion scattering is +σ4pnt +SI += +m4 +π +πSff4πs +�6κSI +16 + κSI,2(m2 +πp2 + 5 +6p4) +� +≈ 3κSIξ4 +64πm2π +(B.12) +with Sf = 2 for two identical particles in the final state, and p = |p| the incoming momentum +of either pion in the CM frame. +The last expression is valid in the non-relativistic limit +p2 ≪ m2 +π in which the velocity-independent s-wave contribution dominates, and s ≈ 4m2 +π. +The result agrees with [28], but differs a factor of 8 with [4] (after rescaling fthem +π += 2fπ). +For Nf = 2 it reproduces the results of [27] (except for the sign in front of the 5 +6p4 term) but +not for other Nf. +Dark photon mediated self-interaction +The diagram for photon exchange is negligble +except near the s-channel resonance, where the amplitude can be approximated by the s- +channel diagrams, and +| ¯ +Mres +SI |2 = 256C2 +4g4 +d +N2π +p4 cos2 θ +(s − m2 +V )2 + m2 +V Γ2 +⇒ +σres +SI = +16C2 +4g4 +d +3πsSfN2π +p4 +(s − m2 +V )2 + m2 +V Γ(p)2 +(B.13) +with Sf = 2 amd Γ(p) the running photon decay width, computed in appendix C. The color +factor eq. (B.8) summed over flavors is C2 +4 = � |Cabcd|2 = 42 (82) for Nf = 3 (4). +The cross section can be rewritten in the more familiar Breit-Wigner form. To do so, +expand the momentum as p = 1 +2mπv = 1 +4mV v + O(δm) with v the relative velocity. The +resonance velocity follows from eq. (C.4), which gives vR ≈ 2 +√ +δm for dark photon masses +eq. (2.3). We further define the velocity dependent and resonant kinetic energies [29] +E(v) = p2 +mπ += 1 +4mπv2, +E(vR) = mV − 2mπ = mπδm. +(B.14) +We can then rewrite the resonant cross section in Breit-Wigner form +σres +SI = +4πS +mπE(v) +Γd(v)2/4 +(E(v) − E(vR))2 + Γ(v)2/4, +S = 3Sf +N2π +(B.15) +where we used the non-relativistic approximation s = m2 +V + O(v2). The numerator is written +in terms of the decay width into dark pions Γd using the explicit expression eq. (C.3). +Ref. [29] list a numerical factor S = NV /N 2 +π for the ratio of mulitplicities of the resonance +dark photon and DM particles; we find here an additional Sf = 2 symmetry factor. +B.3 +Thermally averaged cross section +The thermally averaged self-interaction cross section is [29] +⟨σv⟩SI = σ4pnt +SI +⟨v⟩ + ⟨σres +SI v⟩ = σ4pnt⟨v⟩ + +� vmax +0 +f(v, v0)(σres +SI v)dv, +(B.16) +with the velocity distribution f(v, v0) = (4v2)/(√πv3 +0)e−v2/v2 +0 taken as a Maxwell-Boltzmann +distribution truncated at the halo escape velocity vmax, and v0 a constant implicitly defined +– 21 – + +via ⟨v⟩ ≃ 2v0/√π. Here we used that the self-interaction cross section is the sum of the (dom- +inantly) s-wave 4pnt interaction and s-channel resonance contribution eqs. (B.12) and (B.15). +The resonance contribution can be calculated in the narrow width approximation eq. (B.14): +⟨σv⟩res +SI = 64π3/2S Γd(vR)Bd(vR) +m3π +e−v2 +R/v2 +0 +v3 +0 +. +(B.17) +with Bd = Γd/Γ the branching ratio for decay into dark sector pions. The decay rate into +dark pions (p-wave) eq. (C.3) and SM fermions (s-wave) eq. (C.7) can be parameterized as +Γi(v) = mV γivni with i = d, v for decay into dark and visible sectors, and mV ≈ 2mπ. This +factors out the explicit velocity dependence of the (ΓdBd)-factor in the thermally averaged +cross section. +C +Dark photon decay rate +The dark photon can decay into dark pions, and in SM fermions and pions via kinetic mixing +with the SM photon. +C.1 +Dark photon decay rate into dark pions ΓV →ππ. +The dark photon decay rate in dark sector particles is Γd = Γ(V → ππ). The decay into dark +pions is mediated by the vertex interaction eq. (B.6), with amplitude +Mµ +ab = −2gd(Pa − Pb)µCab, +Cab = Tr([T a, T b]Q) +(C.1) +with Pa, Pb the 4-momenta of the outgoing pions. The amplitude squared (summed over final +states, and averaged over intial photon polarizations states) in the CM frame is +| ¯ +M|2 +d = −1 +3gµν +� +ab +Mµ +abMν∗ +ab = 16C4 +3 +g2 +dp2 +out +(C.2) +with pout = |pa| = |pb| the final state 3-momentum, and color factor C2 +2 = � +ab |Cab|2 = C4. +The decay rate becomes +Γd(pout) = +pout +Sf8πm2 +V +| ¯ +M|2 +d = 8C4αdp3 +out +3Sfm2 +V +(C.3) +with αd = g2 +d/(4π) and Sf = 2 for two identical particles in the final state. +The resonance momentum (taking the intial state photon at rest) is +p2 +R = 1 +4m2 +V +� +1 − 4m2 +π +m2 +V +� += m2 +πδm + O(δm2) +(C.4) +where we used the parameterization eq. (2.3) in the last step, and assumed the dark photon +mass is close to resonance δm2 ≪ m2 +π. Evaluating the decay rate at resonance pout = pR gives +Γd(pR) = 2C4αd +3Sf +mπδm3/2. +(C.5) +– 22 – + +C.2 +Dark photon decay rate in SM particles. +The dark photon can decay into SM electrons via kinetic mixing and Γs = Γ(V → ¯ff) with +f the electron; for large enough DM mass the decay into muons and charged SM pions also +becomes kinematically accessible. The decay rate into SM pions will be of the form eq. (C.3) +with αd → ϵ2α and instead of C4 → CQCD +4 += 1/2 the appropiate color factor for QCD. It is +velocity suppressed compared to decay into fermions, which is s-wave, and we neglect it in +the following. +The amplitude for dark photon decay into a SM fermion is +Mµ +v = (ieϵ)¯u(P2)γµv(P1) +(C.6) +with P1, P2 the 4-momenta of the outgoing fermions, ϵ the kinetic mixing paramer appearing +in the Lagrangian eq. (2.4), and e the electric charge of the electron. The amplitude squared +(summed over final states, and averaged over photon polarizations) in the CM frame and +decay rate are +| ¯ +M|2 +v = 1 +3(eϵ)24m2 +V +⇒ +Γv = αϵ2 +Sf3πmV +(C.7) +with α = e2/(4π), symmetry factor Sf = 2 for two identical particles in the final state, and +s = m2 +V . +Decay into the darks sector pions dominates and Γd ≫ Γv or +ϵ ≲ +� +παdC4 +α +δm3/4 = 8.5 × 10−6 +�C4 +4 +� �αd +α +� +δm +3 × 10−8 +�3/4 +, +(C.8) +where we set v → vR. +D +Dark pion annilation and scattering +Dark pions can annihilate into and scatter off SM fermions. These processes are important +for the final relic density and for keeping the dark sector in thermal equilibrium with the SM +sector. +D.1 +Dark pion annihiliation into SM fermions +Consider the annihiliation of dark pions into Standard Model electrons πa(P1)πb(P2) → +¯f(K1)f(K2). For large enough dark pion mass, also the decay channel into muons (mπ > +105 MeV) and SM charged pions (mπ > 139.6 MeV) opens up. +The amplitude for annihilation in a SM Dirac fermion pair ¯ff is mediated by the vertex +eq. (B.6) and the coupling to the SM fermion current via kinetic mixing. +iMππ→ ¯ff = −i2gD(P1 − P2)µTr([T a, T b]Q) +−i +s − m2 +V + iϵ(ieϵ) +� +f +¯u(K1)qfγµv(K2) +(D.1) +– 23 – + +Averaging over intitial states and summing over the spins of the final state fermions gives +|M|2 +ππ→ ¯ff = +(2gDeϵ)2C4 +N2π(s − m2 +V )2 +� +spin +� +¯u(K1)(/P 1 − /P 2)v(K2)¯v(K2)(/P 1 − /P 2)u(K1) +� += +(2gDeϵ)2C4 +N2π(s − m2 +V )2 32p2 � +k2(1 − cos2 θ) + m2 +f +� +(D.2) +with color factor C4 = C2 +2 ≡ � +ab |Tr([T a, T b]Q|2. Here we denoted the CM 3-momentum of +the incoming pions with p and that of the outgoing fermions with k, and θ the scattering +angle. The cross section eq. (A.1) becomes +σππ→ ¯ff = 4Aann +� +s − 4m2π +� +s − 4m2 +f(s + 2m2 +f) +s(s − m2 +V )2 +mf→0 += +Aann +m2π +√˜s − 1 +√ +˜s +(˜s − m2 +V /(4m2π))2 +(D.3) +with Aann = 4πC4ϵ2αDα/(3N2 +π). In the last line we neglected the SM fermion mass, and +rewrote the cross section in terms of the dimensionless variable ˜s = s/(4m2 +π). +The amplitude for annihilation in a pair of SM pions π± +SM with is +iMπaπb→πc +SMπd +SM = i2gD(P1 − P2)µTr([T c, T c]Q) +1 +s − m2 +V + iϵ(eϵ)(K1 − K2)µTr([T a, T b]QQCD) +(D.4) +Averaging over intitial states and summing over the spins of the final state pions – c, d running +over the generators corresponding to π± +SM – gives +|M|2 +ππ→2πSM = (2gDeϵ)2C4CQCD +4 +N2π(s − m2 +V )2 +16k2p2 cos2 θ +(D.5) +with color factor CQCD +4 += 1/2. The cross section can then be expressed as +σππ→2πSM = CQCD +4 +4 +� +1 − +4mπ2 +SM +s3/2 +�3/2 +σππ→e¯e +(D.6) +where we neglected the electron mass. +Substituting eq. (D.3) in the thermally averaged cross section eq. (A.8) for annihilation +into SM electrons is +⟨σv⟩ππ→¯ee = ∆ann4xAann +Sαm2πK2(x)2 +� ∞ +1 +d˜s ˜s(˜s − 1)3/2K1(2x +√ +˜s) +(˜s − m2 +V /(4m2π))2 +(D.7) +with ∆ann/Sα = 1. We are interested in the thermally averaged cross section during freeze- +out, in the limit x = mπ/T ≫ 1. This will be dominated by the resonance contribution +which we can calculate in the narrow width approximation eq. (A.10). Including the decay +– 24 – + +width in the propagator eq. (A.9) and defining ˜Γ = Γ/(2mπ), ˜mV = mV /(2mπ) the thermally +averaged cross section eq. (D.7) becomes +⟨σv⟩res +ππ→¯ee ≈ +4xAann +m2πK2(x)2 +� ∞ +1 +d˜s˜s(˜s − 1)3/2K1(2x +√ +˜s) +π +˜mV ˜Γ +δ(˜s − ˜m2 +V ) +x≫1 +≈ 8√πx3/2Aann +mπΓ +δm3/2e−δm x +(D.8) +where in the last step we used that K2(x) = K1(x) = +� +π/(2x)e−x + ... for x ≫ 1, and +expanded ( ˜m2 +V − 1) ≈ δm. +Now expand the dark photon mass in small δm defined in eq. (2.3), and use the explicit +expression for the decay width eq. (C.5) to get +⟨σv⟩res +ππ→¯ee = 32π3/2ϵ2αx3/2Bd +N2πm2π +e−δm x + O(δm) +(D.9) +with Bd = Γd/Γ the branching ratio for decay into the dark sector. We can then write the +full thermally averaged annihilation cross section as +⟨σv⟩ann = gann +32π3/2ϵ2αx3/2Bd +N2πm2π +e−δm x + O(δm) +(D.10) +with gann incorporating the degrees of freedom the dark pion can annihilate in. Below the +muon threshold this is only electrons and gann = 1. Including muons and SM pions +gann = 1+Θ(mπ−mµ) +� +1 − m2µ +m2π +� +1 + m2 +µ +2m2π +� ++Θ(mπ−mπSM)CQCD +4 +4 +� +1 − m2 +πSM +m2π +�3/2 +(D.11) +with Θ the Heaviside step function. +D.2 +Pion-electron scattering +Consider scattering of dark pions with SM particles, which for light DM is dominated by +electron scattering via the reaction πa(P1)f(K1) → πa(P2)f(Kf). +iMπf→πf = −i2gd(P1 + P2)µTr([T a, T b]Q) +−i +t − m2 +V + iϵ(ieϵ)¯u(K2)γµu(K1) +(D.12) +Averaging over intitial and summing over final states gives +|M|2 +πf→πf = +1 +2Nπ +(2gDeϵ)2C4 +(t − m2 +V )2 +� +spin +� +¯u(K2)(/P 1 + /P 2)u(K1)¯u(K2)(/P 1 + /P 2)u(K2) +� +(D.13) +The spinor sum now becomes +� +(...) = 4 +� +2(P1 + P2).K1(P1 + P2).K2 − (P1 + P2)2(K1.K2 − m2 +f) +� +≈ 16m2 +πp2(1 + cos θ) + O(p2) +(D.14) +– 25 – + +with p the CM 3-momenta of the incoming particles and k of the outgoing particles, and θ the +scattering angle between the ingoing and outgoing pion. In the last step we set the fermion +mass to zero and took the non-relativistic limit. The non-relativistic cross section becomes +[5] +σπf→πf = 8C4αDαϵ2m2 +πp2 +Nπsm4 +V +� +dΩ (1 + cos θ) = ϵ2Ascat +p2 +m4π +(D.15) +with Ascat = 2πC4αDα/Nπ and mV ≈ 2mπ. +The total scattering cross section is written as +σscat = gscatϵ2Ascat +p2 +m4π +, +(D.16) +with gscat = 1 + Θ(mπ − mµ) + Θ(mπ − mπSM)CQCD +4 +, where for simplicity we have neglected +the muon and SM pion masses. +E +Cross section for 3 → 2 dark pion interactions +The 3 → 2 dark pion amplitude has a contribution from the 5pnt pion interaction and from +dark photon exchange. The necessary vertices with an odd number of pions arise from the +WZW Lagrangian eq. (2.4). +E.1 +Pion 5pnt interaction from the WZW-term +The 5pnt pion interaction from the WZW term can be written as [4] +LWZW ⊃ A5 +f5π +ϵµνρσTr [π∂µπ∂νπ∂ρπ∂σπ] , +A5 = 2Nc +15π2 +(E.1) +with Nc the number of colors of the dark QCD-like gauge group. +There are 5! different +contributions to the amplitude Mabc→de. We can group them by their momentum dependence +iMabc→de = iA5 +f5π +ϵµνρσ� +P a +µP b +νP c +ρP d +σfe + P b +µP c +νP d +ρ P e +σfa + P c +µP d +ν P e +ρP a +σ fb ++ P d +µP e +ν P a +ρ P b +σfc + P e +µP a +ν P b +ρP c +σfd +� +(E.2) +with fa coefficients that each are the sum of 4! terms, and all momenta are taken as incoming. +The color-coefficient fe is defined as +fe = Teabcd − Teabdc − Teacbd + Teacdb + Teadbc − Teadcb +(E.3) +with +Teabcd ≡ Tr +� +T eT aT bT cT d� ++ cycl. of {a, b, c, d} += Tr +� +T eT aT bT cT d� +− Tr +� +T eT bT cT dT a� ++ Tr +� +T eT cT dT aT b� +− Tr +� +T eT dT aT bT c� +(E.4) +– 26 – + +That is, fe is the sum of all traces of five generators with Te fixed in the first position, and all +possible permutations of the other generators (of the {a, b, c, d} indices); the sign of each term +is determined by the number of permutations away from the {a, b, c, d, e}-sequence. Likewise, +all other fi coefficients can be defined. +The matrix element squared averaged over initial and summed over final states is (the +calculation is done with the Mathematica package FeynCalc) +| ¯ +M3→2|2 = +1 +N3π +� +abcde +|Mabc→de|2 = +A2 +5Nf(N2 +f − 4)(N2 +f − 1) +N3π +F(Pi) +f10 +π +≡ +¯A2F(Pi) +f10 +π +(E.5) +with Nπ = N2 +f − 1 the number of pions and Nf the number of flavors. F is a complicated +momentum-dependent function, which we will evaluate for non-relativistic incoming mo- +menta. We parameterize the momenta in the CM frame (we set P d → −P d and P e → −P e to +make them outgoing momenta with positive energy as the zeroth component of the 4-vector): +P a = (E1, p1, 0, 0), +P b = (E2, p2 cos θ, p2 sin θ cos ϕ, p2 sin θ sin ϕ), +P c = (E3, −p1 − p2 cos θ, −p2 sin θ cos ϕ, −p2 sin θ sin ϕ), +P d = (1/2(E1 + E2 + E3), p4 cos ¯θ, p4 sin ¯θ cos ¯ϕ, p4 sin ¯θ sin ¯ϕ), +P e = (1/2(E1 + E2 + E3), −p4 cos ¯θ, −p4 sin ¯θ cos ¯ϕ, −p4 sin ¯θ sin ¯ϕ), +(E.6) +The momentum p1 is aligned with the z-axis. Ω2 and Ω4 are then the solid angle of p2 and +p4 respectively. p3 and p5 are fixed in center of mass frame, and E4, E5 are fixed by energy +conservation. The energies are Ei = +� +p2 +i + m2π for i = 1, 2, E2 +3 = (p2 +1 +2p1p2 cos θ +p2 +2)+m2 +π, +and +� +p2 +4 + m2π = 1/2(E1 + E2 + E3). We can thus express Ei, p4 in terms of p1, p2. +To get the leading order term in the limit of non-relativistic momentum for the incoming +particles set pi = ϵpi for i = 1, 2 and expand in small ϵ. The first non-zero term arises at 4th +order: F = ϵ4F4(p1, p2, θ, ϕ, ¯θ, ¯ϕ) + O(ϵ6) with +F4 = 3375m4 +πp2 +1p2 +2 +16 +sin2(θ) sin2(¯θ) sin2(φ − ¯φ) +(E.7) +The transition amplitude eqs. (A.3) and (A.4) then becomes +γ3→2 = +¯A2 +SαSβ(2π)11f10 +π +� �d3p3d3p4 +� +i(2Ei) +� +p2 +1dp1p2 +2dp2N3 +πfeq +1 feq +2 feq +3 δ(Eα − Eβ) +� +dΩd¯ΩF4 += +125 +√ +5 ¯A2mπ +1024π5SαSβf10 +π +� +d3p3 +(2π)3 Nπfeq +3 +� +dp1p4 +1Nπfeq +1 +� +dp2p4 +2Nπfeq +2 +(E.8) +with Sα = 3! and Sβ = 2! to account for identical particles in initial and final states. The +integral +� +dΩd¯ΩF4 = 750π2m4 +πp2 +1p2 +2. +To get the final expression, we further used that in +the non-relativistic limit Ei ≈ mπ for i = 1, 2, 3 and Ei ≈ 3 +2mπ for i = 4, 5, which yields +– 27 – + +� +i(2Ei) ≈ 2332m5 +π. Moreover +� +d3p4δ(Eα − Eβ) = +3 +2 +√ +5 +� +d3p4δ(p4 − +√ +5/2mπ) = 3π +√ +5/2m2 +π. +The integrations over the phase space densities give in the non-rel limit +Nπ +� +d3p3 +(2π)3 feq +3 = neq +π , +Nπ +� +dp1p4 +1feq +1 = 6π2mπTneq +π +(E.9) +The thermally averaged cross section eq. (A.5) is then +⟨σv2⟩5pnt +3→2 = α3→2 +x2m5π +, +α3→2 = N2 +c κ3→2 +Nf +5 +√ +5ξ10 +1536π5 , +κ3→2 = +N2 +f (N2 +f − 4) +(N2 +f − 1)2 += 1 + O(1/Nf) +(E.10) +with x = mπ/T and ξ = mπ/fπ. This result matches the result in Ref [4] (taking into account +the different definitions fthem +π += 2fπ). +E.2 +Dark photon interactions from WZW term +In the presence of a dark photon, there are additional diagrams with photon exchange con- +tributing to the 3 → 2 cross section. The WZW term also contains photon interactions with +an odd number of pions eq. (2.4). The A(3π) and (2A)π interactions are +LWZW ⊃ Ncgd +3π2f3π +ϵµνρσAµP a +ν P b +ρP c +σTQabc − Ncg2 +d +4π2fπ +ϵµνρσP (Aν) +µ +P a +σ AνAρπaTQa +(E.11) +with TQabc = Tr +� +QT aT bT c� +and TQa = Tr +� +Q2T a� +. +Including the 5pnt pion interaction +discussed in the previous subsection and the A(2π)-interaciton from the chiral Lagrangian, +the relevant photon-pion couplings vertices are +Aabcde ≡ iA5 +f5π +¯ +Aabcde = iA5 +f5π +ϵµνρσ� +P a +µP b +νP c +ρP d +σfe + P b +µP c +νP d +ρ P e +σfa + P c +µP d +ν P e +ρP a +σ fb ++ P d +µP e +ν P a +ρ P b +σfc + P e +µP a +ν P b +ρP c +σfd +� +Aµ +abc ≡ iA3 +f3π +¯ +Aµ +abc = iA3 +f3π +ϵµνρσP a +ν P b +ρP c +σ (TQabc + TQbca + TQcab − TQacb − TQcba − TQbac) +Aνρ +a ≡ −iA1 +fπ +¯ +Aνρ +a = −iA1 +fπ +ϵµνρσ(P (Aν) +µ +− P (Aρ) +µ +)P a +σ TQa +Aµ +ab ≡ iA2 ¯ +Aµ +ab = iA2π(Pa − Pb)µTr([T a, T b]Q) +(E.12) +with +A5 = 2Nc +15π2 , +A3 = Ncgd +3π2 , +A1 = Ncg2 +d +8π2 , +A2 = 2gd +(E.13) +and all momenta are taken as incoming. For the Aabc vertex we used that there are 3! different +terms, corresponding to abc + cycl.. We have symmetrized the Aa vertex in the two photon +legs. Q2 = 1 For our choice of charge matrix eq. (2.2), and thus TQa = Tr(Q2T a) = Tr(T a) = +0, and the (2A)π interaction vanishes Aµρ +a = 0. +The propagator is +Dµν(P) = +−igµν +P 2 − m2 +V + imV Γ = −igµν +∆(P) = +−igµν +(4m2π) ˜∆(P) +(E.14) +where in the last step we introduced the dimensionless propagator ˜∆ = ∆/(4m2 +π). +– 28 – + +E.2.1 +Amplitude +The photon interactions give rise to 6 additional diagrams contributing to the 3 → 2 in- +teractions [28]. For our charge matrix eq. (2.2) the Aµρ +a += 0 vertex vanishes, and only the +first three diagrams contribute. We will calculate the amplitude for Nc = Nf = 3. The full +amplitude is +iM = Aabcde + a1 +−iAµ +abcAdeµ +∆(de) ++ a2Pabc +�−iAµ +abAcdeµ +∆(ab) +� ++ a3Pabc;de +�−iAµ +ceAabdµ +∆(ce) +� +(E.15) +with ∆(ij) = ∆(Pi + Pj) the propagator of momenta Pi and Pj. Pabc means all cyclic per- +mutations of (abc) – hence there are three diagrams contributing to a2 –, and Pabc;de cyclic +permutations of (abc) times cyclic permutations of (de) – hence there are six diagrams con- +tributing to a3. The ai are included as note keeping devices and can be set to unity at any +time during the calculation.The amplitude in terms of the ¯ +A-vertices becomes +M = A5 +f5π +� +¯ +Aabcde + f2 +πA2A3 +(4m2π)A5 +� +a1 +¯ +Aµ +abc ¯ +Adeν +˜∆(de) ++ a2Pabc +� ¯ +Aµ +ab ¯ +Acdeν +˜∆(ab) +� ++ a3Pabc;de +� ¯ +Aµ +ce ¯ +Aabdν +˜∆(ce) +�� � +(E.16) +Diagram a2 has an s-channel resonance for mV ≈ 2mπ, as the propagators ˜∆ij with i, j = +a, b, c go nearly on shell. Diagram a1 can become resonant for mV ≥ 3mπ, and the propagator +˜∆de can be put on shell. This case was analysed in [59]. We will here not consider it any +further, and instead focus on lighter dark photon masses. +Using the same momentum parameterization as before eq. (E.6), the amplitude squared +can be written as +| ¯ +M|2 = 3375 ¯A2m4 +π +16f10 +π +p2 +1p2 +2 sin2(θ) sin2(¯θ) sin2(φ − ¯φ)(1 + X) = | ¯ +M|2 +5pnt(1 + X) +(E.17) +with as before ¯A2 = A2 +5N−2 +π Nf(N2 +f −4) +Nf=3 += +15 +64A2 +5, and X parameterizing the photon-exhange +corrections. +E.3 +Resonance contribution from mV ≈ 2mπ +In the limit that mV ≈ 2mπ the propagators ˜∆(ij) with i, j = a, b, c are resonantly enhanced. +The resonance contribution is dominated by the ˜∆−2 +(ij) terms in the amplitude squared pro- +portional to a2 +2. Dropping all subdominant terms the correction eq. (E.17) becomes +Xres = +�παd +ξ2 +�2 +a2 +2 +� +�128 +45 +� +I +1 +˜∆2 +(I) +− 4 +27 +� +I̸=J +1 +˜∆(I) ˜∆(J) +� +� +(E.18) +with I, J = ab, bc, ca. Defining +F(h) ≡ +� +dp1 p4 +1Nπfeq +1 +� +dp2 p4 +2Nπfeq +2 +� +d cos θ sin2(θ)h(p1, p2, cos θ) +(E.19) +– 29 – + +The transition amplitude can be written as +γres +3→2 +γ5pnt +3→2 += F(Xres) +F(1) +, +F(1) = 6πN2 +πe−2xm10 +π +x5 +(E.20) +Consider first the ˜∆−2 +(I) terms, which can be evaluated in the narrow width approximation +eq. (A.10) +1 +˜∆2 +(I) += +(4m2 +π)2 +(sI − m2γ)2 + m2 +V Γ2 ≈ π(4m2 +π)2 +mV Γ +δ(sI − m2 +V ) +(E.21) +with I = ab, bc, ca and sab = (Pa + Pb)2 etc. The x = cos θ integral in F( ˜∆−2 +(I)) becomes of +the form +� +dx (1 − x2)δ(sI − m2 +V ) = +� (1 − x2 +0) +|s′ +I(x0)| , +(E.22) +where the sum is over the roots x0 of sI − m2 +V = 0. It will be useful to redefine the momenta +(ab) : +p± = +1 +√ +2(p1 ± p2); +(bc) : +p± = +1 +√ +2(p1 ± 2p2); +(ac) : +p± = +1 +√ +2(2p1 ± p2). +(E.23) +for I = ab, bc, ac respectively. Then for all I we get +|x0| = p2 ++ + p2 +− − 4m2 +πδm +p2 ++ − p2 +− +, +|s′ +I(x0)| = p2 ++ − p2 +−, +|p+| ≥ mπ +√ +2δm ≥ |p−| +(E.24) +up to O(δm2) corrections. +The constraint on the momentum range arises from requiring +| cos θ| ≤ 1; only small p−-momenta can hit the resonance. A further suppression comes from +the (1 − x2 +0) ∝ δm in eq. (E.22), in the limit that |p−| ≪ p+. Putting it all together +F( ˜∆−2 +I ) = π(4m2 +π)2 +mV Γ +� +dp1 p4 +1Nπfeq +1 +� +dp2 p4 +2Nπfeq +2 +(1 − x2 +0) +|s′ +I(x0)| +p+≫|p−| +≈ +ρI +8πm6 +πδm +mV Γ +� ∞ +dp+p4 ++ +� mπ +√ +2δm +−mπ +√ +2δm +dp−N2 +πfeq +1 feq +2 += ˜ρI +48π√πN2 +πm12 +π (δm)3/2e−2x +mV Γx5/2 += ˜ρI +36SfBdπ√πN2 +πm10 +π e−2x +C4αdx5/2 +(E.25) +On the 2nd line ρI = 1 for I = ab and ρI = 2−5 for I = bc, ac; the suppression of the latter +terms come from the factors of 2 in the definition of p± in eq. (E.23) (including a factor 1/2 +from the Jacobian). On the last line ˜ρi = 1 for I = ab, and ˜ρi = ρ1(128/25) +� +2/5 ≈ 0.1 for +I = bc, ac, with the additional factor for I = bc, ac arising from the different p±-dependence +of feq +i . The final expression uses the explicit decay width eq. (C.5) into pions, Bd = Γd/Γ the +branching ratio, and mV ≈ 2mπ. A careful inclusion of the integration boundary replaces (to +first order in δm) +e−2x → e−2x +√ +˜s = e−2x−δmx, +(E.26) +– 30 – + +which suppresses the interactions when the center of mass energy drops below the temper- +ature. This is the same exponential factor as found in the thermally averaged annihilation +cross section eq. (D.9). +The I = bc, ac contributions are subdominant. Expecting the mixed terms in eq. (E.18), +which already come with a small coefficient, likewise to be subdominant, we can approximate +the resonant interaction by the ˜∆−2 +(ab)-term. 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Seo, Cosmic abundances of SIMP dark matter, JHEP 04 +(2017) 154 [1702.07860]. +– 34 – + diff --git a/V9E3T4oBgHgl3EQfbQol/content/tmp_files/load_file.txt b/V9E3T4oBgHgl3EQfbQol/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4adce10b70a6fd64361b1c41ee5e595a5f40eca --- /dev/null +++ b/V9E3T4oBgHgl3EQfbQol/content/tmp_files/load_file.txt @@ -0,0 +1,1386 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf,len=1385 +page_content='Prepared for submission to JHEP Nikhef-2022-025 SIMPly add a dark photon Pieter Braata,b and Marieke Postmaa,c aNikhef, Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands bInstitute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Nether- lands cInstitute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyen- daalseweg 135, Nijmegen, the Netherlands E-mail: pbraat@nikhef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='nl, mpostma@nikhef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='nl Abstract: Pions of a dark sector gauge group can be strongly interacting massive particle (SIMP) dark matter, produced by the freeze-out of 3 → 2 interactions, with naturally large self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We study if adding a dark photon to the set-up can do it all: i) main- tain thermalization with the visible sector, ii) resonantly enhance the 3 → 2 interactions, thus allowing for a perturbative pion description, and iii) provide a velocity dependent self- interaction that can affect small scale structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We find that this is marginally excluded, as the required kinetic mixing is too small to maintain thermal equilibrium with the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Dropping the small scale structure requirement iii), a viable setup is reproduced for dark charges of αd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='01 − 1 and a dark pion mass mπ ≥ 30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Late time annihilations are non-negligible making the SIMP dark pion a bit WIMPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='04513v1 [hep-ph] 11 Jan 2023 Contents 1 Introduction 1 2 Lagrangian 2 3 Self-interactions 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Bullet cluster bound 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Resonant self-interactions 5 4 Relic density 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 SIMP freeze-out 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Annihilation scenario 8 5 Kinetic mixing 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Thermal equilibrium between the dark and visible sector 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Annihilations subdominant during freeze-out 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 CMB bound and other bounds on kinetic mixing 10 6 SIMP scenarios 11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Self-interacting resonant SIMP DM 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 SIMP DM 14 7 Conclusion 16 A Cross sections 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Scattering/annihilation cross section 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Thermally averaged cross section 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 S-channel resonance and narrow width approximation 19 B Pion self-interactions 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Amplitude 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Cross section 20 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 Thermally averaged cross section 21 C Dark photon decay rate 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Dark photon decay rate into dark pions ΓV →ππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Dark photon decay rate in SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 23 D Dark pion annilation and scattering 23 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Dark pion annihiliation into SM fermions 23 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Pion-electron scattering 25 – i – E Cross section for 3 → 2 dark pion interactions 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Pion 5pnt interaction from the WZW-term 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Dark photon interactions from WZW term 28 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Amplitude 29 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 Resonance contribution from mV ≈ 2mπ 29 1 Introduction Despite ongoing experimental and theoretical efforts, the nature of DM remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' An attractive possibility is that DM is a thermal relic, and its abundance is determined by freeze-out from the thermal plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Although most attention has been on weakly interact- ing particles (WIMPs), their parameter space is increasingly constrained [1–3], and other explanations have come to prominence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' One such alternative is strongly interactive massive particles (SIMPs) [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the SIMP scenario, dark matter freeze-out occurs in a dark sector via 3 → 2 interactions, which are typically stronger than in the WIMP scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' As a result, SIMPs have a lower mass (typically MeV-GeV scale), to which direct detection experiments are less sensitive [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To avoid overheating the dark sector during freeze-out, a portal coupling maintains thermal equilibrium with the visible Standard Model sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Cosmological observations question the collisionless dark matter (CDM) paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For example, observations of dark matter halo density profiles do not match the expected NFW- profile [8, 9] of CDM [10–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This discrepancey is known as the cusp vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' core problem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' [16] for a recent review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Although the inclusion of baryonic effects in the numerical simulations may resolve the discrepancy [17–21], it is also possible that the resolution lies in the dark matter properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Indeed, dark matter with strong self-interactions naturally alleviate the problem by transferring heat from the inner to the outer parts of the halo, thus smoothening the density profile [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The required interactions are scale dependent – a factor 10 difference between galaxies and galaxy clusters – pointing to a velocity-dependent self-interaction [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The archetypical SIMPs are the pseudo Nambu-Goldstone bosons – the dark pions – of a condensed dark Yang-Mills theory, with the Wess-Zumino-Witten term providing the five-point interactions [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark pions have naturally large self-interactions, and may address the small scale problems of CDM as well [27], except that the interactions are not velocity dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Moreover, satisfying both the relic density and the self-interaction constraints (or more conservatively, the upper bound on the self-interactions from the Bullet cluster observations), is only possible for non-perturbatively large pion couplings, invalidating the chiral perturbation theory approach [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Both of these issues can be overcome with extra vector bosons in the model [28], and in this paper we will consider adding a massive dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For finetuned dark photon mass, almost twice the dark pion mass, the photon mediated self-interactions can be on resonance, thus giving rise to a velocity-dependent effect – 1 – [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In addition, the WZW-interactions may likewise be resonantly enhanced, bringing the freeze-out interactions back in the resonant regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In addition, the dark photon can maintain equilibrium with the SM through kinetic mixing [30, 31] with the SM photon [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In this case dark pion annihilations to SM final states should be included in the relic density calculations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The annihilation rate grows at late times, as the dark pions lose kinetic energy and the annihilation cross section gets more and more resonantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' As a result, even after freeze-out of the (resonantly enhanced) WZW interactions, the annihilations can still be important and affect the final relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Late time annihilations in photons and electrons are bounded by nucleosynthesis and cosmic microwave background observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In this paper we will study if one particle – the dark photon – can do it all, solve the small scale structure problems of CDM, enhance the freeze-out interactions such that the relic density is obtained for non-perturbative pion couplings, and maintain thermal equilibrium with the SM – this is dubbed the resonant self-interacting dark matter (RSIDM) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We also include collider bounds on dark photon kinetic mixing and millicharged particles, as well as cosmological constraints from nucleosynthesis and the cosmic microwave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We find that RSIDM cannot affect small scale structure formation while maintaining thermal equilibrium with the SM – although this possibility is only marginally excluded and more precise calculations may be needed to make definite statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We will also study how parameter space opens up if the requirement that dark matter affects small scale structure formation is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Section 2 introduces the dark sector, with the dark pions and dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This is followed by a discussion of the dark matter self-interactions and Bullet cluster bound in section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' analytical estimates for the freeze-out temperature and final relic density, including both WZW interactions and annihilations, in section 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' and the constraints on the kinetic mixing parameter in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In section 6 we then discuss the parameter space for which the the relic density can be obtained in a perturbative set-up, the self-interactions can be resonantly enhanced to address the small scale structure problems of CDM, and the dark photon can keep the dark and visible sector in thermal equilibrium during freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In addition to the analytical estimates we will also provide numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We end with concluding remarks in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For completeness, we have also added the computation of the various (thermally averaged) cross sections in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Our results for the WZW and pion self-interactions agree with the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' new is the photon mediated resonant contributions to the various cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 2 Lagrangian Strongly interacting massive particles (SIMPs) freeze out via 3 → 2 dark matter interactions [7, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The large required number changing interactions can naturally be obtained in a dark sector with a non-abelian symmetry, with dark pions playing the role of the DM [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In this paper we study the phenomenology of this set-up if we add a dark photon [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark – 2 – photon can provide a portal between the dark and SM sectors, and – for tuned masses – can resonantly enhance the dark pion interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We thus consider a dark sector with an SU(Nc) × U(1) gauge symmetry, with Nf dark quarks in the fundamental representation of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The quark mass matrix is assumed diagonal M = mq1, allowing for a Wess-Zumino-Witten (WZW) term in the action for Nc ≥ 3 colors [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' A dimension-four kinetic mixing operator connects the dark U(1) group with the SM hypercharge [30, 31], and provides a portal between the dark and visible sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' At a scale Λ the non-abelian gauge group condenses, and the approximate flavor symmetry of the light left- and right-handed quarks is broken down to the diagonal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The (N2 f − 1) dark pions are the pseudo-Goldstone bosons of the this symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' They can be naturally lighter than the scale Λ, which sets the mass of the baryons in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Depending on their couplings to other sectors, including the SM, the dark pions can be stable on cosmological timescales and thus are a good dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' At energies below the condensation scale the effective action is S = � d4x �f2 π 4 Tr(DµU)†DµU + ζf3 π 2 Tr(MU + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=') − 1 4V 2 µν − 1 2mV V 2 µ − ϵVµJµ SM � + ΓWZW (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) with ζ = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The first two terms are the leading order operators of the chiral effective Lagrangian describing the dark pion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Here U = e2iπ/fπ, π = πaT a with T a gen- erators of SU(Nf), and fπ ∼ √NcΛ/(4π) the pion decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The covariant derivative is DµU = ∂µU + igd[Q, U]Vµ, with Vµ the dark photon field and gd the dark U(1) gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We choose the charge matrix [6, 33] Q = Diag(1, −1, 1, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) with Nf entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' With this charge assignment Tr(Q2T a) = 0 and the mixed anomaly vanishes, avoiding decay of the neutral dark pion into to two (dark) photons [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The next two terms are the dark photon kinetic term, where we defined the gauge field strength Vµν = ∂µVν−∂νVµ, and the St¨uckelberg mass term for the dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To avoid pion decay the dark photon mass should exceed twice the pion mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' in the limit that the photon mass is close to that threshold, the photon-pion interactions can be resonantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We parameterize1 mV = mπ(2 + δm) > 2mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) The last term in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) between the square brackes couples the dark photon to the SM vector current Jµ SM = � qf ¯fγµf, and qf the electric charge of the SM fermion f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This term arises as a consequence of kinetic mixing between the dark U(1) group and SM hypercharge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' after redefining the fields to make the kinetic terms canonical, and diagonalizing the mass 1Here mπ is the mass of the charged – with unit charge – dark pions, which interact with the dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The mass of the charged dark pions receive loop corrections, and is slightly larger than the mass of the neutral pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 3 – matrix, the result is the coupling in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Here we used that ϵ ≪ 1 and have dropped the O(ϵ2) terms, and we have neglected the coupling to the Z-boson, valid if the dark photon mass is small compared to the electroweak scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Finally, the last term is the WZW action, which is present if the 5th homotopy group of the coset space π5(G/H) is non-trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' this is the case for Nf ≥ 3 mass degenerate flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Expanding the action in pion fields, the Lagrangian is the sum of the Lagrangian for chiral perturbation theory (χPT), the dark photon Lagrangian, and the terms from the WZW action: L = LχPT + LV + LWZW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The relevant dark pion and and dark photon interactions are: LχPT = Tr(∂π)2 − m2 πTr(π2) + 1 3f2π Tr � (2π∂π)(π∂π) − 2(ππ)(∂π∂π) + m2 ππ4� + 2igdV µTr ((∂µπ)[Q, π]) , LV = −1 4V 2 µν − 1 2mV V 2 µ − ϵVµ � f qf ¯fγµf LWZW = 2Nc 15π2f5π ϵµνρσTr [π∂µπ∂νπ∂ρπ∂σπ] − i Ncgd 3π2f3π ϵµνρσVµTr (Q∂µπ∂ρπ∂σπ) , + Ncg2 d 4π2fπ ϵµνρσ(∂µVν)VρTr � Q2∂σπ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) The first two terms in the chiral lagrangian are the kinetic and mass term for the pion fields, with mass m2 π = 2ζfπmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Chiral perturbation theory is perturbative for ξ ≡ mπ fπ ≲ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) The 3rd term gives the 4pnt pion interactions, and the last term the pion-dark photon coupling (V 2π interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The first two terms in the dark photon Lagrangian are the kinetic and mass term for the dark photon field, and the last term the coupling of the dark photon to the SM fermions from kinetic mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Finally, the WZW lagrangian contains the 5pnt pion interaction, and additional dark photon-pion couplings with an odd number of pions (V 3π and 2V π interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We have only included the most relevant, lowest dimensional operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 3 Self-interactions The dark pion can scatter via a 4-point contact interaction appearing in the χPT Lagrangian, and via the exchange of a dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The cross sections for these contributions are cal- culated in the non-relativistic limit in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The photon exchange contribution is subdominant, unless enhanced by an s-channel resonance which can appear for fine-tuned dark photon masses eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The cross section is can then be approximated by a sum of the velocity independent contact interaction and velocity dependent resonance contribution σSI ≈ σ4pnt SI + σres SI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 4 – For SIMP dark matter the self-interactions are naturally large, and the Bullet clus- ter observations [37–39] put a strong constraint on the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the resonant self- interacting dark matter (RSIDM) scenario [29, 40], with a judicial choice of parameters, the self-interactions can affect structure formation on small scales in a velocity dependent way, and thus may address the putative problems of collisionless dark matter [22, 41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Bullet cluster bound The s-wave part of the dark pion self-interaction cross section is eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12) σ4pnt SI mπ = 3κSIξ4 64πm3π = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 × 105 cm2 g �MeV mπ �3 3ξ4κSI 64π ≤ aint cm2 g , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) with κSI = (N4 f − 2 3N2 f + 2)/(N2 f (N2 f − 1)) = 1 + O(N−2 f ) and MeV−3 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 × 105 cm2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The cross section is not very sensitive to the number of quark flavors Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The Bullet cluster observation puts an absolute upper bound on the self-interaction cross section, given by aint ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the RSIDM scenario, where the resonant interactions from dark photon exchange become important at small scales, the data is fit by a smaller s-wave contribution and aint ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11 (and the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) becomes an equality) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The bound on the cross section can be translated in a condition on the dark pion mass mπ ≥ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9 MeV �ξ4κSI aint �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Resonant self-interactions The velocity dependent contribution to the self-interactions from nearly on-shell dark photon exchange can be written in Breit-Wigner form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='15) σres SI = 4πS mπE(v) Γd(v)2/4 (E(v) − E(vR))2 + Γ(v)2/4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with S = 3Sf/N 2 π the ratio of multiplicities of the resonance dark photon (3 polarizations) and DM particles (Nπ = N2 f − 1) times a symmetry factor Sf = 2 that takes into account that there are two identical particles in the final state of both the self-interaction process and in dark photon decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' E(v) = mπv2/4 is the kinetic energy in terms of the relative velocity v, and E(vR) = mπδm the resonant kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Further, Γd(v) is the running decay rate of the dark photon into dark sector pions, and Γ = Γd + Γv the total running decay rate, which includes the decay into the visible SM sector fermions and pions via kinetic mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The velocity dependence of the decay widths can be parameterized as Γi = mV γivni with i = d, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' explicitly (see appendix C) Γd(v) = mV �C4αd 24Sf � v3, Γv(v) = mV � αϵ2 3πSf � v0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) with αd = g2 d/(4π) and α = e2/(4π) the dark sector and SM fine-structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 5 – The resonance is peaked for v = vR, with vR = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6 × 10−4c from small scale structure data [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This determines δm δm = �vR 2 �2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) The height of the peak is fit by mπ = 4000 MeVS1/3(Bdγd)1/3, with Bd = Γd/Γ the branching ratio for decay into the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This determines the dark photon gauge coupling: αd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 × 10−4 � mπ 100 MeV �3 N2 π BdC4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) The resonant enhancement of the cross section is large, and small αd is required to avoid too large self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For RSIDM both the mass and the dark gauge coupling are given in terms of ξ, which together with the kinetic mixing parameter ϵ are the only free parameters left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The combined constraints eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) with aint = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11 and eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) give mπ = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='0 MeV � ξ4κSI �1/3 , αd = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7 × 10−6ξ4 N2 πκSI C4Bd , δm = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 × 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) 4 Relic density The dark pions can freeze out via 3 → 2 number changing SIMP interactions and via annihi- lation into SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the SIMP scenario thermal equilibrium with the SM sector should be maintained through freeze-out, to avoid entropy production and heating up of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We will assume this is the case in this section, and return to the question whether the dark photon can be responsible for this in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The SIMP interactions get a contribution form the 5-point coupling in the WZW La- grangian, and from diagrams with dark photon exchange allowed by the (3π)V coupling in the WZW Lagrangian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' with our choice of dark charges eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) the π(2V ) coupling vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The thermally averaged cross section is calculated in appendix E, and is given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Introducing the ‘time’ variable x = mπ T , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) it can be written in the form ⟨σv2⟩3→2 ≈ ⟨σv2⟩5pnt 3→2 + ⟨σv2⟩res 3→2 = α3→2 x2m5π (1 + αresx5/2e−δm x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) The first term comes from the 5pnt pointlike interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The 2nd term is from dark photon exchange, which is dominated by an s-channel resonance if the dark photon is nearly on shell δm ≪ 1, and we have we used the narrow width approximation to evaluate the thermally – 6 – averaged cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We have calculated the latter term only for Nf = 3 flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The effective couplings are α3→2 = 5 √ 5 1536π5 N2 c κ3→2ξ10 Nf , αres|Nf=3 = 128π5/2 15 αd ξ4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with κ3→2 = N2 f (N2 f −4)/N 2 π = 1+O(1/Nf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The resonant contribution dominates at freeze- out x = xf for αd/ξ4 ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8 × 10−6xf20 with xf20 = xf/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For RSIDM eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) this is the case for ξ ≲ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Dark pions can annihilate into SM electrons (and depending on their mass, into heavier charged SM particles) via the kinetic mixing portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Annihilation is also dominated by the s-channel resonance and the thermally averaged cross section is eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) ⟨σv⟩ann = αann x3/2e−δm x m2π , αann = 32π√πϵ2αBd N2π (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) with as before Bd = Γd/Γ the branching ratio for decay into dark sector states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We note that both the resonant part of the WZW interactions and the resonant annihilations are independent of the mass splitting δm, except for the exponential factor, which determines below which temperature x ≳ 1/δm these interactions are ‘turned off’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The Boltzmann equation for the dark pions reads ˙n + 3Hn = −⟨σv⟩ann(n2 − n2 eq) − ⟨σv2⟩3→2(n3 − n2neq), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) which in terms of the number density fraction Y = n/s becomes dY dx = −λ3→2 x7 (1 + αresx5/2e−δm x)(Y 3 − YeqY 2) − λanne−δm x √x (Y 2 − Y 2 eq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) Here λ3→2 = s(mπ)2α3→2 m5πH(mπ) , λann = s(mπ)αann m2πH(mπ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) with s(mπ) = (2π2g∗sm3 π)/45 and H(mπ) = (π√g∗m2 π)/(3 √ 10mpl) the entropy density and Hubble constant at T = mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The relic dark matter density matches observations [44] for Ωπ,0 = Nπmπs0Y∞ ρc ⇒ mπY∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4 × 10−6 MeV (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) with Y∞ = n/s the asymptotic number density fraction, and s0 the entropy density today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 SIMP freeze-out Consider first the case that freeze out of the dark pion is determined by the WZW interactions, either by the 5pnt contact interaction or the dark photon mediated contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This means annihilation are subdominant at freeze out ⟨σv⟩ann ≲ ⟨σv2⟩3→2 neq at x = xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We will – 7 – estimate the bound this condition gives on ϵ in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Note, however, that the annihilation cross section grows at late times, and even if negligible at freeze out, annihilation may still affect the final relic density significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To describe freeze out we thus set the annihilation contribution to zero in the Boltzmann eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' At late times the equilibrium distributions can be dropped, which allows to solve for the asymptotic distribution lim x→∞ dY dx = −λ3→2 x7 (1 + αresx5/2)Y 3 ⇒ Yf ≃ √ 3x3 f √λ3→2 1 � 1 + 12 7 αresx5/2 f (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) where we introduced the notation Yf for the asymptotic number density after freeze-out of the SIMP reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The freeze-out temperature can be estimated from n2 π⟨σv2⟩3→2 ≃ H which gives x3e2x�� x=xf ≃ N2 πα3→2mπ (2π)3H(mπ)(1 + αresx5/2) ≡ C3→2(1 + αresx5/2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) where we used that the non-relativistic number density is n = Nπm3 π(2πx)−3/2e−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the limit that the 5pnt interaction respectively the resonant contribution dominates the cross section we can estimate the freeze-out temperature using that xne2x = c gives x ≈ ln √c − n 2 ln(ln √c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The SIMP interactions are negligible after freeze out, but annihilations may still have an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To estimate this we solve the Boltzmann equation with the boundary condition Y (xf) = Yf: dY dx = −λanne−δm x √x Y 2, ⇒ Y∞ ≈ Yf √ δm √ δm + √πYfλann (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11) where we assumed that (δm xf) ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The annihilation rate increases at late time, and is most efficient just before the exponential cutoff at x = 1/δm kicks in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' this is how the δm- dependence appears in the estimate for the relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' It follows that annihilations are negligible if √πYfλann √ δm < 1 ⇒ ϵ < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6 × 10−9Nπ � Bdδm � mπ MeV � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12) that is, only for very small kinetic mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Annihilation scenario In the opposite limit that 3 → 2 interactions are always subdominant, ⟨σv⟩ann ≥ ⟨σv2⟩3→2 neq at freeze-out, the relic density is set by annihilation reactions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We can still use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11) for the relic density, but now Yf(xf) is the number density as annihilations freeze out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The freeze-out temperature can be estimated from nπ⟨σv⟩ann ≃ H which gives x−2ex ≃ Nπαannmπ (2π)3/2H(mπ) ≡ Cann ⇒ xf ≃ ln Cann + 2 ln(ln Cann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='13) – 8 – We estimate the freeze out density Yf ≈ n s ��� xf = x7/2 f λann (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14) where we used eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' If xf in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14) is smaller than that for SIMP reactions eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10), it follows freeze-out is dominated by annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' An earlier freeze out means a larger density Yf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Hence we can write the asymptotic number density as Y∞ ≈ Yf √ δm √ δm + √πYfλann , Yf = max � Yf �� ann, Yf �� 3→2 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='15) with the freeze out density for 3 → 2 reactions and annihilation given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10), and eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 5 Kinetic mixing Kinetic mixing between the dark and SM photons provides a portal between the dark and visible sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the SIMP scenario, in which the relic density is determined by the freeze- out of 3 → 2 dark pion number changing interactions, both sectors need to be in thermal equilibrium during freeze-out to avoid heating up the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The SIMP scenario further requires that dark pion annihilation into SM particles is subdominant during freeze-out – although, as we have seen in the previous subsection, annihilation still may affect the relic density at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In this subsection we determine the constraints on the mixing parameter ϵ that these two requirements give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The relevant cross sections are computed in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We also quickly review the relevant cosmological and collider bounds on the kinetic mixing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Thermal equilibrium between the dark and visible sector The dark and visible sector can be kept in thermal equilibrium via pion scattering with SM electrons and positrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The non-relativistic cross section for this process is eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='16) σscat = Ascatϵ2 p2 m4π , Ascat = 2πC4αDα Nπ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) This can be straightforwardly generalized to include muon and SM pion scattering as well,2 see below eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='16), in case freeze-out occurs at temperatures exceeding the muon and pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Here p ≈ Ee is the incoming electron momentum, and C4 = 4 (8) for Nf = 3 (4) the same color factor as appearing in the dark photon decay rate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The scattering rate 2Scattering off muons was included in the numerical results, but its effect for thermalisation was found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 9 – can be estimated as Γscat ≈ ⟨neE2 e⟩(σv)scat/E2 e [5], and demanding that it exceeds the Hubble rate Γscat > H at the time of freeze out gives the bound ϵ > � H(Tf)m4π ⟨neE2e⟩Ascat = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6 × 10−8x3/2 f20 � mπ 100 MeV 4 ge Nπ C4αd , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) where we used ⟨neE2 e⟩|x=xf = ge45ζ(5)m5 π 4π2x5 f , H(T) = H(mπ)/x2 f, and as before xf20 = xf/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Further, ge = 4 are the degrees of freedom of the electron/positron pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To get the numerical value we used α = 1/137 and gs = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For RSIDM the bounds eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) eliminate the dark pion mass and gauge coupling dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' If the dark photon is to maintain thermal equilibrium with the SM, annihilations cannot be neglected for the relic density calculation if (comparing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) mπ ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7 × 10−2 MeV αdδm 10 C4Nπ x3 f20 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) = 3600 MeV �8xf20 Nπ �3/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) The last equality applies to RSIDM, for which annihilations thus always play a role, depleting the dark matter abundance at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Annihilations subdominant during freeze-out Annihilations are subdominant at freeze-out if ⟨σv⟩ann < ⟨σv2⟩3→2 nπ at x = xf eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10), which translates to � α3→2(1 + αresx5/2 f )H(mπ)/mπ > αannx7/2 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This gives an upper bound on the kinetic mixing parameter ϵ < 4 × 10−8 � mπ 100 MeV �1/4 ξ5/2 x7/4 f20 Nπ √Nc N1/4 f � 1 + 3 × 105 αdx5/2 f20 ξ4 �1/4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) where we set κ3→2, Bd to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 CMB bound and other bounds on kinetic mixing BBN and CMB observations place bounds on the energy injected in the photon fluid from late time dark pion annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' These constraints can be stringent as the thermally averaged cross section grows as ⟨σv⟩ ∝ x3/2e−δmx at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For self-interacting resonant DM the mass splitting δm ∼ 3 × 10−8 is small eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5), and the exponential suppression of the cross section only kicks in shortly before or after the CMB is formed, depending on the mass of the dark pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The thermally averaged cross section thus peaks at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' CMB observations are more stringent for s-wave annihilations (as opposed to more stringent BBN bounds for p-wave annihilations) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Since our thermally averaged cross sections scales inversely proportional to the velocity, CMB bounds are stronger than bounds from BBN, and we thus only consider the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The CMB bound is given by pann < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 × 10−31 cm3s−1MeV−1, where pann ≡ f(z) ⟨σv⟩ mπ and f(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='01−1 is a function that quantifies the efficiency of energy injection in the CMB – 10 – [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Recasting this to a bound on the mixing parameter ϵ, the constraint is given by (setting f(z) = 1) ϵ ≲ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4 × 10−14� mπ 10 MeV �3/4 exp � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 mπ 10 MeV � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) which vanishes above dark pion masses of 150 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This bound was derived for s-wave scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In our model, the annihilation cross section increases for later times until the exponential cut-off kicks in, so the bound is expected to be stronger at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' At the same time, the energy injection into the CMB is maximized at z ∼ 600, where f(z) ≈ 1 [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Although applying the CMB bound naively for our case at the different choices of z affects the exact constraint on the mixing parameter somewhat, this has no consequences on parameter space of the SIMP scenarios discussed in the next section, as this is dominated by the constraintes from thermalisation and beam dump experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For simplicity, then, we imposed the CMB bound at z ∼ 600, and that is what is shown in our figures in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For larger δm only the BBN bound applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Following [45], energy injection during BBN is most efficient in the range 1/T ∼ 102 − 103 MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Requiring the annihilations to be suppressed at this time (δmx ≳ 1), the BBN bound can evaded for mass splittings δm ≳ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Finally, there are bounds from dark photon searches at beam dump or fixed target ex- periments at electron or proton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In these experiments large number of dark photons can be produced from Bremsstrahlung or secondary meson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The experiments typi- cally search for highly displaced vertices in the detector [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For our region of interest, 10−4 ≲ ϵ ≲ 10−8 and 10 MeV ≲ mγ′ ≲ 1 GeV we consider bounds from NuCal[50, 51], CHARM[52], and E137 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Given the p-wave nature of our scattering cross section, scatter- ing at late times is heavily suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Bounds on millicharged particles from direct detection experiments like XENON are therefore too weak to constrain the kinetic mixing parameters and are therefore not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 6 SIMP scenarios In the ‘standard’ SIMP scenario the dark pion relic density results from the freeze out of the 5pnt 3 → 2 WZW processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We reproduce this set-up by turning off the dark photon interactions αd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The Bullet cluster observation puts an upper bound on the pion self- interactions, and consequently on the pion mass eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The correct relic density is obtained for ξ ∼ 4π for Nf = Nc = 3, uncomfortably close to the perturbativity bound ξ ≲ 4π [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Increasing the number of colors improves the situation slightly, but a large number of colors is needed to be within the perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The problem with only the 3 → 2 interactions, and no resonance enhancement, is that dark matter is over produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark pions contribute a fraction to the DM density eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) R ≡ Ωπ ΩDM ≈ 9 × 102x3 f20 ξ3Nc � Nf aint .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) – 11 – Bullet cluster perturbativity WZW WZW+res 10 50 100 500 1000 5000 104 0 2 4 6 8 10 12 14 Figure 1: Relic density constraint on ξ for different values of the dark pion mass for the WZW-term only (red), and including the resonance via the dark photon (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dashed lines correspond to the estimate of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) (setting αd = 0 for the WZW estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The gray shaded areas are excluded by ξ > 4π, where the χPT description breaks down, and the Bullet cluster bound on the DM self-interaction σ/m ≤ 1 cm2/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' All lines are for Nf = Nc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The freeze-out temperature only depends logarithmicly on the model parameters, and ranges from xf20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='68 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 for ξ = 1 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The relic density is reproduced for R = 1, which requires large ξ ≳ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This is illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 1, which shows the numerical solution to the Boltzmann equation without (red) and with (blue) adding the 3 → 2 resonance, where αd is chosen from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dashed lines correspond to the estimate of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) with αres = 0 (αres ̸= 0) for the red (blue) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The shaded areas are excluded by the perturbativity cutoff on ξ and the Bullet cluster bound on the self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Without the dark photon resonance, the required value of ξ is at or above the perturbativ- ity cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Including the resonance, but not considering annihilations, the situation improves significantly as the increased 3 → 2 interactions reduce the dark matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The observed relic density is obtained for a larger dark pion mass for a fixed ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For such a large pion mass, however, the kinetic mixing parameter is too small to maintain kinetic equilibrium with the SM eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) and either an additional portal interaction is required or annihilations should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the following subsections we discuss two SIMP scenarios that give the correct relic density, satisfy self-interaction constraints, and the kinetic mixing portal interaction maintains thermal equilibrium with the SM during freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' First, we focus on the possibility that the – 12 – dark pions are RSIDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The 3 → 2 freeze-out interactions can be resonantly enhanced for large enough αd/ξ4 ≳ 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Given the small mass difference δm in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5), annihilations become important at late times, and reduce the relic density further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Second, we consider the more classical SIMP scenario, and drop the requirement that the self-interactions can affect small scale structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The self-interactions should still satisfy the upper bound from the bullet cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Both 3 → 2 interactions and annihilations may be resonantly enhanced, by tuning the dark photon mass, but now have more freedom in the resonant condition δm and the dark gauge coupling αd to satisfy all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Self-interacting resonant SIMP DM Consider first the RSIDM scenario that the relic density is produced via the SIMP mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' via the freeze out of 3 → 2 reactions, and that resonant DM self-interactions can address the small scale structure problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The requirements on the self-interactions eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) fixes the parameters mπ, αd, δm in terms of ξ, which in turn is determined from the correct relic density eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For large enough kinetic mixing the relic density will be reduced by (late-time) annihi- lations, which reduces the required ξ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Assuming annihilations are subdominant during freeze-out and Bd ≈ 1, then Y∞ is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11) and R now becomes R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7 × 103x3 f20 ξ3 1 Rres + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='0 × 1018x3 f20ϵ2/ξ17/3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) where Rres ≡ � 1 + 30x5/2 f20 accounts for the resonant enhancement of the WZW interactions, and the ϵ-dependent term for late-time annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Annihilations significantly reduce R for ϵ > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6 × 10−12ξ17/6x−3/2 f20 for sufficiently small ξ ≲ O(1), but rapidly shut off for larger ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Solving for the observed relic density R = 1 gives ϵ = � 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 × 10−16x3 f20ξ8/3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4 × 10−19Rresξ17/3 x−3/2 f20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) Figure 2 shows the value of ϵ as a function of the dark pion mass mπ for which R = 1 from numerically solving the Boltzmann equations (red curve), as well as the analytical estimate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) above (dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The observed relic density can be obtained with perturbative couplings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' ξ = 1 for kinetic mixing ϵ = 3×10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Note, however, that there is a maximum value ϵ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9 × 10−7 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) to get the observed relic density, at a dark pion mass mπ ∼ 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For larger dark pion masses, the 3 → 2 interactions alone underproduce DM given the imposed relations on the self-interaction eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7), so additional annihilations are not of any help;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' this explains the sharp cutoff of the red curve at large pion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For such small kinetic mixing annihilations are subdominant during SIMP freeze-out eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4), and decay is predominantly into dark sector pions Bd ≈ 1, validating our assumptions for the freeze out calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 13 – Thermalisation CMB Direct detection 10 50 100 500 1000 10-9 10-8 10-7 10-6 10-5 Figure 2: Kinetic mixing parameter space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The red (numerically) and dashed (estimate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3)) lines show the values of ϵ for which the correct DM relic density is produced, and for which the dark pions have to required self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The blue shaded area excludes those values of ϵ for which the dark photon cannot maintain kinetic equilibrium with the SM sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The purple shaded area depicts the CMB constraint on DM annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The grey shaded area is excluded from beam dump searches from E137, nuCAL and CHARM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In addition to the relic density constraint fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 2 also shows the bounds from the CMB, colliders, and from the requirement of thermal equilibrium between the dark and visible sector during SIMP freeze out eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' It is the latter requirement that rules out most of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Because the annihilations are highly efficient, and only a small portion of the DM should be depleted, the annihilation cross section should be suppressed by small values of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For these small ϵ, the heat transfer to the SM is not fast enough to prevent the dark bath from heating up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' It is thus not possible for the dark pions to be resonant self-interacting DM, and affect small scale structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Around mπ ∼ 500 MeV kinetic mixing is marginally too small to maintain thermal equilibrium with the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Additional effects might affect this part of parameter space which could render the model viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Alternatively, one could allow for (partial) heating of the dark bath to study the effect on the dark bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This is left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 SIMP DM We now consider the SIMP scenario, in which the relic density is determined by 3 → 2 inter- actions and possibly additional annihilations, but self-interactions are too weak to affect small scale structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' As we have seen eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1), with just the 5pnt WZW interactions – 14 – and given the Bullet cluster bound, too much DM is produced for perturbative couplings ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The DM density can be reduced by a resonant enhancement of the WZW interactions and by annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' No longer constrained by the small scale structure data, the value of δm can now be larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This immediately avoids CMB and BBN constraints as the thermally averaged annihilation cross section ∝ e−δm x eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) is exponentially suppressed in these eras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Moreover, for larger δm dark photons will predominantly decay to dark pions, thus evading dark photon searches at beam dump experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For concreteness we will fix the mass splitting to δm = 10−3 throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This choice avoids the cosmological constraints, while it can still give rise to interesting phenomenology at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the parameter space region where the dark photon maintains thermal equilibrium with the SM sector during freeze-out of the WZW-interactions, the kinetic mixing parameter is always large enough that late time – after freeze-out – annihilations cannot be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For general mπ, ξ, δm and αd the required mixing parameter that reproduces the correct relic density is ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5 × 10−12 Nπ Yf �√ δm � mπ MeV � � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8 × 107 � mπ MeV � Yf − 15 � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) with Yf from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Parameter space opens up significantly compared to the RSIDM scenario, as all other parameters are free except for the Bullet cluster bound on the dark matter self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Figure 3 shows the numerical result for the kinetic mixing parameter as a function of the dark pion mass that reproduces the observed relic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The red, orange and purple curves correspond to different values of ξ = 2, 5, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In all plots the mass splitting is δm = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The numerical results are in excellent agreement with the analytical estimate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Along the curves three different regions can be identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' i) a part where the self-interactions are larger than allowed by the Bullet cluster constraint, which is excluded (dotted);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' ii) a part where the 3 → 2 interactions dominate freeze-out, and annihilations are only important at later times times (solid);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' and iii) annihilations are the dominant freeze-out process (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The blue region in the plots is excluded by the thermalisation requirement eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For larger αd a smaller kinetic mixing parameter is required to maintain thermal equilibrium with the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The left top plot shows the result for αd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For such small gauge couplings, the resonance enhancement of the WZW interaction is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' DM is over produced unless annihilations are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In fact, we see that for the parameter space allowed by the Bullet cluster constraints, the annihilations are actually so large that they always dominate freeze out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Hence, in this scenario the dark pions are WIMP rather than SIMP dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the right top plot the gauge coupling is increased αd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1, but still resonance effects on the WZW interactions are small except for large ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Indeed, for ξ ∼ 10 the dark pion can be SIMP DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Although late time annihilations reduce the relic density somewhat – this is what generates the slope of the solid curve as a function of mixing parameter ϵ – the effect is not strong enough to allow for much smaller ξ than in the ‘standard’ scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 15 – 10 50 100 500 1000 10-8 10-7 10-6 10-5 10 50 100 500 1000 10-8 10-7 10-6 10-5 10 50 100 500 1000 10-8 10-7 10-6 10-5 Figure 3: The values of ϵ for which the correct relic density is reproduced as a function of the pion mass, for different values of ξ (coloured lines) and αd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The blue shaded area is excluded by the thermalisation requirement eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Dotted lines violate the Bullet cluster constraint on the self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The solid lines represent the part of parameter space where the 3 → 2 interactions are the dominant freeze-out interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dashed lines are when 2 → 2 annihilations via the dark photon are important at freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Finally, the bottom plot is for seizable couplings αd, and the WZW interactions are sig- nicficantly enhanced for smaller ξ = 1−5 as well, allowing SIMP dark matter in a perturbative set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The slope of the solid part of the curves shows the impact of late time annihilations as a function of kinetic mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The curves asymptote to a constant value for small mixing and annihilations are negligible at all times, thus providing a lower bound on the dark pion mass for a given ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 7 Conclusion We have studied a dark sector containing dark pions and a dark photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark pions are stable and can be SIMP dark matter, that is produced by freeze-out of 3 → 2 WZW- interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark photon mixes kinetically with the SM sector, and can maintain thermal – 16 – equilibrium during freeze out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For a fine-tuned dark photon mass mV ≈ 2mπ the WZW are resonantly enhanced, which opens up the possibility that 1) the observed relic density is produced in the perturbative regime ξ = mπ/fπ ≲ 4π of the effective chiral Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In addition, the pion self-interactions are resonantly enhanced and become velocity depen- dent, which opens up the possibility that it can address the small scale structure problems of collisionless dark matter – this scenario is dubbed resonant self-interaction dark matter RSIDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We found that the RSIDM scenario is not possible for all dark pion masses considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Because of the highly efficient dark pion annihilations, the value of the mixing parameter required to reproduce the relic density is too small to maintain kinetic equilibrium with the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For dark pion masses of mπ ∼ 500 MeV the difference is marginal, and more precise calculations are required to assess the viability of the model in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In particular, one could allow for (partial) heating of the dark bath to study the effect on the dark bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This is left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' If we give up the demand that the self-interactions have an effect on small scale structure formation, and consequently are only constraint by an upper bound from observations of the Bullet cluster, then parameter space opens up and smaller ξ-values become possible for sufficiently dark gauge couplings αd ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Acknowledgments The authors thank Jordy de Vries for very useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This work was funded by an NWO-klein2 grant (OCENW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='KLEIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='427).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' A Cross sections We list here the definitions of the (thermally averaged) cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This appendix also serves to set the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Scattering/annihilation cross section The cross section for scattering with two particles in both the initial and final state, labeled by α and β respectively, is σα→β = 1 4FSβ � � � β=1,2 � pβ � � (2π)4δ4(Pα − Pβ)| ¯ Mα→β|2 CM = � dΩ 1 (8π)2sSβ pout pin | ¯ Mα→β|2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) with � p = � d3p/(2Ep(2π)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We use P µ for 4-momenta, and pi for 3-momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The second expression is valid in the center of mass frame (CM), with pin = |pα| (pout = |pβ|) the absolute value of the three-momentum of either incoming (outgoing) particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Sβ = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' for N identical particles in the final state, to avoid overcounting in the phase space integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' | ¯ M|2 is the – 17 – amplitude averaged over initial and summed over final state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The flux factor can be written as F = E1E2|v1 − v2| = � (p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='p2)2 − m2 1m2 2 CM = pin √s with s = (E1 + E2)2 the center of mass energy squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Thermally averaged cross section The thermally averaged cross sections can be defined in terms of the scattering rates appearing in the Boltzmann equation for the DM particle [32] ˙n + 3Hn = − � α,β ∆αβ(˜γα→β − ˜γβ→α) = −⟨σv⟩ann(n2 − n2 eq) − ⟨σv2⟩3→2(n3 − n2neq), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) with n = nDM the number density of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We have included both DM annihilation and 3 → 2 interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' ∆αβ = (Ndm α − Ndm β ) is the difference between the number of DM particles in the initial (Ndm α ) and final state (Ndm β ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The collision terms are ˆγα→β(fα) = 1 SαSβ �� α � α Nαfα � � �� β � β � � (2π)4δ4(Pα − Pβ)| ¯ Mα→β|2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with fα, Nα the distribution functions and degrees of freedom of the initial states, and Sα (Sβ) = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' for N identical particles in the initial (final) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Assuming kinematic equi- librium for the DM and chemical equilibrium for all other particles gives the relations ˆγα→β(fi) = � n neq �Ndm α γα→β(feq i ), γα→β = γβ→α (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) with γα→β ≡ ˆγα→β(feq α ) and feq α = e−Eα/T the Maxwell-Boltzmann distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We can then express the thermally averaged cross sections appearing in the Boltzmann equation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) in terms of the collision rates as follows ⟨σv⟩ann = 2γann n2eq , ⟨σv2⟩3→2 = γ3→2 n3eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) For annihilations the momentum integrations in γann can be partially done, and the final expression is given in terms of one remaining integral over the center of mass energy [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The thermally averaged cross section is ⟨σv⟩ann = 1 2Tm2 1m2 2K2( m1 T )K2( m2 T ) � ∞ (m1+m2)2 ds K1( √s T )(pinE1E2vmølσ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) with the Møller velocity related to the flux factor as F = (E1E2)vmøl, and the factor ∆ann/Sα = 1 is set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The equilibrium number density is defined as neq α = Nα (2π)3 � d3pαfeq α = Nαm2 αT 2π2 K2(mα T ) mα≫T = Nα �mαT 2π �3/2 e−mα/T , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) with the last expression valid in the non-relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For m1 = m2 ≡ m we can rewrite this in dimensionless variables ⟨σv⟩ann = 4x∆ SαK2(x)2 � ∞ 1 d˜s √ ˜s(˜s − 1)K1(2 √ ˜sx)σ(˜s), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) with ˜s = s/(4m2) and x = m/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 18 – A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 S-channel resonance and narrow width approximation If the DM interactions are mediated by a massive meditor particle – in our case, the dark photon – there will be an s-channel resonance for momenta that the mediator is nearly on shell s ≈ m2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To incorporate this effect, we include the decay rate in the dark photon propagator Dµν(P 2) = −igµν P 2 − m2 V + iϵ → −igµν P 2 − (mV − i 1 2Γ)2 ≈ −igµν P 2 − m2 V + imV Γ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) where we used that Γ2 ≪ m2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Γ = Γd + Γv is the total decay width of the dark photon, which is the sum of the decay rate into pions and decay rate into SM fermions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' into the dark and visible sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the resonance limit the most enhanced terms in the cross section will be ∝ Dµν(s)2, which can be evaluated in the narrow width approximation 1 (s − mV )2 + m2 V Γ2 ≈ π mV Γδ(s − m2 V ) + O(Γ2/m2 V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) B Pion self-interactions In this appendix we calculate the pion self-interaction cross section σSI = σ(ππ → ππ), which has contributions from 4pnt self-interactions and from dark photon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Amplitude Dark photon mediated self-interaction The 4pnt pion interaction follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) L ⊃ 1 3f2π � 2πa∂πbπc∂πd − 2πaπb∂πc∂πd + m2 ππaπbπcπd� � Tr[T aT bT cT d] + perm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) There are 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' possible orderings of the pions a, b, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Consider first the amplitude for the {acbd}-term plus the cyclic permutations: M4pnt {acbd} = −4Tr[T aT cT bT d] 3f2π � (Pc · Pd + Pa · Pb) + 1 2(Pb · Pd + Pc · Pb + Pa · Pc + Pd · Pa) − m2 π � = − 2 f2π Tr[T aT cT bT d] � s − 2m2 π � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) where we took Pa, Pb as incoming momenta, and Pc, Pd as outgoing (∂πa → −iPa, ∂πc → iPc), and on the 2nd line we used the Mandelstam variables s = (Pa + Pb)2, t = (Pa − Pc)2, u = (Pa − Pd)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) The results are similar for the other possible permutations, and the total amplitude is [27] M4pnt abcd = M(πaπb → πcπd) = − 2 f2π � Tr[T aT bT cT d] + (b ↔ d) � � t − 2m2 π � − 2 f2π � Tr[T aT cT bT d] + (c ↔ d) � � s − 2m2 π � − 2 f2π � Tr[T aT cT dT b] + (b ↔ c) � � u − 2m2 π � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) – 19 – Dark photon mediated self-interaction The pion-dark photon interactions that follow from the covariant derivatives in the chiral Lagangian eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) can be written in the form L ⊃ = −2igdVµ � πa(∂πb) − (∂πa)πb� Tr � [T a, T b]Q] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) The (V 2π)-vertex interaction and dark photon propagator (in Lorentz gauge) are then Aµ ab = 2igd(Pa − Pb)µTr([T a, T b]Q), Dµν(P 2) = −igµν P 2 − m2 V + iϵ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) with both momenta Pa, Pb incoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude for ππ → ππ scattering via dark photon exchange is MV abcd = iAµ abDµν(s)Aν cd − iAµ acDµν(t)Aν bd − iAµ adDµν(u)Aν bc = 4g2 d � (t − u) s − m2 V + iϵCabcd + (s − u) t − m2 V + iϵCacbd + (s − t) u − m2 V + iϵCadbc � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) with color factor Cabcd ≡ Tr([T a, T b]Q)Tr([T c, T d]Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) Resonant scattering arises for P 2 ≈ m2 V , and we include the decay width in the propagator eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) to describe this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Cross section The matrix element squared summed over final and averaged over initial states is | ¯ M|2 SI = 1 N2π � abcd |Mabcd|2 with Nπ = (N2 f − 1) the number of pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude is the sum of the 4pnt interaction eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) and the photon exchange contribution eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The latter is subdominant, except for momenta near the s-channel resonance s ≈ m2 V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' we can then neglect interference terms and the non-resonant contributions, and approximate the amplitude | ¯ MSI|2 ≈ 1 N2π � abcd � |M4pnt abcd|2 + |Mres abcd|2� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) with Mres = MV |s≈m2 V the s-channel resonance contribution from the photon exchange diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' 4pnt self-interaction Starting with the 4-pnt contribution, the amplitude squared can be calculated using the Feyncalc Mathematica program [56–58] | ¯ M4pnt SI |2 = 6κSI m4 π f4π − κSI,2 (st + tu + us) f4π , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) with κSI = (3N4 f − 2N2 f + 6) 3N2 f (N2 f − 1) = 1 + O(N−1 f ), κSI,2 = N2 f (N2 f − 1) = 1 + O(N−1 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11) – 20 – The cross section eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) for pion scattering is σ4pnt SI = m4 π πSff4πs �6κSI 16 + κSI,2(m2 πp2 + 5 6p4) � ≈ 3κSIξ4 64πm2π (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12) with Sf = 2 for two identical particles in the final state, and p = |p| the incoming momentum of either pion in the CM frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The last expression is valid in the non-relativistic limit p2 ≪ m2 π in which the velocity-independent s-wave contribution dominates, and s ≈ 4m2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The result agrees with [28], but differs a factor of 8 with [4] (after rescaling fthem π = 2fπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For Nf = 2 it reproduces the results of [27] (except for the sign in front of the 5 6p4 term) but not for other Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Dark photon mediated self-interaction The diagram for photon exchange is negligble except near the s-channel resonance, where the amplitude can be approximated by the s- channel diagrams, and | ¯ Mres SI |2 = 256C2 4g4 d N2π p4 cos2 θ (s − m2 V )2 + m2 V Γ2 ⇒ σres SI = 16C2 4g4 d 3πsSfN2π p4 (s − m2 V )2 + m2 V Γ(p)2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='13) with Sf = 2 amd Γ(p) the running photon decay width, computed in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The color factor eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) summed over flavors is C2 4 = � |Cabcd|2 = 42 (82) for Nf = 3 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The cross section can be rewritten in the more familiar Breit-Wigner form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To do so, expand the momentum as p = 1 2mπv = 1 4mV v + O(δm) with v the relative velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The resonance velocity follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4), which gives vR ≈ 2 √ δm for dark photon masses eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We further define the velocity dependent and resonant kinetic energies [29] E(v) = p2 mπ = 1 4mπv2, E(vR) = mV − 2mπ = mπδm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14) We can then rewrite the resonant cross section in Breit-Wigner form σres SI = 4πS mπE(v) Γd(v)2/4 (E(v) − E(vR))2 + Γ(v)2/4, S = 3Sf N2π (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='15) where we used the non-relativistic approximation s = m2 V + O(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The numerator is written in terms of the decay width into dark pions Γd using the explicit expression eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' [29] list a numerical factor S = NV /N 2 π for the ratio of mulitplicities of the resonance dark photon and DM particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' we find here an additional Sf = 2 symmetry factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 Thermally averaged cross section The thermally averaged self-interaction cross section is [29] ⟨σv⟩SI = σ4pnt SI ⟨v⟩ + ⟨σres SI v⟩ = σ4pnt⟨v⟩ + � vmax 0 f(v, v0)(σres SI v)dv, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='16) with the velocity distribution f(v, v0) = (4v2)/(√πv3 0)e−v2/v2 0 taken as a Maxwell-Boltzmann distribution truncated at the halo escape velocity vmax, and v0 a constant implicitly defined – 21 – via ⟨v⟩ ≃ 2v0/√π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Here we used that the self-interaction cross section is the sum of the (dom- inantly) s-wave 4pnt interaction and s-channel resonance contribution eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The resonance contribution can be calculated in the narrow width approximation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14): ⟨σv⟩res SI = 64π3/2S Γd(vR)Bd(vR) m3π e−v2 R/v2 0 v3 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='17) with Bd = Γd/Γ the branching ratio for decay into dark sector pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The decay rate into dark pions (p-wave) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) and SM fermions (s-wave) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) can be parameterized as Γi(v) = mV γivni with i = d, v for decay into dark and visible sectors, and mV ≈ 2mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This factors out the explicit velocity dependence of the (ΓdBd)-factor in the thermally averaged cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' C Dark photon decay rate The dark photon can decay into dark pions, and in SM fermions and pions via kinetic mixing with the SM photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Dark photon decay rate into dark pions ΓV →ππ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark photon decay rate in dark sector particles is Γd = Γ(V → ππ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The decay into dark pions is mediated by the vertex interaction eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6), with amplitude Mµ ab = −2gd(Pa − Pb)µCab, Cab = Tr([T a, T b]Q) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) with Pa, Pb the 4-momenta of the outgoing pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude squared (summed over final states, and averaged over intial photon polarizations states) in the CM frame is | ¯ M|2 d = −1 3gµν � ab Mµ abMν∗ ab = 16C4 3 g2 dp2 out (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) with pout = |pa| = |pb| the final state 3-momentum, and color factor C2 2 = � ab |Cab|2 = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The decay rate becomes Γd(pout) = pout Sf8πm2 V | ¯ M|2 d = 8C4αdp3 out 3Sfm2 V (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with αd = g2 d/(4π) and Sf = 2 for two identical particles in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The resonance momentum (taking the intial state photon at rest) is p2 R = 1 4m2 V � 1 − 4m2 π m2 V � = m2 πδm + O(δm2) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) where we used the parameterization eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) in the last step, and assumed the dark photon mass is close to resonance δm2 ≪ m2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Evaluating the decay rate at resonance pout = pR gives Γd(pR) = 2C4αd 3Sf mπδm3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) – 22 – C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Dark photon decay rate in SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The dark photon can decay into SM electrons via kinetic mixing and Γs = Γ(V → ¯ff) with f the electron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' for large enough DM mass the decay into muons and charged SM pions also becomes kinematically accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The decay rate into SM pions will be of the form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with αd → ϵ2α and instead of C4 → CQCD 4 = 1/2 the appropiate color factor for QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' It is velocity suppressed compared to decay into fermions, which is s-wave, and we neglect it in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude for dark photon decay into a SM fermion is Mµ v = (ieϵ)¯u(P2)γµv(P1) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) with P1, P2 the 4-momenta of the outgoing fermions, ϵ the kinetic mixing paramer appearing in the Lagrangian eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4), and e the electric charge of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude squared (summed over final states, and averaged over photon polarizations) in the CM frame and decay rate are | ¯ M|2 v = 1 3(eϵ)24m2 V ⇒ Γv = αϵ2 Sf3πmV (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) with α = e2/(4π), symmetry factor Sf = 2 for two identical particles in the final state, and s = m2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Decay into the darks sector pions dominates and Γd ≫ Γv or ϵ ≲ � παdC4 α δm3/4 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5 × 10−6 �C4 4 � �αd α � δm 3 × 10−8 �3/4 , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) where we set v → vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' D Dark pion annilation and scattering Dark pions can annihilate into and scatter off SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' These processes are important for the final relic density and for keeping the dark sector in thermal equilibrium with the SM sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Dark pion annihiliation into SM fermions Consider the annihiliation of dark pions into Standard Model electrons πa(P1)πb(P2) → ¯f(K1)f(K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For large enough dark pion mass, also the decay channel into muons (mπ > 105 MeV) and SM charged pions (mπ > 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6 MeV) opens up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude for annihilation in a SM Dirac fermion pair ¯ff is mediated by the vertex eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) and the coupling to the SM fermion current via kinetic mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' iMππ→ ¯ff = −i2gD(P1 − P2)µTr([T a, T b]Q) −i s − m2 V + iϵ(ieϵ) � f ¯u(K1)qfγµv(K2) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) – 23 – Averaging over intitial states and summing over the spins of the final state fermions gives |M|2 ππ→ ¯ff = (2gDeϵ)2C4 N2π(s − m2 V )2 � spin � ¯u(K1)(/P 1 − /P 2)v(K2)¯v(K2)(/P 1 − /P 2)u(K1) � = (2gDeϵ)2C4 N2π(s − m2 V )2 32p2 � k2(1 − cos2 θ) + m2 f � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) with color factor C4 = C2 2 ≡ � ab |Tr([T a, T b]Q|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Here we denoted the CM 3-momentum of the incoming pions with p and that of the outgoing fermions with k, and θ the scattering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The cross section eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) becomes σππ→ ¯ff = 4Aann � s − 4m2π � s − 4m2 f(s + 2m2 f) s(s − m2 V )2 mf→0 = Aann m2π √˜s − 1 √ ˜s (˜s − m2 V /(4m2π))2 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with Aann = 4πC4ϵ2αDα/(3N2 π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the last line we neglected the SM fermion mass, and rewrote the cross section in terms of the dimensionless variable ˜s = s/(4m2 π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The amplitude for annihilation in a pair of SM pions π± SM with is iMπaπb→πc SMπd SM = i2gD(P1 − P2)µTr([T c, T c]Q) 1 s − m2 V + iϵ(eϵ)(K1 − K2)µTr([T a, T b]QQCD) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) Averaging over intitial states and summing over the spins of the final state pions – c, d running over the generators corresponding to π± SM – gives |M|2 ππ→2πSM = (2gDeϵ)2C4CQCD 4 N2π(s − m2 V )2 16k2p2 cos2 θ (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) with color factor CQCD 4 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The cross section can then be expressed as σππ→2πSM = CQCD 4 4 � 1 − 4mπ2 SM s3/2 �3/2 σππ→e¯e (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) where we neglected the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) in the thermally averaged cross section eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) for annihilation into SM electrons is ⟨σv⟩ππ→¯ee = ∆ann4xAann Sαm2πK2(x)2 � ∞ 1 d˜s ˜s(˜s − 1)3/2K1(2x √ ˜s) (˜s − m2 V /(4m2π))2 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) with ∆ann/Sα = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We are interested in the thermally averaged cross section during freeze- out, in the limit x = mπ/T ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This will be dominated by the resonance contribution which we can calculate in the narrow width approximation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Including the decay – 24 – width in the propagator eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) and defining ˜Γ = Γ/(2mπ), ˜mV = mV /(2mπ) the thermally averaged cross section eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) becomes ⟨σv⟩res ππ→¯ee ≈ 4xAann m2πK2(x)2 � ∞ 1 d˜s˜s(˜s − 1)3/2K1(2x √ ˜s) π ˜mV ˜Γ δ(˜s − ˜m2 V ) x≫1 ≈ 8√πx3/2Aann mπΓ δm3/2e−δm x (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) where in the last step we used that K2(x) = K1(x) = � π/(2x)e−x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' for x ≫ 1, and expanded ( ˜m2 V − 1) ≈ δm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Now expand the dark photon mass in small δm defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3), and use the explicit expression for the decay width eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) to get ⟨σv⟩res ππ→¯ee = 32π3/2ϵ2αx3/2Bd N2πm2π e−δm x + O(δm) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) with Bd = Γd/Γ the branching ratio for decay into the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We can then write the full thermally averaged annihilation cross section as ⟨σv⟩ann = gann 32π3/2ϵ2αx3/2Bd N2πm2π e−δm x + O(δm) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) with gann incorporating the degrees of freedom the dark pion can annihilate in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Below the muon threshold this is only electrons and gann = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Including muons and SM pions gann = 1+Θ(mπ−mµ) � 1 − m2µ m2π � 1 + m2 µ 2m2π � +Θ(mπ−mπSM)CQCD 4 4 � 1 − m2 πSM m2π �3/2 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11) with Θ the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Pion-electron scattering Consider scattering of dark pions with SM particles, which for light DM is dominated by electron scattering via the reaction πa(P1)f(K1) → πa(P2)f(Kf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' iMπf→πf = −i2gd(P1 + P2)µTr([T a, T b]Q) −i t − m2 V + iϵ(ieϵ)¯u(K2)γµu(K1) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12) Averaging over intitial and summing over final states gives |M|2 πf→πf = 1 2Nπ (2gDeϵ)2C4 (t − m2 V )2 � spin � ¯u(K2)(/P 1 + /P 2)u(K1)¯u(K2)(/P 1 + /P 2)u(K2) � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='13) The spinor sum now becomes � (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=') = 4 � 2(P1 + P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='K1(P1 + P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='K2 − (P1 + P2)2(K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='K2 − m2 f) � ≈ 16m2 πp2(1 + cos θ) + O(p2) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14) – 25 – with p the CM 3-momenta of the incoming particles and k of the outgoing particles, and θ the scattering angle between the ingoing and outgoing pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' In the last step we set the fermion mass to zero and took the non-relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The non-relativistic cross section becomes [5] σπf→πf = 8C4αDαϵ2m2 πp2 Nπsm4 V � dΩ (1 + cos θ) = ϵ2Ascat p2 m4π (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='15) with Ascat = 2πC4αDα/Nπ and mV ≈ 2mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The total scattering cross section is written as σscat = gscatϵ2Ascat p2 m4π , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='16) with gscat = 1 + Θ(mπ − mµ) + Θ(mπ − mπSM)CQCD 4 , where for simplicity we have neglected the muon and SM pion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' E Cross section for 3 → 2 dark pion interactions The 3 → 2 dark pion amplitude has a contribution from the 5pnt pion interaction and from dark photon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The necessary vertices with an odd number of pions arise from the WZW Lagrangian eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Pion 5pnt interaction from the WZW-term The 5pnt pion interaction from the WZW term can be written as [4] LWZW ⊃ A5 f5π ϵµνρσTr [π∂µπ∂νπ∂ρπ∂σπ] , A5 = 2Nc 15π2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1) with Nc the number of colors of the dark QCD-like gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' There are 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' different contributions to the amplitude Mabc→de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We can group them by their momentum dependence iMabc→de = iA5 f5π ϵµνρσ� P a µP b νP c ρP d σfe + P b µP c νP d ρ P e σfa + P c µP d ν P e ρP a σ fb + P d µP e ν P a ρ P b σfc + P e µP a ν P b ρP c σfd � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) with fa coefficients that each are the sum of 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' terms, and all momenta are taken as incoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The color-coefficient fe is defined as fe = Teabcd − Teabdc − Teacbd + Teacdb + Teadbc − Teadcb (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) with Teabcd ≡ Tr � T eT aT bT cT d� + cycl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' of {a, b, c, d} = Tr � T eT aT bT cT d� − Tr � T eT bT cT dT a� + Tr � T eT cT dT aT b� − Tr � T eT dT aT bT c� (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) – 26 – That is, fe is the sum of all traces of five generators with Te fixed in the first position, and all possible permutations of the other generators (of the {a, b, c, d} indices);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' the sign of each term is determined by the number of permutations away from the {a, b, c, d, e}-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Likewise, all other fi coefficients can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The matrix element squared averaged over initial and summed over final states is (the calculation is done with the Mathematica package FeynCalc) | ¯ M3→2|2 = 1 N3π � abcde |Mabc→de|2 = A2 5Nf(N2 f − 4)(N2 f − 1) N3π F(Pi) f10 π ≡ ¯A2F(Pi) f10 π (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) with Nπ = N2 f − 1 the number of pions and Nf the number of flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' F is a complicated momentum-dependent function, which we will evaluate for non-relativistic incoming mo- menta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We parameterize the momenta in the CM frame (we set P d → −P d and P e → −P e to make them outgoing momenta with positive energy as the zeroth component of the 4-vector): P a = (E1, p1, 0, 0), P b = (E2, p2 cos θ, p2 sin θ cos ϕ, p2 sin θ sin ϕ), P c = (E3, −p1 − p2 cos θ, −p2 sin θ cos ϕ, −p2 sin θ sin ϕ), P d = (1/2(E1 + E2 + E3), p4 cos ¯θ, p4 sin ¯θ cos ¯ϕ, p4 sin ¯θ sin ¯ϕ), P e = (1/2(E1 + E2 + E3), −p4 cos ¯θ, −p4 sin ¯θ cos ¯ϕ, −p4 sin ¯θ sin ¯ϕ), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6) The momentum p1 is aligned with the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Ω2 and Ω4 are then the solid angle of p2 and p4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' p3 and p5 are fixed in center of mass frame, and E4, E5 are fixed by energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The energies are Ei = � p2 i + m2π for i = 1, 2, E2 3 = (p2 1 +2p1p2 cos θ +p2 2)+m2 π, and � p2 4 + m2π = 1/2(E1 + E2 + E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We can thus express Ei, p4 in terms of p1, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To get the leading order term in the limit of non-relativistic momentum for the incoming particles set pi = ϵpi for i = 1, 2 and expand in small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The first non-zero term arises at 4th order: F = ϵ4F4(p1, p2, θ, ϕ, ¯θ, ¯ϕ) + O(ϵ6) with F4 = 3375m4 πp2 1p2 2 16 sin2(θ) sin2(¯θ) sin2(φ − ¯φ) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='7) The transition amplitude eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4) then becomes γ3→2 = ¯A2 SαSβ(2π)11f10 π � �d3p3d3p4 � i(2Ei) � p2 1dp1p2 2dp2N3 πfeq 1 feq 2 feq 3 δ(Eα − Eβ) � dΩd¯ΩF4 = 125 √ 5 ¯A2mπ 1024π5SαSβf10 π � d3p3 (2π)3 Nπfeq 3 � dp1p4 1Nπfeq 1 � dp2p4 2Nπfeq 2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='8) with Sα = 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' and Sβ = 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' to account for identical particles in initial and final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The integral � dΩd¯ΩF4 = 750π2m4 πp2 1p2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' To get the final expression, we further used that in the non-relativistic limit Ei ≈ mπ for i = 1, 2, 3 and Ei ≈ 3 2mπ for i = 4, 5, which yields – 27 – � i(2Ei) ≈ 2332m5 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Moreover � d3p4δ(Eα − Eβ) = 3 2 √ 5 � d3p4δ(p4 − √ 5/2mπ) = 3π √ 5/2m2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The integrations over the phase space densities give in the non-rel limit Nπ � d3p3 (2π)3 feq 3 = neq π , Nπ � dp1p4 1feq 1 = 6π2mπTneq π (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9) The thermally averaged cross section eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) is then ⟨σv2⟩5pnt 3→2 = α3→2 x2m5π , α3→2 = N2 c κ3→2 Nf 5 √ 5ξ10 1536π5 , κ3→2 = N2 f (N2 f − 4) (N2 f − 1)2 = 1 + O(1/Nf) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) with x = mπ/T and ξ = mπ/fπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This result matches the result in Ref [4] (taking into account the different definitions fthem π = 2fπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2 Dark photon interactions from WZW term In the presence of a dark photon, there are additional diagrams with photon exchange con- tributing to the 3 → 2 cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The WZW term also contains photon interactions with an odd number of pions eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The A(3π) and (2A)π interactions are LWZW ⊃ Ncgd 3π2f3π ϵµνρσAµP a ν P b ρP c σTQabc − Ncg2 d 4π2fπ ϵµνρσP (Aν) µ P a σ AνAρπaTQa (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='11) with TQabc = Tr � QT aT bT c� and TQa = Tr � Q2T a� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Including the 5pnt pion interaction discussed in the previous subsection and the A(2π)-interaciton from the chiral Lagrangian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='the relevant photon-pion couplings vertices are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aabcde ≡ iA5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='f5π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aabcde = iA5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='f5π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ϵµνρσ� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='P a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µP b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='νP c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ρP d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σfe + P b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µP c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='νP d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ρ P e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σfa + P c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µP d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ν P e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ρP a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σ fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='+ P d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µP e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ν P a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ρ P b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σfc + P e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µP a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ν P b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ρP c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σfd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='abc ≡ iA3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='f3π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='abc = iA3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='f3π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ϵµνρσP a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ν P b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ρP c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σ (TQabc + TQbca + TQcab − TQacb − TQcba − TQbac) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aνρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='a ≡ −iA1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='fπ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aνρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='a = −iA1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='fπ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ϵµνρσ(P (Aν) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='− P (Aρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=')P a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='σ TQa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ab ≡ iA2 ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='Aµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='ab = iA2π(Pa − Pb)µTr([T a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' T b]Q) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='12) with A5 = 2Nc 15π2 , A3 = Ncgd 3π2 , A1 = Ncg2 d 8π2 , A2 = 2gd (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='13) and all momenta are taken as incoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For the Aabc vertex we used that there are 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' different terms, corresponding to abc + cycl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='. We have symmetrized the Aa vertex in the two photon legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Q2 = 1 For our choice of charge matrix eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2), and thus TQa = Tr(Q2T a) = Tr(T a) = 0, and the (2A)π interaction vanishes Aµρ a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The propagator is Dµν(P) = −igµν P 2 − m2 V + imV Γ = −igµν ∆(P) = −igµν (4m2π) ˜∆(P) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='14) where in the last step we introduced the dimensionless propagator ˜∆ = ∆/(4m2 π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' – 28 – E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 Amplitude The photon interactions give rise to 6 additional diagrams contributing to the 3 → 2 in- teractions [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' For our charge matrix eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='2) the Aµρ a = 0 vertex vanishes, and only the first three diagrams contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We will calculate the amplitude for Nc = Nf = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The full amplitude is iM = Aabcde + a1 −iAµ abcAdeµ ∆(de) + a2Pabc �−iAµ abAcdeµ ∆(ab) � + a3Pabc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='de �−iAµ ceAabdµ ∆(ce) � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='15) with ∆(ij) = ∆(Pi + Pj) the propagator of momenta Pi and Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Pabc means all cyclic per- mutations of (abc) – hence there are three diagrams contributing to a2 –, and Pabc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='de cyclic permutations of (abc) times cyclic permutations of (de) – hence there are six diagrams con- tributing to a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The ai are included as note keeping devices and can be set to unity at any time during the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='The amplitude in terms of the ¯ A-vertices becomes M = A5 f5π � ¯ Aabcde + f2 πA2A3 (4m2π)A5 � a1 ¯ Aµ abc ¯ Adeν ˜∆(de) + a2Pabc � ¯ Aµ ab ¯ Acdeν ˜∆(ab) � + a3Pabc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='de � ¯ Aµ ce ¯ Aabdν ˜∆(ce) �� � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='16) Diagram a2 has an s-channel resonance for mV ≈ 2mπ, as the propagators ˜∆ij with i, j = a, b, c go nearly on shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Diagram a1 can become resonant for mV ≥ 3mπ, and the propagator ˜∆de can be put on shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This case was analysed in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' We will here not consider it any further, and instead focus on lighter dark photon masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Using the same momentum parameterization as before eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='6), the amplitude squared can be written as | ¯ M|2 = 3375 ¯A2m4 π 16f10 π p2 1p2 2 sin2(θ) sin2(¯θ) sin2(φ − ¯φ)(1 + X) = | ¯ M|2 5pnt(1 + X) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='17) with as before ¯A2 = A2 5N−2 π Nf(N2 f −4) Nf=3 = 15 64A2 5, and X parameterizing the photon-exhange corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='3 Resonance contribution from mV ≈ 2mπ In the limit that mV ≈ 2mπ the propagators ˜∆(ij) with i, j = a, b, c are resonantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The resonance contribution is dominated by the ˜∆−2 (ij) terms in the amplitude squared pro- portional to a2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Dropping all subdominant terms the correction eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='17) becomes Xres = �παd ξ2 �2 a2 2 � �128 45 � I 1 ˜∆2 (I) − 4 27 � I̸=J 1 ˜∆(I) ˜∆(J) � � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='18) with I, J = ab, bc, ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Defining F(h) ≡ � dp1 p4 1Nπfeq 1 � dp2 p4 2Nπfeq 2 � d cos θ sin2(θ)h(p1, p2, cos θ) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='19) – 29 – The transition amplitude can be written as γres 3→2 γ5pnt 3→2 = F(Xres) F(1) , F(1) = 6πN2 πe−2xm10 π x5 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='20) Consider first the ˜∆−2 (I) terms, which can be evaluated in the narrow width approximation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='10) 1 ˜∆2 (I) = (4m2 π)2 (sI − m2γ)2 + m2 V Γ2 ≈ π(4m2 π)2 mV Γ δ(sI − m2 V ) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='21) with I = ab, bc, ca and sab = (Pa + Pb)2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The x = cos θ integral in F( ˜∆−2 (I)) becomes of the form � dx (1 − x2)δ(sI − m2 V ) = � (1 − x2 0) |s′ I(x0)| , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='22) where the sum is over the roots x0 of sI − m2 V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' It will be useful to redefine the momenta (ab) : p± = 1 √ 2(p1 ± p2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (bc) : p± = 1 √ 2(p1 ± 2p2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (ac) : p± = 1 √ 2(2p1 ± p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='23) for I = ab, bc, ac respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Then for all I we get |x0| = p2 + + p2 − − 4m2 πδm p2 + − p2 − , |s′ I(x0)| = p2 + − p2 −, |p+| ≥ mπ √ 2δm ≥ |p−| (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='24) up to O(δm2) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The constraint on the momentum range arises from requiring | cos θ| ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' only small p−-momenta can hit the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' A further suppression comes from the (1 − x2 0) ∝ δm in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='22), in the limit that |p−| ≪ p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Putting it all together F( ˜∆−2 I ) = π(4m2 π)2 mV Γ � dp1 p4 1Nπfeq 1 � dp2 p4 2Nπfeq 2 (1 − x2 0) |s′ I(x0)| p+≫|p−| ≈ ρI 8πm6 πδm mV Γ � ∞ dp+p4 + � mπ √ 2δm −mπ √ 2δm dp−N2 πfeq 1 feq 2 = ˜ρI 48π√πN2 πm12 π (δm)3/2e−2x mV Γx5/2 = ˜ρI 36SfBdπ√πN2 πm10 π e−2x C4αdx5/2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='25) On the 2nd line ρI = 1 for I = ab and ρI = 2−5 for I = bc, ac;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' the suppression of the latter terms come from the factors of 2 in the definition of p± in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='23) (including a factor 1/2 from the Jacobian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' On the last line ˜ρi = 1 for I = ab, and ˜ρi = ρ1(128/25) � 2/5 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='1 for I = bc, ac, with the additional factor for I = bc, ac arising from the different p±-dependence of feq i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The final expression uses the explicit decay width eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='5) into pions, Bd = Γd/Γ the branching ratio, and mV ≈ 2mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' A careful inclusion of the integration boundary replaces (to first order in δm) e−2x → e−2x √ ˜s = e−2x−δmx, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='26) – 30 – which suppresses the interactions when the center of mass energy drops below the temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' This is the same exponential factor as found in the thermally averaged annihilation cross section eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' The I = bc, ac contributions are subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Expecting the mixed terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='18), which already come with a small coefficient, likewise to be subdominant, we can approximate the resonant interaction by the ˜∆−2 (ab)-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' Then ⟨σv2⟩res 3→2 ⟨σv2⟩5pnt 3→2 = γres 3→2 γ5pnt 3→2 ≈ �παd ξ2 �2 128 45 F( ˜∆−2 (ab)) F(1) = 256Sfπ2√π 15C4 αdx5/2 ξ4 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='27) where we have set a2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content=' References [1] LZ collaboration, First Dark Matter Search Results from the LUX-ZEPLIN (LZ) Experiment, 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} +page_content='03764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E3T4oBgHgl3EQfbQol/content/2301.04513v1.pdf'} 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We propose in this paper a new minimization algorithm based on a slightly +modified version of the scalar auxialiary variable (SAV) approach coupled with a relaxation +step and an adaptive strategy. It enjoys several distinct advantages over popular gradient +based methods: (i) it is unconditionally energy diminishing with a modified energy which is +intrinsically related to the original energy, thus no parameter tuning is needed for stability; +(ii) it allows the use of large step-sizes which can effectively accelerate the convergence rate. +We also present a convergence analysis for some SAV based algorithms, which include our +new algorithm without the relaxation step as a special case. We apply our new algorithm +to several illustrative and benchmark problems, and compare its performance with several +popular gradient based methods. The numerical results indicate that the new algorithm +is very robust, and its adaptive version usually converges significantly faster than those +popular gradient descent based methods. +1. Introduction +Minimization plays an important role in many branches of science and engineering. In +particular, how to accelerate the convergence rate of the minimization process is a cen- +tral issue in data science and machine learning problems. We consider in this paper an +unconstrained minimization problem +min +θ∈RN f(θ) +(1.1) +which arises in many applications, including particularly machine learning problems. For +large scale minimization problems, the first order methods such as gradient descent, its +variants such as stochastic gradient descent [21], Nesterov’s accelerated gradient descent +[17], adaptive momentum estimation method [15, 24, 10], are popular choices. We refer +to [19, 18, 23], and the references therein, for more detail on the design and analysis of +gradient descent method and its various variants. +The vanilla gradient descent method for (1.1) is +θk+1 = θk − ηk∇f(θk), +(1.2) +2020 Mathematics Subject Classification. 65K10,49M05,90C26. +Key words and phrases. +discrete gradient system; optimization; scalar auxiliary variable; adaptive; +stability; convergence. +1Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA (liu1957@purdue.edu, +shen7@purdue.edu, zhan1966@purdue.edu). +1 +arXiv:2301.02942v1 [math.NA] 7 Jan 2023 + +2 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +which can also be regarded as a numerical scheme for the gradient flow +θt = −∇f(θ), +(1.3) +with time step ηk. The gradient flow (1.3) is energy diminishing in the sense that +d +dtf(θ) = (∇f(θ), θt) = −(θt, θt) = −∥θt∥2 ≤ 0, +where (·, ·) (resp. ∥·∥) denotes the l2 inner product (resp. norm). However, gradient decent +type schemes are not necessarily energy diminishing, and may blow up if the time step is +too large. Although the stability of gradient descent based methods is well understood, +the main challenge in practice is how to choose the step-size, i.e. learning rate, to balance +between stability and efficiency [22]. +We propose in this paper a different class of minimization algorithms inspired from the +recently developed sclar auxiliary variable (SAV) approach for gradient flows [25, 26]. The +SAV approach enjoys a particular advantage of unconditional energy diminishing compared +to popular gradient decent based methods. This advantage avoids tuning step sizes and +allows the use of large step sizes, which may effectively accelerate the convergence rate. +Assume the cost function has a splitting +f(θ) = 1 +2(Lθ, θ) + [f(θ) − 1 +2(Lθ, θ)] := 1 +2(Lθ, θ) + g(θ), +(1.4) +where L is a self-adjoint positive semi-definite linear operator. Note that L = 0 is a trivial +splitting. Then, the gradient flow (1.3) becomes +θt + Lθ + ∇g(θ) = 0. +(1.5) +Inspired by the SAV approach [25], assuming there exists C > 0 such that g(θ) > −C for +all θ, we introduce a scalar auxiliary variable r(t) = +� +g(θ) + C, and expand (1.5) to: +� +� +� +θt + Lθ + +∇g(θ) +√ +g(θ)+C r = 0, +rt = +1 +2√ +g(θ)+C (∇g(θ), θt). +(1.6) +Obviously, with r(0) = +� +g(θ|t=0) + C, the above system admits a solution r(t) = +� +g(θ) + C +with θ being the solution of (1.5). The main advantage of the expanded system, which in- +cludes an energy evolution equation, is that it allows us to construct simple numerical +schemes with modified energy diminishing. For example, the following scheme +� +� +� +� +� +θk+1−θk +δt ++ Lθk+1 + +∇g(θk) +√ +g(θk)+C rk+1 = 0, +rk+1−rk +δt += +� +∇g(θk) +2√ +g(θk)+C , θk+1−θk +δt +� +, +(1.7) +can be easily implemented by solving only two linear systems of the form (I + δtL)x = b, +and is unconditionally energy stable with a modified energy [26]. +While the scheme (1.7) has been shown to be very effective for gradient flows, it is not +particularly suitable for the minimization problem (1.1). Indeed, for any fixed δt, assuming + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +3 +θk → θ∗ and rk → r∗, then +rk +√ +g(θk)+C → +r∗ +√ +g(θ∗)+C which is usually not equal to 1, and +consequently, the first equation of (1.7) leads to +0 = Lθ∗ + +r∗ +� +g(θ∗) + C +∇g(θ∗) = Lθ∗ + +r∗ +� +g(θ∗) + C +(−Lθ∗ + ∇f(θ∗)) . +If L ̸= 0, we observe from the above that in general ∇f(θ∗) ̸= 0 thus θ∗ is not a solution +for (1.1). Another complication of this approach is that it is not obvious how to choose L +such that g(θ) is bounded from below for all θ. The main goal of this paper is to design +suitable SAV based schemes for (1.1), develop their convergence theory, and validate them +through extensive numerical experiments. +The rest of the paper is organized as follows. In Section 2, we first discuss a suitable SAV +algorithm for minimization, introduce its relaxed version RSAV, and discuss the adaptive +rule and choices for the non-negative operator L(θ). Then, we present in Section 3 several +numerical results to show the performance of the RSAV in different optimization problems. +We provide a convergence study in Section 4, some concluding remarks in Section 5. +2. A new SAV approach and its relaxed version +We have observed in the last section that the standard SAV approach is not suitable for +the minimization problem (1.1). In this section, we propose a different SAV approach and +its related version which are well suited for (1.1). +2.1. A modified SAV approach. We still assume the splitting (1.4), and rewrite (1.3) +as +θt + Lθ + ∇f(θ) − Lθ = 0. +(2.1) +Since f(θ) in a minimization problem should always be bounded from below, there exists +C > 0 such that f(θ) > −C for all θ. We introduce r(t) = +� +f(θ) + C, and expand (2.1) +to: +� +� +� +θt + Lθ + +∇f(θ) +√ +f(θ)+C r − Lθ = 0, +rt = +1 +2√ +f(θ)+C (∇f(θ), θt). +(2.2) +Then, a simple SAV scheme to approximate the above is +� +� +� +� +� +θk+1−θk +δt ++ Lθk+1 + +∇f(θk) +√ +f(θk)+C rk+1 − Lθk = 0, +rk+1−rk +δt += +� +∇f(θk) +2√ +f(θk)+C , θk+1−θk +δt +� +. +(2.3) +Note that if θk → θ∗ and rk → r∗, then (2.3) leads to ∇f(θ∗) = 0, and consequently θ∗ is a +solution of (1.1). +The scheme (2.3) leads to a coupled linear system for (θk+1, rk+1), but it can be imple- +mented explicitly after solving a linear system as will be shown in the Section 4.1. Let + +4 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +A = I + δtL, then (2.3) can be equivalently implemented as +rk+1 = +1 +1 + δt (∇f(θk),A−1∇f(θk)) +2[f(θk)+C] +rk, +θk+1 = θk − +rk+1 +� +f(θk) + C +δtA−1∇f(θk). +Moreover, taking the discrete inner product of the first (resp. second) equation in (2.3) +with θk+1 − θk (resp. 2rk+1), summing up the results, we obtain the following: +Theorem 1. If L is non-negative, then for any δt > 0, the modified energy r2 in the scheme +(2.7) is unconditionally diminishing in the sense that +r2 +k+1 − r2 +k = − 1 +δt∥θk+1 − θk∥2 − (L(θk+1 − θk), (θk+1 − θk)) − (rk+1 − rk)2. +The above result shows the key advantage of (2.3): the energy dissipation holds for any +δt > 0 and any splitting with non-negative L. +As we shall demonstrate in numerical tests in Section 3, when the cost functional f(θ) has +a suitable splitting, the above algorithm usually converge much faster the vanilla gradient +decent method. When the cost function does not have any obvious quadratic part, we can +either choose any suitable non-negative linear operator L in (1.4), or simply take L = 0, +which results in a fully explicit method: +� +� +� +� +� +θk+1−θk +δt ++ +∇f(θk) +√ +f(θk)+C rk+1 = 0, +rk+1−rk +δt += +� +∇f(θk) +2√ +f(θk)+C , θk+1−θk +δt +� +, +(2.4) +which, we refer as the SAV gradient descent method. As will be shown in the Section 4.1, +the scheme (2.4) can be decoupled and implemented as +rk+1 = +rk +1 + δt (∇f(θk),∇f(θk)) +2(f(θk)+C) +, +θk+1 = θk − δt +rk+1 +� +f(θk) + C +∇f(θk). +(2.5) +Compared with the vanilla gradient descent method (1.2), there are extra computational +costs of +computing both f(θk) and (∇f(θk), ∇f(θk)) in (2.5), but Theorem 1 ensures +stability for any δt. In contrast, δt in (1.2) needs to be small enough to ensure stability. +2.2. A relaxed version of the modified SAV approach. While for fixed δt, the so- +lution of the SAV scheme (2.3) converges to a solution of the minimization problem (1.1), +the evolution of rk+1 is not directly linked to +� +f(θk+1) + C, and its value may decrease +rapidly to ensure stability when ∥∇f(θk) − Lθk∥ becomes large. In this case, the ratio +rk+1 +√ +f(θk+1)+C may deviate significantly from 1, which indicates that rk+1 is no longer a good +approximation of +� +f(θ(tk+1)) + C, thus θk+1 will not be a good approximation of θ(tk+1). +For dynamic simulation of gradient flows, a remedy is to monitor the ratio +rk+1 +√ +f(θk+1)+C and + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +5 +adjust the time step so that it stays close to 1. For the minimization problem (1.1), since +we are mainly interested in the steady steady state solutions of (1.3), it is found in [30] +that setting rk+1 = +� +f(θk+1) + C at each time step is also very effective. More precisely, +we can use the following modified SAV scheme: +� +� +� +� +� +� +� +� +� +θk+1−θk +δt ++ Lθk+1 + +∇f(θk) +√ +f(θk)+C ˜rk+1 − Lθk = 0, +˜rk+1−rk +δt += +� +∇f(θk) +2√ +f(θk)+C , θk+1−θk +δt +� +, +rk+1 = +� +f(θk+1) + C. +(2.6) +However, the above modified SAV scheme is no longer energy diminishing. Recently, an- +other way to link rk+1 with +� +f(θk+1) + C while still being energy diminishing is proposed +in [14] (see also [28]). When applied to (2.1), the relaxed SAV method takes the following +form: +� +� +� +� +� +� +� +� +� +θk+1−θk +δt ++ Lθk+1 + +∇f(θk) +√ +f(θk)+C ˜rk+1 − Lθk = 0, +˜rk+1−rk +δt += +� +∇f(θk) +2√ +f(θk)+C , θk+1−θk +δt +� +, +rk+1 = ξ˜rk+1 + (1 − ξ) +� +f(θk+1) + C. +(2.7) +Here, the relaxation parameter ξ is a scalar chosen from the set +V = {ξ ∈ [0, 1] : (rk+1)2 − (˜rk+1)2 − (˜rk+1 − rk)2 ≤ ηG(θk+1, θk)} +(2.8) +where G(θk+1, θk) = +1 +δt ((θk+1 − θk), A(θk+1 − θk)) with A = I + δtL, and η ∈ [0, 1] is an +artificial parameter with default value η = 0.99. In particular, it is shown in [14] that we +can choose +ξ = max{0, −b − +√ +b2 − 4ac +2a +}, +(2.9) +with the coefficients that +a = (˜rk+1 − +� +f(θk+1) + C)2 +(2.10) +b = 2 +� +˜rk+1 − +� +f(θk+1) + C +� � +f(θk+1) + C +(2.11) +c = f(θk+1) + C − (˜rk+1)2 − (˜rk+1 − rk)2 − ηG(θk+1, θk). +(2.12) +Taking the discrete inner product of the first (resp. +second) equation in (2.7) with +θk+1 − θk (resp. 2˜rk+1), summing up the results, we get +G(θk+1, θk) = 1 +δt ((θk+1 − θk), A(θk+1 − θk)) = −2(˜rk+1 − rk)˜rk+1, +(2.13) +then the non-zero choice of ξ can be rewritten as +ξ += +−b − +√ +b2 − 4ac +2a += +� +f(θk+1) + C − +� +(˜rk+1)2 + (˜rk+1 − rk)2 + ηG(θk+1, θk) +� +f(θk+1) + C − ˜rk+1 += +� +f(θk+1) + C − +� +(1 − η)˜r2 +k+1 + ηr2 +k + (1 − η)(˜rk+1 − rk)2 +� +f(θk+1) + C − ˜rk+1 +. + +6 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +The implementation of (2.7) is summarized in Algorithm 1. +Theorem 2. If L is non-negative and linear, then for any δt > 0, the modified energy r2 +in the scheme (2.7) is unconditionally diminishing in the sense that +r2 +k+1 − r2 +k = −(1 − η)G(θk+1, θk) ≤ 0. +Proof. By (2.13), we obtain +˜r2 +k+1 − r2 +k = − 1 +δt∥θk+1 − θk∥2 − (L(θk+1 − θk), (θk+1 − θk)) − (˜rk+1 − rk)2. +Adding r2 +k+1 − ˜r2 +k+1 on both sides, noticing +G(θk+1, θk) = 1 +δt∥θk+1 − θk∥2 + (L(θk+1 − θk), (θk+1 − θk)), +we obtain +r2 +k+1 − r2 +k = −G(θk+1, θk) − (˜rk+1 − rk)2 + r2 +k+1 − ˜r2 +k+1, +which implies the desired result thanks to (2.8). +□ +Algorithm 1 The basic RSAV scheme +1: Inputs: +δt: step-size, +C: constant to guarantee the positivity of f(x) + C, +A = I + δtL : the linear operator, +θ0: initial parameter vector +2: +r0 ← +� +f(θ0) + C +3: for k = 0, 1, 2, ..., M − 1 do +4: +gk = +∇f(θk) +√ +f(θk)+C +5: +ˆgk = A−1gk +6: +˜rk+1 = +rk +1+ δt +2 (gk,ˆgk) +7: +θk+1 = θk − δt˜rk+1ˆgk +8: +ξ = +√ +f(θk+1)+C− +� +(1−η)˜r2 +k+1+ηr2 +k+(1−η)(˜rk+1−rk)2 +√ +f(θk+1)+C−˜rk+1 +9: +ξ = max{0, ξ} +10: +rk+1 = ξ˜rk+1 + (1 − ξ) +� +f(θk+1) + C +return θM +2.3. Selection of the operator L. In the SAV approach for gradient flows [26], it is found +that a proper choice of the splitting (1.4), i.e., the choice of the quadratic term 1 +2(Lθ, θ), +can significantly increase the robustness and efficiency of the SAV schemes. For gradient +flows coming from materials science or fluid dynamics, there are usually obvious candidates +in the free energy. However, for minimization problems, particularly those from machine +learning problems, there are usually no obvious quadratic terms in the energy functions. + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +7 +In these cases, we can artificially choose some simple operators. In this paper, we consider +two simple operators below, for which the inverse operator (I + δtL)−1 can be efficiently +implemented. +2.3.1. Diagonal Matrix. In many optimization problems, an l2 regularization term is often +added into the loss function to avoid overfitting to the data in training sets, namely, +J(x) = f(x) + λ +2 ∥x∥2. +(2.14) +In this case, a natural choice is to set L = λI. More generally, we can use L = D with +D being a diagonal matrix with positive entries, e.g., D can be the diagonal entries of the +Hessian of the cost function. +2.3.2. Discrete Laplacian Matrix. In some machine learning problems, the discrete Lapla- +cian matrix is used as a smoothing operator which can reduce the variance during the +mini-batch training process [20]. This corresponds to L = −σ∆ where σ is a positive pa- +rameter and ∆ is a discrete Laplacian matrix, and (I + δtL)−1 can be efficiently inverted +by FFT based methods. The acceleration by using discrete Laplacian in classical primal +dual algorithms has been also justified in [13]. +2.4. An adaptive algorithm based on the RSAV scheme (2.7). Similar to the mod- +ified SAV scheme (2.3), the RSAV scheme (2.7) is also unconditionally energy diminishing. +A main advantage of unconditionally stable schemes is that one can adaptively adjust the +time step size to achieve faster convergence. In particular, we can use +Ik(r, θ) = +rk +� +f(θk) + C +(2.15) +as an indicator to control the deviation between modified energy and true energy. +For +solving differential equations θt = −∇f(θ), Ik(r, θ) should be as close as to 1 for the sake of +the time accuracy. But for a minimization problem, there is no time accuracy issue thus we +can allow Ik(r, θ) to deviate from 1 to achieve faster convergence. However, Ik(r, θ) needs +to be away from zero to avoid slow convergence, as the SAV and RSAV algorithms may +converge much slower than the vanilla gradient descent when the ratio Ik(r, θ) becomes too +small. +We observe from (2.7) that the true step-size for the gradient ∇f(θk) is +˜rk+1 +√ +f(θk)+C δt. +Thus if the ratio is small i.e. Ik(r, θ) < γ, the true step-size for the gradient would be too +small resulting in slow convergence. To this end, we present a simple adaptive rule with an +adaptive constant ρ > 1 with default value ρ = 1.1 described in Algorithm 2. +Remark 1. Note that in many applications of neural networks and machine learning, the +cost of computing the full batch is generally too high. In these cases, we can adopt the +mini-batch approach commonly used in stochastic gradient decent, and restart the RSAV +scheme at the beginning of each mini-batch. + +8 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +Algorithm 2 The adaptive RSAV scheme +1: Inputs: +δt0: initial step-size, δtmin: the lower bound of step-size, +C: constant to guarantee the positivity of f(x) + C, +A = I + δtL : the linear operator, +θ0: initial parameter vector, +ρ: adaptive constant which is greater than 1, +γ: threshold for the indicator I(r, θ). +2: +r0 ← +� +f(θ0) + C: Initialize the SAV, +3: for k = 0, 1, 2, ..., M − 1 do +4: +if +rk +√ +f(θk)+C < γ and δt > δtmin then +5: +δtk+1 = max{ +rk +√ +f(θk)+C δtk, δtmin} +6: +else +7: +δtk+1 = ρδtk +8: +9: +gk = +∇f(θk) +√ +f(θk)+C +10: +ˆgk = A−1gk +11: +˜rk+1 = +rk +1+ +δtk+1 +2 +(gk,ˆgk) +12: +θk+1 = θk − δtk+1˜rk+1ˆgk +13: +ξ = +√ +f(θk+1)+C− +� +(1−η)˜r2 +k+1+ηr2 +k+(1−η)(˜rk+1−rk)2 +√ +f(θk+1)+C−˜rk+1 +14: +ξ = max{0, ξ} +15: +rk+1 = ξ˜rk+1 + (1 − ξ) +� +f(θk+1) + C +return θM +3. Numerical Results +We present in this section several illustrative numerical experiments by using our RSAV +approach, and compare it with popular gradient based approaches. +In order to present a fair comparison to gradient descent (GD), we consider a composite +gradient method. By abuse of notation, we shall refer to GD with L as the following method +for the splitting (1.4): +θk+1 + δtLθk+1 = θk + δt(Lθk − ∇f(θk)), +which is equivalent to +θk+1 = θk − δt(I + δtL)−1∇f(θk). +(3.1) +The scheme (3.1) can also be regarded as the forward-backward splitting scheme +θk+1 = θk − δt∇F(θk) − δt∇G(θk+1) +for minimizing a composite function F(θ) + G(θ) with F(θ) = f(θ) − 1 +2θT Lθ and G(θ) = +1 +2θT Lθ. + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +9 +Note that the scheme (3.1) reduces to the vanilla gradient descent (1.2) if setting L = 0, +and (3.1) reduces to the vanilla gradient descent (1.2) with step size +δt +1+δt if setting L = I. +Therefore, we do not consider GD with L = I, and we compare the adaptive RSAV in +Algorithm 2 to the following algorithms: +(1) GD with L = 0, which is the vanilla gradient descent (1.2). +(2) GD with L = −σ∆ with discrete Laplacian ∆, which is similar to the Laplacian +Smoothing Gradient Descent [20]. +(3) ADAM [15] +(4) Nesterov accelerated gradient decent (NAG) [16]. +Unless specified otherwise, we use ADAM and NAG with the default parameter settings as +in [22]: NAG with γ = 0.9, and ADAM with β1 = 0.9, β2 = 0.999, ε = 10−8. +3.1. A quadratic cost function. We start with a quadratic loss function from [20]: +f(θ1, θ2, . . . , θ100) = +50 +� +i=1 +θ2 +2i−1 + +50 +� +i=1 +1 +100θ2 +2i. +(3.2) +For this simple function, we take either L = 0 or L = D where the diagonal matrix D is +chosen to be the Hessian ∇2f(θ). +To demonstrate the unconditional stability of SAV-based approaches, in Table 1 and +Figure 1 we show the results of different initial step sizes δt for the vanilla gradient descent, +i.e., GD with L = 0, as well as GD with L = D. We observe that the vanilla gradient +descent blows up for the constant step size δt = 1, while the adaptive RSAV works quite +well. +(initial) step-size δt +0.01 +0.1 +1 +GD (L = 0) +0.3351 +0.009121 +50 +adaptive RSAV (L = 0) +6.34e-12 +5.749e-12 +2.264e-18 +GD (L = D) +0.3352 +0.009194 +3.152e-18 +adaptive RSAV (L = D) +0 +0 +0 +Table 1. Loss of quadratic function after 1000 iterations. + +10 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +Figure 1. Loss curves for GD and adaptive RSAV with different splits and +(initial) step-sizes δt. +Next we consider the gradient perturbed by a Gaussian noise +∇ϵf(x) := ∇f(x) + ϵN(0, I), +(3.3) +where ϵ controls the noise level, N(0, I) is the Gaussian noise vector with zero mean and +unit variance in each coordinate. The comparison is given in Table 2 and Figure 2 where +L = 0 is used for both GD and adaptive RSAV. We observe that the adaptive RSAV +converges much faster than GD. The fast convergence of adaptive RSAV is partly due to +the indicator Ik(r, θ) which can give a proper step size. Especially in the noisy case, the +true step size is given at a proper level to reach a better convergence than GD and reduce +the oscillation in the loss curves. +(initial) step-size δt +0.01 +0.1 +1 +GD (ϵ = 0.01) +0.335 +0.009223 +58.58 +adaptive RSAV (ϵ = 0.01) +0.0002283 +0.0002298 +0.0002251 +GD (ϵ = 0.05) +0.3348 +0.01584 +diverge +adaptive RSAV (ϵ = 0.05) +0.004934 +0.005023 +0.004889 +GD (ϵ = 0.1) +0.3354 +0.03808 +diverge +adaptive RSAV (ϵ = 0.1) +0.01746 +0.01924 +0.0188 +Table 2. Loss of quadratic function after 1000 iterations with different +noise levels ϵ and (initial) step-sizes δt. L = 0 is used for both GD and +adaptive RSAV. + +100 +( +10-5 +一 +(4) +10-10 +GD(C=0.ot=0.01) +GD (L = 0, t = 0.1) +GD (L= 0, St = 1) +adaptive RSAV (L = 0. ot = 0.01) +10-15 +0 +200 +400 +600 +800 +1000 +iterations100 +( +10-5 +(4) +10-10 +GD (C = D. ot = 0.01) +GD (C = D, t = 0.1) +GD (C = D, ot = 1) +adaptive RSAV (C = D, St = 0.01 +10-15 +0 +200 +400 +600 +800 +1000 +iterationsA SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +11 +(a) ϵ = 0.01 +(b) ϵ = 0.05 +(c) ϵ = 0.1 +Figure 2. Loss curves for GD and adaptive RSAV with different noise +levels. The learning rate (lr) refers to the step size δt for GD and the initial +step size δt in the adaptive RSAV. +3.2. Rastrigin function. Consider +f(x) = f(θ1, θ2, . . . , θn) = +n +� +i=1 +θ2 +i + 10n − 10 +n +� +i=1 +cos(2πθi), +(3.4) +which has many local minima. The function can be defined on any input domain but it +is usually evaluated on x ∈ [−5.12, 5.12] for i = 1, 2, . . . , n. The function has a global +minimum at f(x∗) = 0 located at x∗ = (0, 0, . . . , 0). In this example, we compare the +adaptive RSAV with popular optimization methods ADAM and NAG with their default +parameter settings as in [22]: NAG with γ = 0.9, and ADAM with β1 = 0.9, β2 = 0.999, ε = +10−8. We shall keep using these default settings in all following experiments. +initial stepsize δt +0.001 +0.01 +0.1 +1 +GD (L = 0) +12.93 +37.86 +109.2 +diverge +adaptive RSAV (L = 0) +13.05 +13.03 +12.95 +2.608e-9 +NAG +12.93 +152 +505.1 +diverge +ADAM +32.28 +12.94 +12.93 +8.958 +Table 3. Loss of Rastrigin function after 100 iterations in 2D. +We plot in the left of Fig. 3 the convergence curves of different methods, and observe +that the RSAV converges much faster than ADAM with the same initial step-size. We +also plot in the right of Fig. 3 the paths towards the minimum by different methods. We +observe that RSAV enjoys a fast convergence if using a large initial step size δt. +3.3. Rosenbrock function. This is a benchmark problem for optimization of non-convex +functions. We first consider the 2D case with +f(x, y) = (a − x)2 + b(y − x2)2, +(3.5) + +104 +1()f -()fl +100 + 10-2 +10- +GD(C= 0.lr=0.1) +adaptive RSAV (C = 0, lr=1) +0 +200 +400 +600 +800 +1000 +iterations104 +1()f -()fl +100 +10~ +GD（C=0.lr=0.1) +adaptive RSAV (C = 0, lr=1 +0 +200 +400 +600 +800 +1000 +iterations10 +1()f -()fl +100 +10* +GD（C=0.lr=0.1) +adaptive RSAV (C = 0, lr=1 +0 +200 +400 +600 +800 +1000 +iterations12 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +Figure 3. Rastrigin function: Left: Convergence curves with 100 itera- +tions; Right: Paths with first 10 iterations. The learning rate (lr) refers to +the initial step size δt in the adaptive RSAV. +it has a global minimum at (x, y) = (a, a2), which is inside a long narrow, parabolic shaped +flag valley. To find the valley is trivial, but to converge to the global minimum is usually +difficult. +We set a = 1 and b = 100 in the following experiments and other parameters the same +as in [20], and start with the initial point with coordinate (−3, −4). In Table 4, we observe +that a large step size can lead to blow-up for other methods except for RSAV. Thus in +Figure 4, we only show the results with the largest suitable step sizes for other algorithms. +For adaptive RSAV, we just use the same initial step size as ADAM. This example reveals +the benefits of modified energy decreasing property of the RSAV. Although ADAM can +get close to the global minimum at first, it still goes to the wrong direction caused by the +momentum and eventually goes back after wasting many iterations. Only RSAV converges +to the global minimum directly with the guide of decreasing (modified) energy. +step-size δt +10−4 +10−2 +1 +GD (L = 0) +0.7142 +diverge +diverge +adaptive RSAV (L = 0) +0.01086 +0.01122 +0.0107 +NAG +5.326 +diverge +diverge +ADAM +15198 +12.5 +1.2 +Table 4. Loss of Rosenbrock function after 1000 iterations in 2D + +102 +100 +10-2 +(×)} - (x)l +10-4 +10-6 +GD (lr = 10-3) +10-8 +RSAV (lr = 1) +NAG (lr = 10-3) +ADAM (lr = 1) +10-10 +0 +20 +40 +60 +80 +100 +iterations4 +3 +2 +1 +0 +0 +-1 +X +Contour +-2 +GD (lr = 10-3) +"-RSAV (lr = 1) +-3 +NAG (lr = 10-3) +-ADAM (lr = 1) +-4 +global minimum +-4 +-2 +0 +2 +4A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +13 +Figure 4. 2D Rosenbrock problem: +Left: +Convergence curves; Right: +Paths with 1000 iterations in the (θ1, θ2) domain. +Next, we consider the high dimensional cases with +f(x) = +n +� +i=1 +(a − θi)2 + b +n−1 +� +i=1 +(θi+1 − θ2 +i )2, +(3.6) +with the global minimum f(x∗) = 0 at x∗ = (a, a2, a, a2, . . . , a, a2). We take a = 1 and +b = 100, and the initial point (0, . . . , 0). The results with the dimension equal to 10, 100 +and 1000 are shown in Fig. 5. We observe similar convergence behavior for all cases as in +the two dimensional case. +(a) n = 10 +(b) n = 100 +(c) n = 1000 +Figure 5. Loss of Rosenbrock function with dimension n. + +106 +104 +104 +If(xk) - f(x ) +100 +10-2 +GD (lr = 10-4 +10-4 +-RSAV (lr = 1) +NAG (lr = 10-4) +ADAM (lr = 1) +10-6 +0 +2000 +4000 +6000 +8000 +10000 +iterations106 +104 +102 +I( x)} - ("x) +100 +10-2 +GD (lr = 10-4 +10-4 +RSAV (lr = 1) +NAG (lr = 10-4) +ADAM (lr = 1) +10-6 +0 +2000 +4000 +6000 +8000 +10000 +iterations106 +104 +102 +I( ×)} - ("x)l +100 +10-2 +GD （lr = 10-4 +10-4 +RSAV (lr = 1) +NAG (lr = 10-4) +ADAM (lr = 1) +10-6 +0 +2000 +4000 +6000 +8000 +10000 +iterations104 +102 +(×)} - (*x)l +100 +10-2 +-GD (lr = 10-4) +RSAV (lr = 1) +NAG (lr = 10-4) +ADAM (lr = 1) +10-4 +200 +400 +600 +800 +1000 +iterations8 +6 +4 +2 +0 +-2 +Contour +-4 +-GD (lr = 10-4) +-RSAV (lr = 1) +NAG(1r= 10-4 +9- +-ADAM (lr = 1) + global minimum +-8 +-5 +0 +514 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +Remark 2. If we compare the adaptive RSAV with Nesterov accelerated gradient decent, +the results are still quite good especially when the dimension is 1000. Thanks to the adaptive +scheme, the performance of RSAV in this problem independent of the dimension. +3.4. Phase Retrieval. The phase retrieval problem [5] can be formulated as +min +z∈CN f(z) := 1 +2∥A(z∗z) − b∥2 +where z∗ ∈ C1×N is the conjugate transpose of z, b ∈ RM and A : CN×N −→ RM is +a linear operator. For the real-valued function f(z) with complex variable z := a + ib, +where i is the imaginary unit and a, b ∈ R are real and imaginary parts of z, we can define +the Fr´echet derivative induced by the natural choice of real inner product for CN as the +following [29]: +∇f(z) := ∂f(z) +a ++ i∂f(z) +b += 2A∗(A(z∗z − b))z, +where A∗ is the adjoint operator of A. Then the vanilla gradient descent algorithm for +minz∈CN f(z) can be defined as in (1.2) using ∇f(z) above. The gradient descent method +with a suitable step sizing rule is also also referred to as the Wirtinger flow [6]. +In particular, f(z) is a non-convex quartic polynomial function of z. For the theorectical +convergence of minimizing such a non-convex function, with a spectral initialization, i.e., z0 +being the leading eigenvector of A∗(b), the convergence of Wirtinger flow with high proba- +bility can be proven for a very special class of phase retrieval problems [6, 4]. For solving +phase retrieval with random initial guess, the convergence for minimizing a smoothed am- +plitude flow based model was proven in [3]. In terms of practical performance with only +random initialization, state-of-the-art algorithms such as the Riemannian LBFGS method +could be much more efficient than gradient descent algorithms [6]. +We emphasize that we only use such phase retrieval problems as a testing example to +validate the performance of the RSAV method. So we test the algorithms with a random +initialization. +We compare vanilla gradient descent (GD), adaptive RSAV with L = 0, and steepest +descent (SD) [8, 2]. The steepest descent method is to use the optimial step size in (1.2), +and it is possible to compute such an optimal step size for a polynomial cost function. We +test the performance of RSAV algorithm on the following phase retrieval problem. Let +Mi ∈ CN be i.i.d Gaussian and ◦ denote the entrywise product. Let F denote the Fourier +transform. The linear operator A is defined by assigning ∥F(Mi ◦ z)∥2 to b, e.g., M +N = m. +We consider the test case for the true solution z∗ being an image of size n×n with n = 256. +So the size of unknown is N = n2 = 2562. We consider two test cases: +(1) The true minimizer z∗ is a real image of camera man with size 256 × 256 as shown +in Figure 6, m = 6 Gaussian random masks and a random initial guess. +(2) The true minimizer z∗ is a complex image of golden ball with size 256 × 256, see +[12] for details, m = 10 Gaussian random masks and a random initial condition. + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +15 +See Figure 7 for the comparision of the performance of gradient based algorithms. For the +vanilla gradient descent (GD), we use the nearly largest stable constant step size δt = 0.5. +In Figure 8, we list a comparsion of the step size δk in the adaptive RSAV method with the +optimital step size in the steepest descent method. We can see the performance of adaptive +RSAV has the closest performance to the steepest descent. For the real image case, SD +converged after 10000 iterations and RSAV converged after 20000 iterations. However, to +compute the optimal step size in the steepest descent, at least two more evaluations of +A are needed, thus quite expensive. More importantly, for a general cost function, it is +difficult to find the optimal step size. See Section 4.4 for an analysis of the step size in the +explicit SAV gradient descent with restarting rk every iteration for quadratic functions. +(a) GD +(b) adaptive RSAV +(c) SD +Figure 6. Results after 20000 iterations for a phase retrieval problem with +z∗ being a 256 × 256 real image of camera man, m = 6 Gaussian random +masks and a random initial guess. The vanilla gradient descent (GD) uses +nearly largest stable constant step size δt = 0.5 + +16 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +Figure 7. Loss of different optimization algorithms for phase retrieval: +Left: the real image of camera man using 6 Gaussian masks; Right: the +complex image of golden balls using 10 Gaussian masks. The vanilla gradient +descent (GD) uses nearly largest stable step size δt = 0.5. +Figure 8. The value of δk in each iteration for phase retrieval: Left: the +real image of camera man using 6 Gaussian masks; Right: the complex +image of golden balls using 10 Gaussian masks. +3.5. Recommendation System. Consider applying the optimization scheme to train a +recommendation system based on matrix factorization model. Given a rate matrix R ∈ +Rm×n where m is the number of users and n is the number of items, the model learns the user +embedding matrix X ∈ Rm×d and item embedding matrix Y ∈ Rn×d such that the product + +10° +If(xk) - f(x )l +10-5 +10-10 +GD +RSAV +NAG +ADAM +SD +10-15 +0 +0.5 +1 +1.5 +2 +iterations +×105105 +100 +I( ×)} - ("x) +10-5 +GD +10-10 +RSAV +SD +NAG +ADAM +10-15 +100 +105 +iterations104 +103 +stepsize t +102 +101 +SD +RSAV +0 +0.5 +1 +1.5 +2 +2.5 +3 +iterations +×104104 +103 +stepsize +102 +101 +SD +RSAV +0 +200 +400 +600 +800 +1000 +iterationsA SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +17 +XY T is a good approximation for R. Here, d is the embedding dimension and usually much +smaller than m and n. Denote the user and item matrix by X = [X1, . . . , Xu, . . . , Xm]T +and Y = [Y1, . . . , Yi, . . . , Yn]T , we have the loss function as +f(X, Y ) = 1 +Nκ +� +(u,i)∈κ +(Ru,i − XuY T +i )2 + λu +� +u +∥Xu∥2 +2 + λi +� +i +∥Yi∥2 +2, +(3.7) +where κ the training set that the (u, i) pairs for which Ru,i is known, Nκ is the number of +training data, λu and λi are the penalty parameters for embedding matrix. We train the +model with the MovieLens 100K dataset [11] which contains 100, 000 ratings (1 − 5) from +943 users on 1682 movies. There is 80% data split as the training data and the rest date is +used for the testing data, e.g., the training data set κ has size 80, 000. All algorithms use +the mini-batch gradient with batch size 80. For l2 regularization, we set λu = λi = 10−4. +For the linear operator L = λI − σ∆ in GD and RSAV, we let λ = 10−4 and σ = 0.1. In +Figure 9, for the training step, we run 10, 000 iterations for the mini-batch gradient based +methods with batch size 80, which is equal to 10 epochs. The result on the test data is +shown in Table 5. We can see that RSAV performs well in the training step, though its +testing result is not the best, which suggests issues of overfitting during the training step. +This is more or less a modelling issue, rather than the optimizaiton issue. +Figure 9. The training loss curve of different optimization algorithms for +Recommendation System. Here GD refers to GD (L = λI −σ∆) and RSAV +refers to the adaptive RSAV (L = λI − σ∆). + +102 +101 +S +SSO +100 +10 +GD (St = 1) +RSAV (St = 0.01) +NAG (St = 0.1) +ADAM (St = 0.01) +10-2 +2000 +4000 +6000 +8000 +10000 +iterations18 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +step-size δt +0.01 +0.1 +1 +10 +GD (L = λI − σ∆) +4.6504 +2.1465 +1.6438 +diverge +NAG +2.1451 +1.6439 +diverge +diverge +ADAM +1.8820 +4.7194 +diverge +diverge +adaptive RSAV (L = λI − σ∆) +1.9090 +1.9102 +1.9156 +1.9156 +Table 5. The loss function on the test data after 10, 000 training iterations +(10 epochs) with different step-sizes. Here diverge means that the training +step already diverges. +4. Convergence study for some SAV based algorithms +In this section, we consider a more general version of the SAV scheme based on the +following expanded system +� +θt = − +� +r +[f(θ)+C]q ∇f(θ) + Lθ − Lθ +� +rt = q[f(θ) + C]q−1(∇f(θ), θt), +(4.1) +where r(t) = [f(θ) + C]q and q ∈ (0, 1). Note that (2.2) is a special case of the above +formulation with q = 1 +2. Similar to (2.3), we can construct a SAV scheme for (4.1) as +follows: +� θk+1−θk +δt += − +� +rk+1 +[f(θk)+C]q ∇f(θk) + L(θk+1 − θk) +� +rk+1−rk +δt += q[f(θk) + C]q−1(∇f(θk), θk+1−θk +δt +). +(4.2) +4.1. Interpretation of the SAV method as a line search method. Let A = (I +δtL). +The system (4.2) can be rewrite as +� +A +δt +∇f(θk) +[f(θk)+C]q +−q[f(θk) + C]q−1∇f(θk) +1 +� � +θk+1 +rk+1 +� += +� +Aθk +rk − q[f(θk) + C]q−1(∇f(θk), θk) +� +After a simple Gaussian elimination, we obtain an explicit update formula for (4.2): +� +� +� +rk+1 += +1 +1+δtq (∇f(θk),A−1∇f(θk)) +f(θk)+C +rk +θk+1 += θk − +rk+1 +[f(θk)+C]q δtA−1∇f(θk) +. +Notice that the scheme above can be regarded as a line search method: +θk+1 = θk + αkPk +Pk = −A−1∇f(θk) +αk = +δt +1 + δtq (∇f(θk),A−1∇f(θk)) +f(θk)+C +rk +[f(θk) + C]q > 0, +with a search direction Pk and step size αk. +The step size αk is guranteed to be positive. On the other hand, it is difficult to establish +any a priori control of αk, and in practice αk could become very small if rk becomes very + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +19 +small. +To avoid small rk, we consider a special version of SAV method by redefining +rk = [f(θk) + C]q, then we have +αk = +δt +1 + δtq (∇f(θk),A−1∇f(θk)) +f(θk)+C +. +In this case, we can view q as a parameter, and the SAV method with rk = [f(θk) + C]q at +every iteration becomes the following line search method: +θk+1 = θk + αkPk +(4.4a) +Pk = −A−1∇f(θk) +(4.4b) +αk = +δt +1 + δtq (∇f(θk),A−1∇f(θk)) +f(θk)+C +, +(4.4c) +which is equivalent to +� +� +� +� +� +� +� +rk = [f(θk) + C]q +θk+1−θk +δt += − +� +˜rk+1 +[f(θk)+C]q ∇f(θk) + L(θk+1 − θk) +� +˜rk+1−rk +δt += q[f(θk) + C]q−1(∇f(θk), θk+1−θk +δt +). +(4.5) +In particular, for any L ≥ 0, A−1 is always positive definite, thus the search direction +Pk = −A−1∇fk is always a descent direction, i.e., −∇fT (θk)Pk = ∇f(θk)T A−1∇f(θk) > 0. +The Wolfe condition [27] for the line search method (4.4) is: there exists 0 < c1 < c2 < 1 +such that +f(θk + αkPk) ≤ f(θk) + c1αk∇f(θk)T Pk +(4.6a) +∇f(θk + αkPk)T Pk ≥ c2∇f(θk)T Pk. +(4.6b) +We recall first the following result [19]: +Theorem 3. Assume f(θ) ∈ C1 and f(θ) is bounded from below. For any descent direction +Pk, there exist intervals of step lengths satisfying the Wolfe condition. +Notice that α(δt, q) = +δt +1+δtq (∇f(θk),A−1∇f(θk)) +f(θk)+C +is an increasing function of δt and an de- +creasing function of q, thus there exists δt and q such that αk = +δt +1+δtq (∇f(θk),A−1∇f(θk)) +f(θk)+C +satisfies the Wolfe conditions (4.6). +We recall below another result [19]: +Theorem 4. Assume f(θ) ∈ C1 and ∇f(θ) is Lipschitz continuous. Let cos φk = +−∇f(θk)T Pk +∥∇f(θk)∥∥Pk∥. +If Pk is a descent direction and αk satisfies the Wolfe Conditions, then the iteration +θk+1 = θk + αkPk satisfies +� +k≥0 +cos2 φk∥∇f(θk)∥2 < ∞. +(4.7) + +20 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +Let λmin(A) and λmax(A) be the smallest and largest eigenvalues of the real symmetric +positive definite matrix A, then by the Courant-Fischer-Weyl min-max principle [7] and +the spectral norm ∥A∥ = λmax(A), we have +P T +k APk +∥Pk∥2 ≥ λmin(A), +∥APk∥ ≤ ∥A∥∥Pk∥ = λmax(A)∥Pk∥, +thus +cos φk = +P T +k APk +∥APk∥∥Pk∥ = P T +k APk +∥Pk∥2 +∥Pk∥ +∥APk∥ ≥ λmin(A) +λmax(A). +Therefore, the uniform lower bound on cos φk and (4.7) implies that ∥∇f(θk)∥ → 0. +Thus the convergence of the SAV method (4.4) is ensured if using a line search to find +δt, q such that αk satisfies the Wolfe condition (4.6). We observe that the above algorithm +involves computing A−1∇f(θk), and evaluation of f(θk) and f(θk+1). +Remark 3. In practice, one can use backtracking line search on αk to ensure that the +Wolfe conditions are satisfied. This in general it does not seem advantageous over a simple +backtracking line search on α. However, for the SAV gradient descent method (2.4), i.e., +A = I, our numerical observation is that the SAV scheme is often more efficient than the +backtracking line search on α. With A = I, the scheme (4.4) reduces to the following SAV +gradient descent method with two parameters δt > 0 and qk > 0: +θk+1 = θk − αk∇f(θk) +(4.8a) +αk = +δt +1 + δtqk +∥∇f(θk)∥2 +f(θk)+C +(4.8b) +4.2. Standard convergence results. We recall that if αk in the line search method (4.4) +satisfies the Goldstein-Armijo rule [1, 9]: there exists 0 < c1 < c2 < 1 such that +f(θk) − c2αk∥∇fk∥2 ≤ f(θk − αk∇fk) ≤ f(θk) − c1αk∥∇fk∥2, +(4.9) +then it is shown (cf. Theorem 2.1.13 in [17]) that ∥∇fk∥ → 0. +Theorem 2.1.13 in [17] can be easily adapted to prove the following result for the line +search method (4.4): +Theorem 5. Assume that ∇f is Lipshitz continuous with the Lipshitz constant L, i.e., +∥∇f(y) − ∇f(x)∥ ≤ L∥x − y∥. If ∇f(θ∗) = 0 and αk ∈ (0, 2 +L), then +∥θk+1 − θ∗∥2 ≤ ∥θk − θ∗∥2 − αk( 2 +L − αk)∥∇f(θk)∥2 +and +f(θk) − f(θ∗) ≤ +1 +[f(θ0) − f(θ∗)]−1 + ∥θ0 − θ∗∥−2 � +k αk(1 − L +2 αk). +So for convergence, we need +∞ +� +k=0 +αk(1− L +2 αk) = +∞, which can be ensured if αk ∈ [a, b] ∈ +(0, 2 +L) for constant bounds a > 0 and b < 2 +L. Also, αk < 2 +L will ensure f(θk+1) < f(θk). + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +21 +4.3. Decreasing step sizes for the SAV gradient descent method. We derive from +(4.8) that +αk = +δt +1 + δtqk +∥∇fk∥2 +fk += +1 +1/δt + qk +∥∇fk∥2 +fk +≤ min +� +δt, +fk +qk∥∇fk∥2 +� +. +For fixed δt, the above is often sufficient to ensure f(θk+1) < f(θk) when θk is far away +from the minimizer. It can be understood as follows. +Theorem 6. Assume that f is strongly convex, i.e., (∇f(y) − ∇f(x), y − x) ≥ m∥x − y∥2 +with m > 0, and ∇f is Lipshitz continuous with the Lipshitz constant L. Let θ∗ be the +minimizer and assume f(θ∗) = 0. Then the following are sufficient conditions to ensure +αk < 2 +L for the SAV gradient descent method with two parameters (4.8): +(1) For any δt > 0, qk > +L2 +4m2 . +(2) Let δt ≡ a 2 +L where a > 0, qk > a−1 +a +L2 +4m2 . +Remark 4. The first sufficient condition implies that αk = +δt +1+δtqk +∥∇fk∥2 +fk +will be a decreasing +step size for any δt if qk ≡ q > +L2 +4m2 . Of course, finding 1 +q < 4m2 +L2 +in general is not easier +than finding δt < 2 +L. But if 2m2 > L, then 4m2 +L2 > 2 +L implies 1 +q < 4m2 +L2 is easier to achieve. +Remark 5. As an example of the second sufficient condition, if we pick qk ≡ 1 +2, and a = 2, +then 1 +2 > 1 +2 +L2 +4m2 ⇔ L < 2m is sufficient to ensure the SAV gradient descent method with +qk ≡ 1 +2 is decreasing with δt = 4 +L, instead of δt < 2 +L in (1.2). +Proof. First, by the strong convexity and Lipschitz continuity, we have +f(x) ≥ f(y) + (∇f(y), x − y) + m +2 ∥x − y∥2, +f(x) ≤ f(y) + (∇f(y), x − y) + L +2 ∥x − y∥2. +Since θ∗ is the minimizer, ∇f(θ∗) = 0. For any θ, we have +(∇f(θ) − ∇f(θ∗), θ − θ∗) ≥ m∥θ − θ∗∥2 ⇒ (∇f(θ), θ − θ∗) ≥ m∥θ − θ∗∥2 +⇒ m ∥θ − θ∗∥ +∥∇f(θ)∥ = (∇f(θ), θ − θ∗) +∥θ − θ∗∥∥∇f(θ)∥ ≤ 1 ⇒ ∥∇f(θ)∥ ≥ m∥θ − θ∗∥. +Hence, +m∥θ − θ∗∥ ≤ ∥∇f(θ)∥ ≤ L∥θ − θ∗∥, +and +m +2 ∥θ − θ∗∥2 ≤ f(θ) − f(θ∗) ≤ L +2 ∥θ − θ∗∥2. +With strong convexity m > 0, we have +2m2 +L +≤ +∥∇f(θ)∥2 +f(θ) − f(θ∗) ≤ 2L2 +m + +22 +XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 +thus +2m2 +L +≤ ∥∇f(θ)∥2 +f(θ) +≤ 2L2 +m . +Finally, +fk +qk∥∇fk∥2 < 2 +L ⇔ 1 +qk +< ∥∇f(θk)∥2 +f(θk) +2 +L ⇐ 1 +qk +< 4m2 +L2 . +□ +Remark 6. In general, if f(θ) is only convex but not strong convex, i.e., m = 0, and +f(θ) + C > 0, then we only have +∥∇f(θ)∥2 +f(θ) + C ≤ L2∥θ − θ∗∥2 +f(θ∗) + C . +This gives a lower bound control on step size: +αk = +δt +1 + δtqk +∥∇fk∥2 +fk+C +≥ +δt +1 + qkδt L2∥θk−θ∗∥2 +f(θ∗)+C +. +In this case, we can set qk ≡ q and do back tracking on δt for αk to satisfy the convergence +condition or Goldstein-Armijo rule. +4.4. The step size for quadratic functions. To see why the step size αk = +δtk +1+δtkqk +∥∇fk∥2 +fk+C +could be a good step size to use, at least for a quadratic cost function, consider a cost +function f(θ) = 1 +2∥Aθ − b∥2 with a square and positive definite matrix A > 0. The steepest +descent algorithm can be written as +θk+1 = θk − βk∇f(θk), +βk = +∥AT (Aθk − b)∥2 +[AT (Aθk − b)]T AT A[AT (Aθk − b)]. +The method (4.8) with C = 0 is +θk+1 = θk − αk∇f(θk), +αk = +1 +1 +δtk + qk +∥AT (Aθk−b)∥2 +1 +2 (Aθk−b)T (Aθk−b) +. +Then for a very large δtk and qk ≡ 1 +2, we have +αk ≈ +(Aθk − b)T (Aθk − b) +(Aθk − b)T AAT (Aθk − b), +βk = +(Aθk − b)T AAT (Aθk − b) +(Aθk − b)T AAT AAT (Aθk − b). +Let vi be orthornormal eigenvectors of A with eigenvalues of λi. Since vi form a basis for +RN, let Aθk − b = r = � +i rivi. Let z = AT (Aθk − b), then z = AT � +i rivi = � +i riλivi and +Az = � +i riλ2 +i vi . We get +αk ≈ +rT r +rT AAT r = rT r +zT z = +[� +i rivi]T [� +i rivi] +[� +i riλivi]T [� +i riλivi] = +� +i r2 +i +� +i λ2 +i r2 +i +, +βk = +zT z +(Az)T (Az) = +� +i λ2 +i r2 +i +� +i λ4 +i r2 +i +. +We can see that αk is very similar to the optimal step size βk but not the same. In practice, +a random initial guess θ0 usually makes αk a descent step size in the first few or many +iterations for δtk ≡ 1 and qk ≡ q = 1 +2. + +A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS +23 +5. Concluding remarks +We proposed in this paper a new minimization algorithm inspired by the scalar auxil- +iary variable (SAV) approach for gradient flows. Since the direct application of the SAV +approach to minimization problems may converge to wrong solutions, we developed a mod- +ified version of the SAV approach coupled with a relaxation step and an adaptive stradegy. +The new algorithm enjoys several distinct advantages, including unconditionally energy di- +minishing with a modified energy, and empirical better performance than many first order +methods. In particular, it overcomes the difficulty in selecting proper step sizes associ- +ated with the usual gradient based algorithms. The energy diminishing property ensures +the convergence, and the relaxation step, built on a connection between the decreasing +modified energy and the original energy, helps to accelerate the convergence. +We also presented a converence analysis for some SAV based algorithms which include the +new algorithm without the relaxation step as a special case. Numerical results for several +illustrative and benchmark problems indicates that the new algorithm is very robust and +usually converges significantly faster than those popular gradient decent based methods. +While we only considered a basic version of the SAV based approach which already +showed its promise, it is clear that this approach can be combined with other techniques +of acceleration and generalization to the gradient decent methods. How to further improve +the robustness and accelerate the convergence rate of the SAV based approach will be the +subject of a future study. +Acknowledgement +This work is partially supported by AFOSR FA9550-16-1-0102, NSF DMS-2012585 and +DMS-2208518. +References +[1] Larry Armijo. Minimization of functions having Lipschitz continuous first partial derivatives. Pacific +Journal of mathematics, 16(1):1–3, 1966. +[2] Arthur Earl Bryson and Walter F Denham. 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Osher, Bao Wang, Penghang Yin, Xiyang Luo, Minh Pham, and Alex Tong Lin. Laplacian +smoothing gradient descent. CoRR, abs/1806.06317, 2018. +[21] Herbert Robbins and Sutton Monro. A stochastic approximation method. The annals of mathematical +statistics, pages 400–407, 1951. +[22] Sebastian +Ruder. +An +overview +of +gradient +descent +optimization +algorithms. +arXiv +preprint +arXiv:1609.04747, 2016. +[23] Ernest K Ryu and Wotao Yin. Large-Scale Convex Optimization: Algorithms & Analyses via Monotone +Operators. Cambridge University Press, 2022. +[24] J Reddi Sashank, Kale Satyen, and Kumar Sanjiv. On the convergence of adam and beyond. In Inter- +national Conference on Learning Representations, volume 5, page 7, 2018. +[25] Jie Shen, Jie Xu, and Jiang Yang. The scalar auxiliary variable (sav) approach for gradient flows. +Journal of Computational Physics, 353, 10 2017. +[26] Jie Shen, Jie Xu, and Jiang Yang. A new class of efficient and robust energy stable schemes for gradient +flows. SIAM Review, 61(3):474–506, 2019. +[27] Philip Wolfe. Convergence conditions for ascent methods. SIAM review, 11(2):226–235, 1969. +[28] Yanrong Zhang and Jie Shen. A generalized sav approach with relaxation for dissipative systems. +Journal of Computational Physics, page 111311, 2022. +[29] Shixin Zheng, Wen Huang, Bart Vandereycken, and Xiangxiong Zhang. Riemannian optimization using +three different metrics for Hermitian PSD fixed-rank constraints: an extended version. arXiv preprint +arXiv:2204.07830, 2022. +[30] Qingqu Zhuang and Jie Shen. Efficient SAV approach for imaginary time gradient flows with applica- +tions to one- and multi-component Bose-Einstein condensates. J. Comput. Phys., 396:72–88, 2019. + diff --git a/VNE1T4oBgHgl3EQfIwMb/content/tmp_files/load_file.txt b/VNE1T4oBgHgl3EQfIwMb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f908658f28fa5c56e566f39970cb201a50d7cce --- /dev/null +++ b/VNE1T4oBgHgl3EQfIwMb/content/tmp_files/load_file.txt @@ -0,0 +1,849 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf,len=848 +page_content='AN EFFICIENT AND ROBUST SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS ARISING FROM OPTIMIZATIONS XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We propose in this paper a new minimization algorithm based on a slightly modified version of the scalar auxialiary variable (SAV) approach coupled with a relaxation step and an adaptive strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' It enjoys several distinct advantages over popular gradient based methods: (i) it is unconditionally energy diminishing with a modified energy which is intrinsically related to the original energy, thus no parameter tuning is needed for stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (ii) it allows the use of large step-sizes which can effectively accelerate the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We also present a convergence analysis for some SAV based algorithms, which include our new algorithm without the relaxation step as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We apply our new algorithm to several illustrative and benchmark problems, and compare its performance with several popular gradient based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The numerical results indicate that the new algorithm is very robust, and its adaptive version usually converges significantly faster than those popular gradient descent based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Introduction Minimization plays an important role in many branches of science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In particular, how to accelerate the convergence rate of the minimization process is a cen- tral issue in data science and machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We consider in this paper an unconstrained minimization problem min θ∈RN f(θ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) which arises in many applications, including particularly machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For large scale minimization problems, the first order methods such as gradient descent, its variants such as stochastic gradient descent [21], Nesterov’s accelerated gradient descent [17], adaptive momentum estimation method [15, 24, 10], are popular choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We refer to [19, 18, 23], and the references therein, for more detail on the design and analysis of gradient descent method and its various variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The vanilla gradient descent method for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) is θk+1 = θk − ηk∇f(θk), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 65K10,49M05,90C26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' discrete gradient system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' scalar auxiliary variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' adaptive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 1Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA (liu1957@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='edu, shen7@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='edu, zhan1966@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='02942v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='NA] 7 Jan 2023 2 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 which can also be regarded as a numerical scheme for the gradient flow θt = −∇f(θ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) with time step ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The gradient flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) is energy diminishing in the sense that d dtf(θ) = (∇f(θ), θt) = −(θt, θt) = −∥θt∥2 ≤ 0, where (·, ·) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' ∥·∥) denotes the l2 inner product (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' However, gradient decent type schemes are not necessarily energy diminishing, and may blow up if the time step is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Although the stability of gradient descent based methods is well understood, the main challenge in practice is how to choose the step-size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' learning rate, to balance between stability and efficiency [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We propose in this paper a different class of minimization algorithms inspired from the recently developed sclar auxiliary variable (SAV) approach for gradient flows [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The SAV approach enjoys a particular advantage of unconditional energy diminishing compared to popular gradient decent based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This advantage avoids tuning step sizes and allows the use of large step sizes, which may effectively accelerate the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Assume the cost function has a splitting f(θ) = 1 2(Lθ, θ) + [f(θ) − 1 2(Lθ, θ)] := 1 2(Lθ, θ) + g(θ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) where L is a self-adjoint positive semi-definite linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Note that L = 0 is a trivial splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Then, the gradient flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) becomes θt + Lθ + ∇g(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5) Inspired by the SAV approach [25], assuming there exists C > 0 such that g(θ) > −C for all θ, we introduce a scalar auxiliary variable r(t) = � g(θ) + C, and expand (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5) to: � � � θt + Lθ + ∇g(θ) √ g(θ)+C r = 0, rt = 1 2√ g(θ)+C (∇g(θ), θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6) Obviously, with r(0) = � g(θ|t=0) + C, the above system admits a solution r(t) = � g(θ) + C with θ being the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The main advantage of the expanded system, which in- cludes an energy evolution equation, is that it allows us to construct simple numerical schemes with modified energy diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For example, the following scheme � � � � � θk+1−θk δt + Lθk+1 + ∇g(θk) √ g(θk)+C rk+1 = 0, rk+1−rk δt = � ∇g(θk) 2√ g(θk)+C , θk+1−θk δt � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) can be easily implemented by solving only two linear systems of the form (I + δtL)x = b, and is unconditionally energy stable with a modified energy [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' While the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) has been shown to be very effective for gradient flows, it is not particularly suitable for the minimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Indeed, for any fixed δt, assuming A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 3 θk → θ∗ and rk → r∗, then rk √ g(θk)+C → r∗ √ g(θ∗)+C which is usually not equal to 1, and consequently, the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) leads to 0 = Lθ∗ + r∗ � g(θ∗) + C ∇g(θ∗) = Lθ∗ + r∗ � g(θ∗) + C (−Lθ∗ + ∇f(θ∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' If L ̸= 0, we observe from the above that in general ∇f(θ∗) ̸= 0 thus θ∗ is not a solution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Another complication of this approach is that it is not obvious how to choose L such that g(θ) is bounded from below for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The main goal of this paper is to design suitable SAV based schemes for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1), develop their convergence theory, and validate them through extensive numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In Section 2, we first discuss a suitable SAV algorithm for minimization, introduce its relaxed version RSAV, and discuss the adaptive rule and choices for the non-negative operator L(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Then, we present in Section 3 several numerical results to show the performance of the RSAV in different optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We provide a convergence study in Section 4, some concluding remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A new SAV approach and its relaxed version We have observed in the last section that the standard SAV approach is not suitable for the minimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In this section, we propose a different SAV approach and its related version which are well suited for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A modified SAV approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We still assume the splitting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4), and rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) as θt + Lθ + ∇f(θ) − Lθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) Since f(θ) in a minimization problem should always be bounded from below, there exists C > 0 such that f(θ) > −C for all θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We introduce r(t) = � f(θ) + C, and expand (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) to: � � � θt + Lθ + ∇f(θ) √ f(θ)+C r − Lθ = 0, rt = 1 2√ f(θ)+C (∇f(θ), θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) Then, a simple SAV scheme to approximate the above is � � � � � θk+1−θk δt + Lθk+1 + ∇f(θk) √ f(θk)+C rk+1 − Lθk = 0, rk+1−rk δt = � ∇f(θk) 2√ f(θk)+C , θk+1−θk δt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) Note that if θk → θ∗ and rk → r∗, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) leads to ∇f(θ∗) = 0, and consequently θ∗ is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) leads to a coupled linear system for (θk+1, rk+1), but it can be imple- mented explicitly after solving a linear system as will be shown in the Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let 4 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 A = I + δtL, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) can be equivalently implemented as rk+1 = 1 1 + δt (∇f(θk),A−1∇f(θk)) 2[f(θk)+C] rk, θk+1 = θk − rk+1 � f(θk) + C δtA−1∇f(θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Moreover, taking the discrete inner product of the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' second) equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) with θk+1 − θk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2rk+1), summing up the results, we obtain the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' If L is non-negative, then for any δt > 0, the modified energy r2 in the scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) is unconditionally diminishing in the sense that r2 k+1 − r2 k = − 1 δt∥θk+1 − θk∥2 − (L(θk+1 − θk), (θk+1 − θk)) − (rk+1 − rk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The above result shows the key advantage of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3): the energy dissipation holds for any δt > 0 and any splitting with non-negative L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' As we shall demonstrate in numerical tests in Section 3, when the cost functional f(θ) has a suitable splitting, the above algorithm usually converge much faster the vanilla gradient decent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' When the cost function does not have any obvious quadratic part, we can either choose any suitable non-negative linear operator L in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4), or simply take L = 0, which results in a fully explicit method: � � � � � θk+1−θk δt + ∇f(θk) √ f(θk)+C rk+1 = 0, rk+1−rk δt = � ∇f(θk) 2√ f(θk)+C , θk+1−θk δt � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) which, we refer as the SAV gradient descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' As will be shown in the Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1, the scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) can be decoupled and implemented as rk+1 = rk 1 + δt (∇f(θk),∇f(θk)) 2(f(θk)+C) , θk+1 = θk − δt rk+1 � f(θk) + C ∇f(θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5) Compared with the vanilla gradient descent method (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2), there are extra computational costs of computing both f(θk) and (∇f(θk), ∇f(θk)) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5), but Theorem 1 ensures stability for any δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In contrast, δt in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) needs to be small enough to ensure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A relaxed version of the modified SAV approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' While for fixed δt, the so- lution of the SAV scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) converges to a solution of the minimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1), the evolution of rk+1 is not directly linked to � f(θk+1) + C, and its value may decrease rapidly to ensure stability when ∥∇f(θk) − Lθk∥ becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In this case, the ratio rk+1 √ f(θk+1)+C may deviate significantly from 1, which indicates that rk+1 is no longer a good approximation of � f(θ(tk+1)) + C, thus θk+1 will not be a good approximation of θ(tk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For dynamic simulation of gradient flows, a remedy is to monitor the ratio rk+1 √ f(θk+1)+C and A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 5 adjust the time step so that it stays close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For the minimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1), since we are mainly interested in the steady steady state solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3), it is found in [30] that setting rk+1 = � f(θk+1) + C at each time step is also very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' More precisely, we can use the following modified SAV scheme: � � � � � � � � � θk+1−θk δt + Lθk+1 + ∇f(θk) √ f(θk)+C ˜rk+1 − Lθk = 0, ˜rk+1−rk δt = � ∇f(θk) 2√ f(θk)+C , θk+1−θk δt � , rk+1 = � f(θk+1) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6) However, the above modified SAV scheme is no longer energy diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Recently, an- other way to link rk+1 with � f(θk+1) + C while still being energy diminishing is proposed in [14] (see also [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' When applied to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1), the relaxed SAV method takes the following form: � � � � � � � � � θk+1−θk δt + Lθk+1 + ∇f(θk) √ f(θk)+C ˜rk+1 − Lθk = 0, ˜rk+1−rk δt = � ∇f(θk) 2√ f(θk)+C , θk+1−θk δt � , rk+1 = ξ˜rk+1 + (1 − ξ) � f(θk+1) + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) Here, the relaxation parameter ξ is a scalar chosen from the set V = {ξ ∈ [0, 1] : (rk+1)2 − (˜rk+1)2 − (˜rk+1 − rk)2 ≤ ηG(θk+1, θk)} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8) where G(θk+1, θk) = 1 δt ((θk+1 − θk), A(θk+1 − θk)) with A = I + δtL, and η ∈ [0, 1] is an artificial parameter with default value η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In particular, it is shown in [14] that we can choose ξ = max{0, −b − √ b2 − 4ac 2a }, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9) with the coefficients that a = (˜rk+1 − � f(θk+1) + C)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10) b = 2 � ˜rk+1 − � f(θk+1) + C � � f(θk+1) + C (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='11) c = f(θk+1) + C − (˜rk+1)2 − (˜rk+1 − rk)2 − ηG(θk+1, θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='12) Taking the discrete inner product of the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' second) equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) with θk+1 − θk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2˜rk+1), summing up the results, we get G(θk+1, θk) = 1 δt ((θk+1 − θk), A(θk+1 − θk)) = −2(˜rk+1 − rk)˜rk+1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='13) then the non-zero choice of ξ can be rewritten as ξ = −b − √ b2 − 4ac 2a = � f(θk+1) + C − � (˜rk+1)2 + (˜rk+1 − rk)2 + ηG(θk+1, θk) � f(θk+1) + C − ˜rk+1 = � f(θk+1) + C − � (1 − η)˜r2 k+1 + ηr2 k + (1 − η)(˜rk+1 − rk)2 � f(θk+1) + C − ˜rk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 6 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 The implementation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' If L is non-negative and linear, then for any δt > 0, the modified energy r2 in the scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) is unconditionally diminishing in the sense that r2 k+1 − r2 k = −(1 − η)G(θk+1, θk) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='13), we obtain ˜r2 k+1 − r2 k = − 1 δt∥θk+1 − θk∥2 − (L(θk+1 − θk), (θk+1 − θk)) − (˜rk+1 − rk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Adding r2 k+1 − ˜r2 k+1 on both sides, noticing G(θk+1, θk) = 1 δt∥θk+1 − θk∥2 + (L(θk+1 − θk), (θk+1 − θk)), we obtain r2 k+1 − r2 k = −G(θk+1, θk) − (˜rk+1 − rk)2 + r2 k+1 − ˜r2 k+1, which implies the desired result thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' □ Algorithm 1 The basic RSAV scheme 1: Inputs: δt: step-size, C: constant to guarantee the positivity of f(x) + C, A = I + δtL : the linear operator, θ0: initial parameter vector 2: r0 ← � f(θ0) + C 3: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', M − 1 do 4: gk = ∇f(θk) √ f(θk)+C 5: ˆgk = A−1gk 6: ˜rk+1 = rk 1+ δt 2 (gk,ˆgk) 7: θk+1 = θk − δt˜rk+1ˆgk 8: ξ = √ f(θk+1)+C− � (1−η)˜r2 k+1+ηr2 k+(1−η)(˜rk+1−rk)2 √ f(θk+1)+C−˜rk+1 9: ξ = max{0, ξ} 10: rk+1 = ξ˜rk+1 + (1 − ξ) � f(θk+1) + C return θM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Selection of the operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In the SAV approach for gradient flows [26], it is found that a proper choice of the splitting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', the choice of the quadratic term 1 2(Lθ, θ), can significantly increase the robustness and efficiency of the SAV schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For gradient flows coming from materials science or fluid dynamics, there are usually obvious candidates in the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' However, for minimization problems, particularly those from machine learning problems, there are usually no obvious quadratic terms in the energy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 7 In these cases, we can artificially choose some simple operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In this paper, we consider two simple operators below, for which the inverse operator (I + δtL)−1 can be efficiently implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Diagonal Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In many optimization problems, an l2 regularization term is often added into the loss function to avoid overfitting to the data in training sets, namely, J(x) = f(x) + λ 2 ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='14) In this case, a natural choice is to set L = λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' More generally, we can use L = D with D being a diagonal matrix with positive entries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', D can be the diagonal entries of the Hessian of the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Discrete Laplacian Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In some machine learning problems, the discrete Lapla- cian matrix is used as a smoothing operator which can reduce the variance during the mini-batch training process [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This corresponds to L = −σ∆ where σ is a positive pa- rameter and ∆ is a discrete Laplacian matrix, and (I + δtL)−1 can be efficiently inverted by FFT based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The acceleration by using discrete Laplacian in classical primal dual algorithms has been also justified in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' An adaptive algorithm based on the RSAV scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Similar to the mod- ified SAV scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3), the RSAV scheme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) is also unconditionally energy diminishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A main advantage of unconditionally stable schemes is that one can adaptively adjust the time step size to achieve faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In particular, we can use Ik(r, θ) = rk � f(θk) + C (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='15) as an indicator to control the deviation between modified energy and true energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For solving differential equations θt = −∇f(θ), Ik(r, θ) should be as close as to 1 for the sake of the time accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' But for a minimization problem, there is no time accuracy issue thus we can allow Ik(r, θ) to deviate from 1 to achieve faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' However, Ik(r, θ) needs to be away from zero to avoid slow convergence, as the SAV and RSAV algorithms may converge much slower than the vanilla gradient descent when the ratio Ik(r, θ) becomes too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We observe from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) that the true step-size for the gradient ∇f(θk) is ˜rk+1 √ f(θk)+C δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Thus if the ratio is small i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Ik(r, θ) < γ, the true step-size for the gradient would be too small resulting in slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' To this end, we present a simple adaptive rule with an adaptive constant ρ > 1 with default value ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Note that in many applications of neural networks and machine learning, the cost of computing the full batch is generally too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In these cases, we can adopt the mini-batch approach commonly used in stochastic gradient decent, and restart the RSAV scheme at the beginning of each mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 8 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 Algorithm 2 The adaptive RSAV scheme 1: Inputs: δt0: initial step-size, δtmin: the lower bound of step-size, C: constant to guarantee the positivity of f(x) + C, A = I + δtL : the linear operator, θ0: initial parameter vector, ρ: adaptive constant which is greater than 1, γ: threshold for the indicator I(r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2: r0 ← � f(θ0) + C: Initialize the SAV, 3: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', M − 1 do 4: if rk √ f(θk)+C < γ and δt > δtmin then 5: δtk+1 = max{ rk √ f(θk)+C δtk, δtmin} 6: else 7: δtk+1 = ρδtk 8: 9: gk = ∇f(θk) √ f(θk)+C 10: ˆgk = A−1gk 11: ˜rk+1 = rk 1+ δtk+1 2 (gk,ˆgk) 12: θk+1 = θk − δtk+1˜rk+1ˆgk 13: ξ = √ f(θk+1)+C− � (1−η)˜r2 k+1+ηr2 k+(1−η)(˜rk+1−rk)2 √ f(θk+1)+C−˜rk+1 14: ξ = max{0, ξ} 15: rk+1 = ξ˜rk+1 + (1 − ξ) � f(θk+1) + C return θM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Numerical Results We present in this section several illustrative numerical experiments by using our RSAV approach, and compare it with popular gradient based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In order to present a fair comparison to gradient descent (GD), we consider a composite gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' By abuse of notation, we shall refer to GD with L as the following method for the splitting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4): θk+1 + δtLθk+1 = θk + δt(Lθk − ∇f(θk)), which is equivalent to θk+1 = θk − δt(I + δtL)−1∇f(θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) The scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) can also be regarded as the forward-backward splitting scheme θk+1 = θk − δt∇F(θk) − δt∇G(θk+1) for minimizing a composite function F(θ) + G(θ) with F(θ) = f(θ) − 1 2θT Lθ and G(θ) = 1 2θT Lθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 9 Note that the scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) reduces to the vanilla gradient descent (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) if setting L = 0, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) reduces to the vanilla gradient descent (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) with step size δt 1+δt if setting L = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Therefore, we do not consider GD with L = I, and we compare the adaptive RSAV in Algorithm 2 to the following algorithms: (1) GD with L = 0, which is the vanilla gradient descent (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2) GD with L = −σ∆ with discrete Laplacian ∆, which is similar to the Laplacian Smoothing Gradient Descent [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (3) ADAM [15] (4) Nesterov accelerated gradient decent (NAG) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Unless specified otherwise, we use ADAM and NAG with the default parameter settings as in [22]: NAG with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9, and ADAM with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='999, ε = 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A quadratic cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We start with a quadratic loss function from [20]: f(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , θ100) = 50 � i=1 θ2 2i−1 + 50 � i=1 1 100θ2 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) For this simple function, we take either L = 0 or L = D where the diagonal matrix D is chosen to be the Hessian ∇2f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' To demonstrate the unconditional stability of SAV-based approaches, in Table 1 and Figure 1 we show the results of different initial step sizes δt for the vanilla gradient descent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', GD with L = 0, as well as GD with L = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We observe that the vanilla gradient descent blows up for the constant step size δt = 1, while the adaptive RSAV works quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (initial) step-size δt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 1 GD (L = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='009121 50 adaptive RSAV (L = 0) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='34e-12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='749e-12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='264e-18 GD (L = D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='009194 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='152e-18 adaptive RSAV (L = D) 0 0 0 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss of quadratic function after 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 10 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss curves for GD and adaptive RSAV with different splits and (initial) step-sizes δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Next we consider the gradient perturbed by a Gaussian noise ∇ϵf(x) := ∇f(x) + ϵN(0, I), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3) where ϵ controls the noise level, N(0, I) is the Gaussian noise vector with zero mean and unit variance in each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The comparison is given in Table 2 and Figure 2 where L = 0 is used for both GD and adaptive RSAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We observe that the adaptive RSAV converges much faster than GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The fast convergence of adaptive RSAV is partly due to the indicator Ik(r, θ) which can give a proper step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Especially in the noisy case, the true step size is given at a proper level to reach a better convergence than GD and reduce the oscillation in the loss curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (initial) step-size δt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 1 GD (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='009223 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='58 adaptive RSAV (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0002283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0002298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0002251 GD (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01584 diverge adaptive RSAV (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='004934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='005023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='004889 GD (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='03808 diverge adaptive RSAV (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0188 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss of quadratic function after 1000 iterations with different noise levels ϵ and (initial) step-sizes δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' L = 0 is used for both GD and adaptive RSAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 100 ( 10-5 一 (4) 10-10 GD(C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='ot=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) GD (L = 0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) GD (L= 0, St = 1) adaptive RSAV (L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' ot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) 10-15 0 200 400 600 800 1000 iterations100 ( 10-5 (4) 10-10 GD (C = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' ot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) GD (C = D, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) GD (C = D, ot = 1) adaptive RSAV (C = D, St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01 10-15 0 200 400 600 800 1000 iterationsA SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 11 (a) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01 (b) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='05 (c) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss curves for GD and adaptive RSAV with different noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The learning rate (lr) refers to the step size δt for GD and the initial step size δt in the adaptive RSAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Rastrigin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Consider f(x) = f(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , θn) = n � i=1 θ2 i + 10n − 10 n � i=1 cos(2πθi), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) which has many local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The function can be defined on any input domain but it is usually evaluated on x ∈ [−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='12, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='12] for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The function has a global minimum at f(x∗) = 0 located at x∗ = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In this example, we compare the adaptive RSAV with popular optimization methods ADAM and NAG with their default parameter settings as in [22]: NAG with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9, and ADAM with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='999, ε = 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We shall keep using these default settings in all following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' initial stepsize δt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 1 GD (L = 0) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='93 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='86 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2 diverge adaptive RSAV (L = 0) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='05 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='03 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='608e-9 NAG 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='93 152 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 diverge ADAM 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='28 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='94 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='93 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='958 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss of Rastrigin function after 100 iterations in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We plot in the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3 the convergence curves of different methods, and observe that the RSAV converges much faster than ADAM with the same initial step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We also plot in the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3 the paths towards the minimum by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We observe that RSAV enjoys a fast convergence if using a large initial step size δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Rosenbrock function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This is a benchmark problem for optimization of non-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We first consider the 2D case with f(x, y) = (a − x)2 + b(y − x2)2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5) 104 1()f -()fl 100 10-2 10- GD(C= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) adaptive RSAV (C = 0, lr=1) 0 200 400 600 800 1000 iterations104 1()f -()fl 100 10~ GD（C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) adaptive RSAV (C = 0, lr=1 0 200 400 600 800 1000 iterations10 1()f -()fl 100 10* GD（C=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) adaptive RSAV (C = 0, lr=1 0 200 400 600 800 1000 iterations12 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Rastrigin function: Left: Convergence curves with 100 itera- tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Right: Paths with first 10 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The learning rate (lr) refers to the initial step size δt in the adaptive RSAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' it has a global minimum at (x, y) = (a, a2), which is inside a long narrow, parabolic shaped flag valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' To find the valley is trivial, but to converge to the global minimum is usually difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We set a = 1 and b = 100 in the following experiments and other parameters the same as in [20], and start with the initial point with coordinate (−3, −4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In Table 4, we observe that a large step size can lead to blow-up for other methods except for RSAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Thus in Figure 4, we only show the results with the largest suitable step sizes for other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For adaptive RSAV, we just use the same initial step size as ADAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This example reveals the benefits of modified energy decreasing property of the RSAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Although ADAM can get close to the global minimum at first, it still goes to the wrong direction caused by the momentum and eventually goes back after wasting many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Only RSAV converges to the global minimum directly with the guide of decreasing (modified) energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' step-size δt 10−4 10−2 1 GD (L = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7142 diverge diverge adaptive RSAV (L = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0107 NAG 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='326 diverge diverge ADAM 15198 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss of Rosenbrock function after 1000 iterations in 2D 102 100 10-2 (×)} - (x)l 10-4 10-6 GD (lr = 10-3) 10-8 RSAV (lr = 1) NAG (lr = 10-3) ADAM (lr = 1) 10-10 0 20 40 60 80 100 iterations4 3 2 1 0 0 1 X Contour 2 GD (lr = 10-3) "-RSAV (lr = 1) 3 NAG (lr = 10-3) ADAM (lr = 1) 4 global minimum 4 2 0 2 4A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 2D Rosenbrock problem: Left: Convergence curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Right: Paths with 1000 iterations in the (θ1, θ2) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Next, we consider the high dimensional cases with f(x) = n � i=1 (a − θi)2 + b n−1 � i=1 (θi+1 − θ2 i )2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6) with the global minimum f(x∗) = 0 at x∗ = (a, a2, a, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , a, a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We take a = 1 and b = 100, and the initial point (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The results with the dimension equal to 10, 100 and 1000 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We observe similar convergence behavior for all cases as in the two dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (a) n = 10 (b) n = 100 (c) n = 1000 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss of Rosenbrock function with dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='If(xk) - f(x ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='GD (lr = 10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='RSAV (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='NAG (lr = 10-4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='ADAM (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='iterations106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='I( x)} - ("x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='GD (lr = 10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='RSAV (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='NAG (lr = 10-4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='ADAM (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2000 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='I( ×)} - ("x)l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='GD （lr = 10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='RSAV (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='NAG (lr = 10-4) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='iterations104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='(×)} - (*x)l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='GD (lr = 10-4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='RSAV (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='NAG (lr = 10-4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='ADAM (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='iterations8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='Contour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='GD (lr = 10-4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='RSAV (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='NAG(1r= 10-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='ADAM (lr = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='global minimum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='514 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='XINYU LIU1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' JIE SHEN1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' AND XIAONGXIONG ZHANG1 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' If we compare the adaptive RSAV with Nesterov accelerated gradient decent, the results are still quite good especially when the dimension is 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Thanks to the adaptive scheme, the performance of RSAV in this problem independent of the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Phase Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The phase retrieval problem [5] can be formulated as min z∈CN f(z) := 1 2∥A(z∗z) − b∥2 where z∗ ∈ C1×N is the conjugate transpose of z, b ∈ RM and A : CN×N −→ RM is a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For the real-valued function f(z) with complex variable z := a + ib, where i is the imaginary unit and a, b ∈ R are real and imaginary parts of z, we can define the Fr´echet derivative induced by the natural choice of real inner product for CN as the following [29]: ∇f(z) := ∂f(z) a + i∂f(z) b = 2A∗(A(z∗z − b))z, where A∗ is the adjoint operator of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Then the vanilla gradient descent algorithm for minz∈CN f(z) can be defined as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) using ∇f(z) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The gradient descent method with a suitable step sizing rule is also also referred to as the Wirtinger flow [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In particular, f(z) is a non-convex quartic polynomial function of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For the theorectical convergence of minimizing such a non-convex function, with a spectral initialization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', z0 being the leading eigenvector of A∗(b), the convergence of Wirtinger flow with high proba- bility can be proven for a very special class of phase retrieval problems [6, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For solving phase retrieval with random initial guess, the convergence for minimizing a smoothed am- plitude flow based model was proven in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In terms of practical performance with only random initialization, state-of-the-art algorithms such as the Riemannian LBFGS method could be much more efficient than gradient descent algorithms [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We emphasize that we only use such phase retrieval problems as a testing example to validate the performance of the RSAV method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' So we test the algorithms with a random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We compare vanilla gradient descent (GD), adaptive RSAV with L = 0, and steepest descent (SD) [8, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The steepest descent method is to use the optimial step size in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2), and it is possible to compute such an optimal step size for a polynomial cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We test the performance of RSAV algorithm on the following phase retrieval problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let Mi ∈ CN be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='d Gaussian and ◦ denote the entrywise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let F denote the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The linear operator A is defined by assigning ∥F(Mi ◦ z)∥2 to b, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', M N = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We consider the test case for the true solution z∗ being an image of size n×n with n = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' So the size of unknown is N = n2 = 2562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We consider two test cases: (1) The true minimizer z∗ is a real image of camera man with size 256 × 256 as shown in Figure 6, m = 6 Gaussian random masks and a random initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2) The true minimizer z∗ is a complex image of golden ball with size 256 × 256, see [12] for details, m = 10 Gaussian random masks and a random initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 15 See Figure 7 for the comparision of the performance of gradient based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For the vanilla gradient descent (GD), we use the nearly largest stable constant step size δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In Figure 8, we list a comparsion of the step size δk in the adaptive RSAV method with the optimital step size in the steepest descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We can see the performance of adaptive RSAV has the closest performance to the steepest descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For the real image case, SD converged after 10000 iterations and RSAV converged after 20000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' However, to compute the optimal step size in the steepest descent, at least two more evaluations of A are needed, thus quite expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' More importantly, for a general cost function, it is difficult to find the optimal step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4 for an analysis of the step size in the explicit SAV gradient descent with restarting rk every iteration for quadratic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (a) GD (b) adaptive RSAV (c) SD Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Results after 20000 iterations for a phase retrieval problem with z∗ being a 256 × 256 real image of camera man, m = 6 Gaussian random masks and a random initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The vanilla gradient descent (GD) uses nearly largest stable constant step size δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 16 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Loss of different optimization algorithms for phase retrieval: Left: the real image of camera man using 6 Gaussian masks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Right: the complex image of golden balls using 10 Gaussian masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The vanilla gradient descent (GD) uses nearly largest stable step size δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The value of δk in each iteration for phase retrieval: Left: the real image of camera man using 6 Gaussian masks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Right: the complex image of golden balls using 10 Gaussian masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Recommendation System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Consider applying the optimization scheme to train a recommendation system based on matrix factorization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Given a rate matrix R ∈ Rm×n where m is the number of users and n is the number of items, the model learns the user embedding matrix X ∈ Rm×d and item embedding matrix Y ∈ Rn×d such that the product 10° If(xk) - f(x )l 10-5 10-10 GD RSAV NAG ADAM SD 10-15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 2 iterations ×105105 100 I( ×)} - ("x) 10-5 GD 10-10 RSAV SD NAG ADAM 10-15 100 105 iterations104 103 stepsize t 102 101 SD RSAV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5 3 iterations ×104104 103 stepsize 102 101 SD RSAV 0 200 400 600 800 1000 iterationsA SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 17 XY T is a good approximation for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Here, d is the embedding dimension and usually much smaller than m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Denote the user and item matrix by X = [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , Xu, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , Xm]T and Y = [Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , Yi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' , Yn]T , we have the loss function as f(X, Y ) = 1 Nκ � (u,i)∈κ (Ru,i − XuY T i )2 + λu � u ∥Xu∥2 2 + λi � i ∥Yi∥2 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) where κ the training set that the (u, i) pairs for which Ru,i is known, Nκ is the number of training data, λu and λi are the penalty parameters for embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We train the model with the MovieLens 100K dataset [11] which contains 100, 000 ratings (1 − 5) from 943 users on 1682 movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' There is 80% data split as the training data and the rest date is used for the testing data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', the training data set κ has size 80, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' All algorithms use the mini-batch gradient with batch size 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For l2 regularization, we set λu = λi = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For the linear operator L = λI − σ∆ in GD and RSAV, we let λ = 10−4 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In Figure 9, for the training step, we run 10, 000 iterations for the mini-batch gradient based methods with batch size 80, which is equal to 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The result on the test data is shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We can see that RSAV performs well in the training step, though its testing result is not the best, which suggests issues of overfitting during the training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This is more or less a modelling issue, rather than the optimizaiton issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The training loss curve of different optimization algorithms for Recommendation System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Here GD refers to GD (L = λI −σ∆) and RSAV refers to the adaptive RSAV (L = λI − σ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 102 101 S SSO 100 10 GD (St = 1) RSAV (St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) NAG (St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) ADAM (St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01) 10-2 2000 4000 6000 8000 10000 iterations18 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 step-size δt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1 1 10 GD (L = λI − σ∆) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6504 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1465 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6438 diverge NAG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1451 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6439 diverge diverge ADAM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8820 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7194 diverge diverge adaptive RSAV (L = λI − σ∆) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9090 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9156 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9156 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The loss function on the test data after 10, 000 training iterations (10 epochs) with different step-sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Here diverge means that the training step already diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Convergence study for some SAV based algorithms In this section, we consider a more general version of the SAV scheme based on the following expanded system � θt = − � r [f(θ)+C]q ∇f(θ) + Lθ − Lθ � rt = q[f(θ) + C]q−1(∇f(θ), θt), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) where r(t) = [f(θ) + C]q and q ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) is a special case of the above formulation with q = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3), we can construct a SAV scheme for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1) as follows: � θk+1−θk δt = − � rk+1 [f(θk)+C]q ∇f(θk) + L(θk+1 − θk) � rk+1−rk δt = q[f(θk) + C]q−1(∇f(θk), θk+1−θk δt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Interpretation of the SAV method as a line search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let A = (I +δtL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2) can be rewrite as � A δt ∇f(θk) [f(θk)+C]q −q[f(θk) + C]q−1∇f(θk) 1 � � θk+1 rk+1 � = � Aθk rk − q[f(θk) + C]q−1(∇f(θk), θk) � After a simple Gaussian elimination, we obtain an explicit update formula for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2): � � � rk+1 = 1 1+δtq (∇f(θk),A−1∇f(θk)) f(θk)+C rk θk+1 = θk − rk+1 [f(θk)+C]q δtA−1∇f(θk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Notice that the scheme above can be regarded as a line search method: θk+1 = θk + αkPk Pk = −A−1∇f(θk) αk = δt 1 + δtq (∇f(θk),A−1∇f(θk)) f(θk)+C rk [f(θk) + C]q > 0, with a search direction Pk and step size αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The step size αk is guranteed to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' On the other hand, it is difficult to establish any a priori control of αk, and in practice αk could become very small if rk becomes very A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 19 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' To avoid small rk, we consider a special version of SAV method by redefining rk = [f(θk) + C]q, then we have αk = δt 1 + δtq (∇f(θk),A−1∇f(θk)) f(θk)+C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In this case, we can view q as a parameter, and the SAV method with rk = [f(θk) + C]q at every iteration becomes the following line search method: θk+1 = θk + αkPk (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4a) Pk = −A−1∇f(θk) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4b) αk = δt 1 + δtq (∇f(θk),A−1∇f(θk)) f(θk)+C , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4c) which is equivalent to � � � � � � � rk = [f(θk) + C]q θk+1−θk δt = − � ˜rk+1 [f(θk)+C]q ∇f(θk) + L(θk+1 − θk) � ˜rk+1−rk δt = q[f(θk) + C]q−1(∇f(θk), θk+1−θk δt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='5) In particular, for any L ≥ 0, A−1 is always positive definite, thus the search direction Pk = −A−1∇fk is always a descent direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', −∇fT (θk)Pk = ∇f(θk)T A−1∇f(θk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The Wolfe condition [27] for the line search method (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) is: there exists 0 < c1 < c2 < 1 such that f(θk + αkPk) ≤ f(θk) + c1αk∇f(θk)T Pk (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6a) ∇f(θk + αkPk)T Pk ≥ c2∇f(θk)T Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6b) We recall first the following result [19]: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Assume f(θ) ∈ C1 and f(θ) is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For any descent direction Pk, there exist intervals of step lengths satisfying the Wolfe condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Notice that α(δt, q) = δt 1+δtq (∇f(θk),A−1∇f(θk)) f(θk)+C is an increasing function of δt and an de- creasing function of q, thus there exists δt and q such that αk = δt 1+δtq (∇f(θk),A−1∇f(θk)) f(θk)+C satisfies the Wolfe conditions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We recall below another result [19]: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Assume f(θ) ∈ C1 and ∇f(θ) is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let cos φk = −∇f(θk)T Pk ∥∇f(θk)∥∥Pk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' If Pk is a descent direction and αk satisfies the Wolfe Conditions, then the iteration θk+1 = θk + αkPk satisfies � k≥0 cos2 φk∥∇f(θk)∥2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) 20 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 Let λmin(A) and λmax(A) be the smallest and largest eigenvalues of the real symmetric positive definite matrix A, then by the Courant-Fischer-Weyl min-max principle [7] and the spectral norm ∥A∥ = λmax(A), we have P T k APk ∥Pk∥2 ≥ λmin(A), ∥APk∥ ≤ ∥A∥∥Pk∥ = λmax(A)∥Pk∥, thus cos φk = P T k APk ∥APk∥∥Pk∥ = P T k APk ∥Pk∥2 ∥Pk∥ ∥APk∥ ≥ λmin(A) λmax(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Therefore, the uniform lower bound on cos φk and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='7) implies that ∥∇f(θk)∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Thus the convergence of the SAV method (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) is ensured if using a line search to find δt, q such that αk satisfies the Wolfe condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We observe that the above algorithm involves computing A−1∇f(θk), and evaluation of f(θk) and f(θk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In practice, one can use backtracking line search on αk to ensure that the Wolfe conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This in general it does not seem advantageous over a simple backtracking line search on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' However, for the SAV gradient descent method (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', A = I, our numerical observation is that the SAV scheme is often more efficient than the backtracking line search on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' With A = I, the scheme (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) reduces to the following SAV gradient descent method with two parameters δt > 0 and qk > 0: θk+1 = θk − αk∇f(θk) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8a) αk = δt 1 + δtqk ∥∇f(θk)∥2 f(θk)+C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8b) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Standard convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We recall that if αk in the line search method (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4) satisfies the Goldstein-Armijo rule [1, 9]: there exists 0 < c1 < c2 < 1 such that f(θk) − c2αk∥∇fk∥2 ≤ f(θk − αk∇fk) ≤ f(θk) − c1αk∥∇fk∥2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='9) then it is shown (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='13 in [17]) that ∥∇fk∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='13 in [17] can be easily adapted to prove the following result for the line search method (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Assume that ∇f is Lipshitz continuous with the Lipshitz constant L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', ∥∇f(y) − ∇f(x)∥ ≤ L∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' If ∇f(θ∗) = 0 and αk ∈ (0, 2 L), then ∥θk+1 − θ∗∥2 ≤ ∥θk − θ∗∥2 − αk( 2 L − αk)∥∇f(θk)∥2 and f(θk) − f(θ∗) ≤ 1 [f(θ0) − f(θ∗)]−1 + ∥θ0 − θ∗∥−2 � k αk(1 − L 2 αk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' So for convergence, we need ∞ � k=0 αk(1− L 2 αk) = +∞, which can be ensured if αk ∈ [a, b] ∈ (0, 2 L) for constant bounds a > 0 and b < 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Also, αk < 2 L will ensure f(θk+1) < f(θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Decreasing step sizes for the SAV gradient descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We derive from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8) that αk = δt 1 + δtqk ∥∇fk∥2 fk = 1 1/δt + qk ∥∇fk∥2 fk ≤ min � δt, fk qk∥∇fk∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For fixed δt, the above is often sufficient to ensure f(θk+1) < f(θk) when θk is far away from the minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' It can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Assume that f is strongly convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', (∇f(y) − ∇f(x), y − x) ≥ m∥x − y∥2 with m > 0, and ∇f is Lipshitz continuous with the Lipshitz constant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let θ∗ be the minimizer and assume f(θ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Then the following are sufficient conditions to ensure αk < 2 L for the SAV gradient descent method with two parameters (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8): (1) For any δt > 0, qk > L2 4m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' (2) Let δt ≡ a 2 L where a > 0, qk > a−1 a L2 4m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The first sufficient condition implies that αk = δt 1+δtqk ∥∇fk∥2 fk will be a decreasing step size for any δt if qk ≡ q > L2 4m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Of course, finding 1 q < 4m2 L2 in general is not easier than finding δt < 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' But if 2m2 > L, then 4m2 L2 > 2 L implies 1 q < 4m2 L2 is easier to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' As an example of the second sufficient condition, if we pick qk ≡ 1 2, and a = 2, then 1 2 > 1 2 L2 4m2 ⇔ L < 2m is sufficient to ensure the SAV gradient descent method with qk ≡ 1 2 is decreasing with δt = 4 L, instead of δt < 2 L in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' First, by the strong convexity and Lipschitz continuity, we have f(x) ≥ f(y) + (∇f(y), x − y) + m 2 ∥x − y∥2, f(x) ≤ f(y) + (∇f(y), x − y) + L 2 ∥x − y∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Since θ∗ is the minimizer, ∇f(θ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' For any θ, we have (∇f(θ) − ∇f(θ∗), θ − θ∗) ≥ m∥θ − θ∗∥2 ⇒ (∇f(θ), θ − θ∗) ≥ m∥θ − θ∗∥2 ⇒ m ∥θ − θ∗∥ ∥∇f(θ)∥ = (∇f(θ), θ − θ∗) ∥θ − θ∗∥∥∇f(θ)∥ ≤ 1 ⇒ ∥∇f(θ)∥ ≥ m∥θ − θ∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Hence, m∥θ − θ∗∥ ≤ ∥∇f(θ)∥ ≤ L∥θ − θ∗∥, and m 2 ∥θ − θ∗∥2 ≤ f(θ) − f(θ∗) ≤ L 2 ∥θ − θ∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' With strong convexity m > 0, we have 2m2 L ≤ ∥∇f(θ)∥2 f(θ) − f(θ∗) ≤ 2L2 m 22 XINYU LIU1, JIE SHEN1, AND XIAONGXIONG ZHANG1 thus 2m2 L ≤ ∥∇f(θ)∥2 f(θ) ≤ 2L2 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Finally, fk qk∥∇fk∥2 < 2 L ⇔ 1 qk < ∥∇f(θk)∥2 f(θk) 2 L ⇐ 1 qk < 4m2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In general, if f(θ) is only convex but not strong convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=', m = 0, and f(θ) + C > 0, then we only have ∥∇f(θ)∥2 f(θ) + C ≤ L2∥θ − θ∗∥2 f(θ∗) + C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' This gives a lower bound control on step size: αk = δt 1 + δtqk ∥∇fk∥2 fk+C ≥ δt 1 + qkδt L2∥θk−θ∗∥2 f(θ∗)+C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In this case, we can set qk ≡ q and do back tracking on δt for αk to satisfy the convergence condition or Goldstein-Armijo rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The step size for quadratic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' To see why the step size αk = δtk 1+δtkqk ∥∇fk∥2 fk+C could be a good step size to use, at least for a quadratic cost function, consider a cost function f(θ) = 1 2∥Aθ − b∥2 with a square and positive definite matrix A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The steepest descent algorithm can be written as θk+1 = θk − βk∇f(θk), βk = ∥AT (Aθk − b)∥2 [AT (Aθk − b)]T AT A[AT (Aθk − b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The method (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content='8) with C = 0 is θk+1 = θk − αk∇f(θk), αk = 1 1 δtk + qk ∥AT (Aθk−b)∥2 1 2 (Aθk−b)T (Aθk−b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Then for a very large δtk and qk ≡ 1 2, we have αk ≈ (Aθk − b)T (Aθk − b) (Aθk − b)T AAT (Aθk − b), βk = (Aθk − b)T AAT (Aθk − b) (Aθk − b)T AAT AAT (Aθk − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let vi be orthornormal eigenvectors of A with eigenvalues of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Since vi form a basis for RN, let Aθk − b = r = � i rivi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Let z = AT (Aθk − b), then z = AT � i rivi = � i riλivi and Az = � i riλ2 i vi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We get αk ≈ rT r rT AAT r = rT r zT z = [� i rivi]T [� i rivi] [� i riλivi]T [� i riλivi] = � i r2 i � i λ2 i r2 i , βk = zT z (Az)T (Az) = � i λ2 i r2 i � i λ4 i r2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We can see that αk is very similar to the optimal step size βk but not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In practice, a random initial guess θ0 usually makes αk a descent step size in the first few or many iterations for δtk ≡ 1 and qk ≡ q = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' A SAV BASED ALGORITHM FOR DISCRETE GRADIENT SYSTEMS 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Concluding remarks We proposed in this paper a new minimization algorithm inspired by the scalar auxil- iary variable (SAV) approach for gradient flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Since the direct application of the SAV approach to minimization problems may converge to wrong solutions, we developed a mod- ified version of the SAV approach coupled with a relaxation step and an adaptive stradegy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The new algorithm enjoys several distinct advantages, including unconditionally energy di- minishing with a modified energy, and empirical better performance than many first order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' In particular, it overcomes the difficulty in selecting proper step sizes associ- ated with the usual gradient based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' The energy diminishing property ensures the convergence, and the relaxation step, built on a connection between the decreasing modified energy and the original energy, helps to accelerate the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' We also presented a converence analysis for some SAV based algorithms which include the new algorithm without the relaxation step as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Numerical results for several illustrative and benchmark problems indicates that the new algorithm is very robust and usually converges significantly faster than those popular gradient decent based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' While we only considered a basic version of the SAV based approach which already showed its promise, it is clear that this approach can be combined with other techniques of acceleration and generalization to the gradient decent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' How to further improve the robustness and accelerate the convergence rate of the SAV based approach will be the subject of a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Acknowledgement This work is partially supported by AFOSR FA9550-16-1-0102, NSF DMS-2012585 and DMS-2208518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' References [1] Larry Armijo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Minimization of functions having Lipschitz continuous first partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' Pacific Journal of mathematics, 16(1):1–3, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfIwMb/content/2301.02942v1.pdf'} +page_content=' [2] Arthur Earl Bryson and 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Wolfe,𝑏 Steven Clayton,𝑏 Mark +Makela,𝑏 C. L. Morris,𝑏 Simon Spannagel,𝑑 Erik Ramberg,𝑒 Juan Estrada,𝑒 Hao Zhu,𝑐 +Jifeng Liu,𝑎 Eric R. Fossum𝑎 and Zhehui Wang𝑏,∗ +𝑎Thayer School of Engineering at Dartmouth, Dartmouth College, +Hanover, NH 03755, USA +𝑏Los Alamos National Laboratory, +Los Alamos, NM 87545, USA +𝑐Chandra Family Department of Electrical & Computer Engineering, The University of Texas at Austin, +Austin, TX, 78712, USA +𝑑Deutsches Elektronen-Synchrotron DESY, +Notkestr. 85, 22607 Hamburg, Germany +𝑒Fermi National Accelerator Laboratory, +Batavia, IL 60510, USA +†Lead authors with equivalent contributions +E-mail: zwang@lanl.gov +We summarize recent progress in ultrafast Complementary Metal Oxide Semiconductor (CMOS) +image sensor development and the application of neural networks for post-processing of CMOS and +charge-coupled device (CCD) image data to achieve sub-pixel resolution (thus ‘super-resolution’). +The combination of novel CMOS pixel designs and data-enabled image post-processing pro- +vides a promising path towards ultrafast high-resolution multi-modal radiographic imaging and +tomography applications. +10th International Workshop on Semiconductor Pixel Detectors for Particles and Imaging (Pixel2022) +12-16 December 2022 +Santa Fe, New Mexico, USA +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.11865v1 [physics.ins-det] 27 Jan 2023 + +Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +1. +Introduction +Using X-rays for imaging and tomography of optically opaque objects dated back to the famous +invention of Wilheim Röntgen in 1895. Multimodal (MM) radiographic imaging and tomography +(RadIT) combines several different forms or energies of ionizing radiation such as X-rays, 𝛾-rays, +neutrons, energetic electrons, protons and others, which can potentially yield more information +about an object than by using a monochromatic (mono-energetic) photons (particles) alone as the +source of illumination. +The technical challenges and opportunities for MM RadIT can be summarized in Figure 1. +Additional information may be found in [1]. The physics framework of radiation-matter interactions +is mostly complete for all practical purposes. Recent advances in high-intensity radiation sources +such as synchrotrons, X-ray free electron lasers, high-current low-emittance charged particle accel- +erators, laser-driven sources open door to MM RadIT. One of the optimization problems in MM +RadIT is to obtain as high temporal, spatial resolution of an object as possible, with a sufficiently +large field of view, and at a certain radiation dose to minimize radiation damage of the object. +Figure 1: A holistic approach to MM RadIT optimization includes at least five branches of effort: Fundamen- +tal physics of radiation-interaction with matter (PHY), radiation sources (SRCE), detectors (DETR), methods +to modulate the radiation field (METH) and data handling (DATA). The fundamental physics principles of +MM RadIT are well established. Some challenges and opportunities for SRCE, DETR, METH and DATA +are listed above for each branch. +Below, we first describe the recent progress in ultrafast Complementary Metal Oxide Semi- +conductor (CMOS) pixelated sensor design and prototyping, followed by the the use of neural +networks for noise emulation in X-ray imaging, and the demonstration of sub-pixel resolution in +neutron detection. Follow-on work includes CMOS image sensor fabrication and extension of the +neural networks to different types of particles or photons, and different noise environment. Our +work demontrates a promising path towards high-speed high-resolution multi-modal radiographic +imaging and tomography applications. +2 + +h +QUANTUM +QED +ELECTROWEAK +MECHANICS +INTERACTIONS +60Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +2. +Ultrafast CMOS image sensor development +Ultra-high-speed (UHS) or ultrafast image sensors are widely used in scientific and industrial +applications. The research on UHS CMOS image sensors for X-ray regimes has been conducted for +years. The recently published works [2, 3] push the frame rate of UHS image sensors to the range +of millions of frames per second (Mfps) by adopting burst-mode operations and advanced CMOS +technology or customized CMOS technology. Three CMOS image sensors were taped out towards +this goal, as shown in Figure 2. +Figure 2: Phase1, 2, 3 CMOS image sensor (CIS) taped out in this research. +During the phase one of this research [4], theoretical modeling and preliminary tests demon- +strated more than 10× quantum efficiency improvement for high-energy X-ray photons (>10 keV) +by depositing a photon-attenuation-layer (PAL) on a CMOS image sensor [5, 6]. In the phase +two of this research, a block-wise compact readout architecture based on unit-length-capacitor and +asynchronous successive-approximation (SAR) analog to digital converter (ADC) [7] was proposed +and implemented, which enabled the image sensor fabricated using a standard 180-nm process to +run at 76 thousand-frames-per-second (kfps). To further boost the frame rate of the image sensor, a +burst mode image sensor based on sequential transfer gates was proposed and taped out in Oct. 2022 +during the phase three of this research. The sequential transfer gates enable the image sensor to run +at least 20 Mfps and achieve the lowest input-referred noise. Some highlights of this burst-mode +image sensor is included below. +Figure 3 shows the conceptual 20-µm pixel layout based on the sequential transfer gates, where +the orange rising-run shape stands for the photodiode, and the green polygon stands for the transfer +gate. Each photodiode finger geometry shape has been carefully calculated and optimized to have +a constant ∼800V/cm strong electrical field pointing from the tip of the photodiode to the center of +the photodiode, which guarantees fast charge transfer without process modification. +As Figure 4 shows, by applying monotonically increasing control voltages and sequential timing +on TX3(Blue), TX2(Green), and TX1(Red) gates, electrons (purple and cyan) in photodiodes can +be fully transferred within 12 ns, with no image lag noticed in simulation. +Figure 5 shows the potential diagram during the charge transfer path. One can see that photon- +generated electrons are first swept into the channel under the TX3 gate due to the strong electrical +field in photodiode fingers. Then TX3, TX2, and TX1 gates turn off sequentially, which pushes +electrons to move toward the floating diffusion node. Because the TX2 gate is entirely off before +the falling transition of the TX1 gate, it is safe to move the floating diffusion node away from TX1, +which will effectively reduce the overlap capacitance between the TX1 gate and floating diffusion +3 + +ImageofPhase1CiS +ImageofPhase2CIS +LavoutofPhase3CiSUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +Figure 3: High-Speed Conceptual Layout of a pixel in a CMOS camera. See the text for further details. +Figure 4: An example of TCAD transient simulation during charge transfer. The text contains further details. +node, increase the conversion gain of the pixel and reduce input-referred noise. At the same time, +the electrical field between the TX1 gate and the floating diffusion node is reduced, which also +reduces the dark current due to Gate-Induced-Drain-leakage (GIDL). A test chip is designed based +on this sequential transfer gate pixel in a standard 180-nm process without any process modification. +Simulations show that the test chip can run at least 20 Mfps with less than 5.8 𝑒− input-referred +noise, the lowest noise reported in the ultrafast burst-mode image sensor category. +3. +Noise emulation using neural network +Noise is ubiquitous in imaging and especially in high-speed and ultrafast imaging, when the +signal-to-noise ratio is limited in part by the source intensity and transients that may be induced in +the electronics. Meanwhile, better understanding of the noise through modeling is useful for noise +4 + +口1e+03 +#(e) +Voltage (V) +tx1OuterVoltage +tx2OuterVoltage +613e- & 574e- +tx3 OuterVoltage +fd OuterVoltage +rst OuterVoltage +110 + IntegrWell(-5,8.5,4.3) eDensity +2.44163V +IntegrWell(-5,1.5,4.3) eDensity +2.2773V +0.1 +0.001 +1e-05 +1e-07 +2e-07 +2.05e-07 +2.1e-07 +timeUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +Figure 5: Electrostatic potential along the charge transfer path. +Figure 6: Comparison of experimental images (Top Row) and images produced using Contrastive Unpaired +Translation (Bottom Row). +reduction. Some examples of noisy images from inertial confinement fusion (ICF) experiments are +shown in Figure 6. Further details may be found for example in [8]. Here we describe the use +of a generative adversarial network (GAN) to emulate image noise from the experiments. Some +examples of the synthetic images are given in Figure 6, which are qualitatively similar to the +experimental data. +The work flow of image synthesis is summarized in Figure 7. The first step is to produce +noise-free synthetic radiographs. Three dimensional (3D) models of ICF shells are generated using +Legendre polynomials for the shell boundaries and constant densities for the shells. These shells +consist of a 𝑆𝑖𝑂2 inner shell, an aluminum outer shell, and a foam between the two shells [8]. These +computer generated 3D shell models are projected to 2D images (or synthetic radiographs free of +noise) using a ray-tracing algorithm implementing the python library TIGRE. +The second step is to ‘add’ noise to the synthetic radiographs. The noise found in experimental +radiographs does not follow a standard distribution such as a Gaussian function and therefore is +difficult to simulate using traditional models. The noise can instead be applied to the synthetically +5 + +Voltage(V) +e +C2(C%(nt87_snapn_0004_ps_0a[6 +ca(C%(nt87_srapn, +T1 Red: +TX3 On, TX2 On, TX1 On +T2 Green: TX3 Off, TX2 On, TX1 On +T3 Blue: +TX3 Off, TX2 Off, TX1 On +T4 Cyan: +Tx3 Off, TX2 Off, TX1 Off + Distance(um) +TX3 +TX2 +TX1 +FDUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +produced noise-free radiographs using a conditional generative adversarial network (cGAN) [9]. +Similar to traditional GANs, a cGAN consists of a generator network and a discriminator network; +however, rather than using a latent noise vector as the input of the generator, an image from +experiment is used instead. We use contrastive unpaired translation (CUT) [10] to model the noise +found in the experimental radiographs. Figure 7 also shows the process of training CUT using +synthetically produced radiographs. +Figure 7: The process of training CUT using synthetically produced radiographs and experimental images +The CUT model consists of a residual convolutional network [11] with nine residual blocks +for the generator network (𝐺), PatchGAN [12] for the discriminator network (𝐷) and a set of +perceptrons (𝐻𝑙). By using feature vectors from the 𝑙th layer of the generator’s encoder 𝐺𝑙 +𝑒𝑛𝑐, +the perceptrons (𝐻𝑙) are used to classify image patches produced by network as patches from +the original image. Classification loss helps to inform the generator through optimization of the +patch contrastive loss, by preserving semantic features of the synthetic data such as the location +of shell boundaries. The noise content of the experimental images is transferred to the synthetic +images by jointly optimizing the generator and discriminator through the least squares GAN loss +(LSGAN) [13]. +The CUT model is trained using 200 synthetically produced images and 66 +experimental images to generate the synthetic images shown in Figure 6. +4. +Super-resolution using neural network +We recently demonstrated sub-pixel resolution or ‘super-resolution’ using neural networks for +post-processing of a boron-coated CCD (bCCD) pixelated images generated by neutrons. Similar +results have also been obtained for data from CMOS sensors, which will be reported elsewhere. +The detection principle for ultracold neutrons (UCNs) using a bCCD was discussed previously. +A scientific grade bCCD was used for UCN detection in our previous work [14]. The bCCD sensor +was built by the Lawrence Berkeley National Laboratory (LBNL) [15] and has been extensively +characterized by Fermilab for the Dark Energy Camera (DECam) project [16]. The detector is a +250 𝜇m thick, fully depleted, back-illuminated sensor fabriacated on high-resistivity silicon and has +8 million pixels (2k × 4k) with a pixel pitch of 15 × 15 𝜇m2. A thin 10B film up to 100 nm thick +is deposited onto the transparent rear window of the bCCD camera to act as a conversion layer. +6 + +Synthetic Image Production +Image Generation +Noisy +Object Parameters +3D Voxel +TIGRE +Synthetic +Generator +(Radii, Materials, etc.) +(Ray-Tracer) +Image +Synthetic +Model +(G) +Data +Losses +Classification +Generator +Perceptrons +Encoder +Experimental +Discriminator +(H) +(Genc) +Image +(D) +Classification +LSGAN +Labels +Loss +LossUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +The UCN hit is captured through the nuclear reactions 10B (n, 𝛼0𝛾) 7Li (6%) and 10B (n, 𝛼1𝛾) 7Li +(94%). The charged particles 𝛼, 7Li, and 𝛾-rays, penetrates the active silicon layer of the detector +to generate electron-hole (e-h) pairs. Influenced by the internal electric field, the generated holes +will travel the full length of the active silicon layer to the potential wells near the poly electrode +gates. The collected charges are converted to digitized value to be readout by the camera to create +an output image of the UCN hit. +4.1 Allpix Squared +High-statistics data samples produced with Monte Carlo simulations are required to train the +neural network. The Allpix Squared semiconductor simulation framework [17, 18] is used to gen- +erate these datasets. It is an open source simulation tool that implements end-to-end simulations +of particle detection from incident radiation to digitized detector output. The framework com- +prises different algorithms for charge transport and front-end simulations as well as an interface to +Geant4 [19] to describe the interaction of the incoming particle with the sensor material. Allpix +Squared works on a first-principles basis, moving individual charge carriers or groups thereof along +the electric field of the sensor using empirical mobility and recombination models. This approach +allows to replicate the sensor response of imaging devices given the detector parameters and electric +field distributions in the sensing element. Previous studies have demonstrated its capabilities of +accurately describing the response of CMOS sensors to minimum ionizing particles [20]. +The simulation is divided into several stages, each of which describes one component of +the signal formation. In the first stage, the interaction of the incoming particle with the sensor +material and the creation of electron-hole pairs is simulated. Subsequently, these charge carriers +are propagated through the sensor in the second stage. The coupling to the front-end electronics is +calculated in the third phase, and the front-end electronics and digitization is simulated in the fourth +and last stage. For each of these stages, Allpix Squared can store the Monte Carlo truth information +which allows to link detector output and initial particle and to trace the complete history of a detected +pixel hit. This information can be exploited when training the neural network by providing both +the true UCN position as well as the generated digital image obtained from the detector simulation. +The built-in multithreading capabilities of Allpix Squared allow to scale the event generation and +to simulate the large datasets required for training the neural network. +4.2 bCCD Modeling in Allpix Squared +While UCN hit images are acquired experimentally using a conversion layer and a silicon +detector, the ground-truth UCN hit position is not available. However, we can obtain synthetically +generated UCN hit images and their corresponding ground-truth hit position by using the silicon +detector framework Allpix Squared as summarized above [17]. +To accurately model the silicon detector physics, Allpix requires the detector to be fully +characterized so that the output physics of Allpix Squared can well match the actual detector. One +physics check we perform is comparing the experimental UCN hit images with Allpix’s synthetically +generated images. Figure 8 shows an example of matching an experimental hit with the closest +generated synthetic image. We utilize a matching algorithm that computes the mean squared error +(MSE) between the experimental and each synthetic image, and returns the synthetic image that +7 + +Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +results in the lowest MSE error. Recall that the generation of Allpix Squared hits is based on Monte +Carlo simulations. Therefore, the more synthetic hits that are generated, the higher the chance of +generating a synthetic hit that better matches the experimental. +Another physics check is the verification of the captured UCN energy spectrum. Due to the +nuclear reaction between a neutron and the 10B film, the detector will capture one of four possible +particle and energy combinations as shown in Table 1. Figure 9a plots the captured energy spectrum +of the experimental hits, while Figure 9b plots the Allpix spectrum. Both energy spectrum plots +are reconstructed to properly center on the 1470 keV 𝛼 peak. Under ideal conditions, the 7Li peak +should naturally occur at 840 keV. However, the 7Li peak is shifted about 70 keV which motivates +the existence of a dead layer between the 10B film and fully depleted silicon layer due to the down- +shift in the energy spectrum peak. The Allpix spectrum was generated using the synthetic UCN +hits while incorporating a dead layer into the bCCD model. The dead layer for the bCCD is fully +characterized by LBNL [21]. With the dead layer modeled, the Allpix energy spectrum distribution +and peaks well matches the experimental, which shows that the energy loss and charge creation +within the detector is well captured by Allpix. +Table 1: Detection probability (𝜔𝑖) and the produced reaction energy of the charged particles from the +neutron capture process. Note that a dead layer exists between the 10B film and the fully depleted region of +the Si sensor, which would reduce the actual energy captured. +Ion +𝜔𝑖 +Energy (keV) +7Li +47% +840 +7Li +3% +1020 +𝛼 +47% +1470 +𝛼 +3% +1780 +Figure 8: We aim to match an experimental UCN hit with the best generated synthetic hit. The matching +algorithm computes the mean squared error (MSE) between the experimental image and a synthetic image. +The smaller the MSE, the closer the synthetic image is to the experimental. The structural similarity index +measure (SSIM) between the experimental and synthetic image is also computed, where a perfect match +corresponds to SSIM = 1. In this example, synthetic image 5 best matches the experimental. +4.3 Deep Learning for position super-resolution +Machine learning techniques are very popular in recent years to learn a predictive model +between input and output labels. Deep learning is a special case of machine learning that is very +8 + +Experimental I Synthetic +1 +2 +3 +4 +5 +MSE +232.59 +61.58 +75.76 +121.63 +2.96 +SSIM +0.59 +0.74 +0.71 +0.71 +0.99Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +Figure 9: The captured UCN energy spectrum by (a) the bCCD and (b) Allpix Squared. (Note: The expected +peaks are 7Li = 840 keV and 𝛼 =1470 keV.) +powerful in learning nonlinear predictive models by using neural networks [22]. We aim to leverage +deep learning to obtain a predictive model that maps from input UCN hit images to the ground-truth +hit position. +Deep learning typically requires a large dataset to attain accurate predictive models. +We +use Allpix Squared to generate a large synthetic dataset consisting of 60,000 images and their +corresponding ground-truth labels. Note that, in addition to the ground-truth hit position, other +types of ground-truth information from the simulation history or prior simulation knowledge can +be included in the output labels. The experimental UCN data is not used to train the neural network +as the ground-truth labels are not available. Using the synthetic data, we propose to train a fully +connected neural network (FCNN). Figure 10a shows the overview of an arbitrary FCNN model +with three hidden layers. In the FCNN architecture, the 2-D input images are first flattened into +a 1-D vector. The flattened layer is then followed by three hidden layers with 124, 125, and 124 +9 + +6000 +a +5000 +4000 +3000 +2000 +1000 +500 +000 +500 +2000 +2500 +energy (keV) +2500 +b +2000 +Count +1500 +1000 +500 ++0 +500 +1000 +1500 +2000 +2500 +Energy (kev)Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +hidden neurons, respectively, and the output layer. While not shown in Figure 10a, dropout layers +are also included in the FCNN during the training and testing process. Dropout layers are popularly +used in neural networks to help mitigate over-fitting issues as well as uncertainty quantification. We +use the Pytorch library to train the FCNN to obtain a predictive mapping between the input UCN hit +images and the ground-truth labels. The trained model can then be used to make predictions on the +hit position of input UCN hit images, both synthetic and experimental images. Figure 10b shows +an example FCNN prediction on the entry point position for a synthetic hit image with sub-pixel +resolution. It is important to note that the accuracy of the predictive model for this arbitrary FCNN +can be improved by tuning the network architecture, number of hidden layers and neurons, and +other neural network parameters including the number of training epochs and learning rate. +Figure 10: (a) Overview of a FCNN model with three hidden layers. The input images of size 14 × 14 pixels +are flattened into a 1-D vector. The output of the neural network is a 1-D vector of size 𝑛, which denotes the +number of ground-truth labels. (b) The ground-truth labels in this example include the (𝑥, 𝑦) position for the +hit entry point, where the red ‘x’ denotes the actual entry point. The blue kernel density estimation (KDE) +plot shows the FCNN prediction, which obtains sub-pixel position resolution. +5. +Summary +A 20 Mfps CMOS image sensor design is described. TCAD simulations showed that the +test chip can run at least 20 Mfps with less than 5.8 𝑒− input-referred noise, the lowest noise +reported in the ultra-fast burst-mode image sensor category. By using neural networks for post data +processing, we demonstrated noise emulation and super position resolution at a fraction of pixel size. +The combination of novel CMOS pixel designs and data-enabled image post-processing provide a +promising path towards ultrafast multi-modal radiographic imaging and tomography applications. +SL and ZW wish to thank Drs. Don Groom and Steve Holland, both from Lawrence Berkeley +National Laboratory, for stimulating discussions. This work is supported in part by the LANL +LDRD, C3, and ICF programs under the Contract No. 89233218CNA000001. Prof. Zhu’s group +at UT Austin would like to acknowledge the NSF support through the Award 1802319. +10 + +1x125 +1x124 +1x196 +X +ActualEntryPoint +b +14x14 +Input +Output +Flatten +3 Hidden +LayersUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and +tomography +Zhehui Wang +References +[1] Z. Wang, Appl. Opt. 16 (2022) RDS1-RDS4. +[2] J. Crooks, B. Marsh, R. Turchetta, K. Taylor, W. Chan, A. Lahav and A. Fenigstein, “Ultra- +high-speed imaging at mega frames per second with a megapixel CIS image sensor”, 865903, +International Society for Optics and Photonics (2013). +[3] M. Suzuki, Y. Sugama, R. Kuroda and S. Sugawa “Over 100 Million Frames per Second 368 +Frames Global Shutter Burst CMOS Image Sensor with Pixel-wise Trench Capacitor Memory +Array”, Sensors 20 (4) (2020) 1806. +[4] Z. Wang, K. Anagnost, C. W. Barnes, D. M. Dattelbaum, E. R. Fossum, E. Lee, J. Liu, J. J. +Ma, W. Z. Meijer, W. Nie, C. M. Sweeney, A. C. Therrien, H. 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Hinton, “Deep learning,” Nature, vol. 521, no. 7553, pp. 436–444, +2015. +12 + diff --git a/a9FKT4oBgHgl3EQfoS5C/content/tmp_files/load_file.txt b/a9FKT4oBgHgl3EQfoS5C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..988d8df5c5adb67deaeaa697b2f6dd8b94c98a09 --- /dev/null +++ b/a9FKT4oBgHgl3EQfoS5C/content/tmp_files/load_file.txt @@ -0,0 +1,419 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf,len=418 +page_content='Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Xin Yue,𝑎,† Shanny Lin,𝑏,𝑐,† Wenting Li,𝑏 Bradley T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Wolfe,𝑏 Steven Clayton,𝑏 Mark Makela,𝑏 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Morris,𝑏 Simon Spannagel,𝑑 Erik Ramberg,𝑒 Juan Estrada,𝑒 Hao Zhu,𝑐 Jifeng Liu,𝑎 Eric R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Fossum𝑎 and Zhehui Wang𝑏,∗ 𝑎Thayer School of Engineering at Dartmouth, Dartmouth College, Hanover, NH 03755, USA 𝑏Los Alamos National Laboratory, Los Alamos, NM 87545, USA 𝑐Chandra Family Department of Electrical & Computer Engineering, The University of Texas at Austin, Austin, TX, 78712, USA 𝑑Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany 𝑒Fermi National Accelerator Laboratory, Batavia, IL 60510, USA †Lead authors with equivalent contributions E-mail: zwang@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='gov We summarize recent progress in ultrafast Complementary Metal Oxide Semiconductor (CMOS) image sensor development and the application of neural networks for post-processing of CMOS and charge-coupled device (CCD) image data to achieve sub-pixel resolution (thus ‘super-resolution’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The combination of novel CMOS pixel designs and data-enabled image post-processing pro- vides a promising path towards ultrafast high-resolution multi-modal radiographic imaging and tomography applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 10th International Workshop on Semiconductor Pixel Detectors for Particles and Imaging (Pixel2022) 12-16 December 2022 Santa Fe, New Mexico, USA ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='11865v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='ins-det] 27 Jan 2023 Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Introduction Using X-rays for imaging and tomography of optically opaque objects dated back to the famous invention of Wilheim Röntgen in 1895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Multimodal (MM) radiographic imaging and tomography (RadIT) combines several different forms or energies of ionizing radiation such as X-rays, 𝛾-rays, neutrons, energetic electrons, protons and others, which can potentially yield more information about an object than by using a monochromatic (mono-energetic) photons (particles) alone as the source of illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The technical challenges and opportunities for MM RadIT can be summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Additional information may be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The physics framework of radiation-matter interactions is mostly complete for all practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Recent advances in high-intensity radiation sources such as synchrotrons, X-ray free electron lasers, high-current low-emittance charged particle accel- erators, laser-driven sources open door to MM RadIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' One of the optimization problems in MM RadIT is to obtain as high temporal, spatial resolution of an object as possible, with a sufficiently large field of view, and at a certain radiation dose to minimize radiation damage of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 1: A holistic approach to MM RadIT optimization includes at least five branches of effort: Fundamen- tal physics of radiation-interaction with matter (PHY), radiation sources (SRCE), detectors (DETR), methods to modulate the radiation field (METH) and data handling (DATA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The fundamental physics principles of MM RadIT are well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Some challenges and opportunities for SRCE, DETR, METH and DATA are listed above for each branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Below, we first describe the recent progress in ultrafast Complementary Metal Oxide Semi- conductor (CMOS) pixelated sensor design and prototyping, followed by the the use of neural networks for noise emulation in X-ray imaging, and the demonstration of sub-pixel resolution in neutron detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Follow-on work includes CMOS image sensor fabrication and extension of the neural networks to different types of particles or photons, and different noise environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Our work demontrates a promising path towards high-speed high-resolution multi-modal radiographic imaging and tomography applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 2 h QUANTUM QED ELECTROWEAK MECHANICS INTERACTIONS 60Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Ultrafast CMOS image sensor development Ultra-high-speed (UHS) or ultrafast image sensors are widely used in scientific and industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The research on UHS CMOS image sensors for X-ray regimes has been conducted for years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The recently published works [2, 3] push the frame rate of UHS image sensors to the range of millions of frames per second (Mfps) by adopting burst-mode operations and advanced CMOS technology or customized CMOS technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Three CMOS image sensors were taped out towards this goal, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 2: Phase1, 2, 3 CMOS image sensor (CIS) taped out in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' During the phase one of this research [4], theoretical modeling and preliminary tests demon- strated more than 10× quantum efficiency improvement for high-energy X-ray photons (>10 keV) by depositing a photon-attenuation-layer (PAL) on a CMOS image sensor [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' In the phase two of this research, a block-wise compact readout architecture based on unit-length-capacitor and asynchronous successive-approximation (SAR) analog to digital converter (ADC) [7] was proposed and implemented, which enabled the image sensor fabricated using a standard 180-nm process to run at 76 thousand-frames-per-second (kfps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' To further boost the frame rate of the image sensor, a burst mode image sensor based on sequential transfer gates was proposed and taped out in Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 2022 during the phase three of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The sequential transfer gates enable the image sensor to run at least 20 Mfps and achieve the lowest input-referred noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Some highlights of this burst-mode image sensor is included below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 3 shows the conceptual 20-µm pixel layout based on the sequential transfer gates, where the orange rising-run shape stands for the photodiode, and the green polygon stands for the transfer gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Each photodiode finger geometry shape has been carefully calculated and optimized to have a constant ∼800V/cm strong electrical field pointing from the tip of the photodiode to the center of the photodiode, which guarantees fast charge transfer without process modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' As Figure 4 shows, by applying monotonically increasing control voltages and sequential timing on TX3(Blue), TX2(Green), and TX1(Red) gates, electrons (purple and cyan) in photodiodes can be fully transferred within 12 ns, with no image lag noticed in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 5 shows the potential diagram during the charge transfer path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' One can see that photon- generated electrons are first swept into the channel under the TX3 gate due to the strong electrical field in photodiode fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Then TX3, TX2, and TX1 gates turn off sequentially, which pushes electrons to move toward the floating diffusion node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Because the TX2 gate is entirely off before the falling transition of the TX1 gate, it is safe to move the floating diffusion node away from TX1, which will effectively reduce the overlap capacitance between the TX1 gate and floating diffusion 3 ImageofPhase1CiS ImageofPhase2CIS LavoutofPhase3CiSUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang Figure 3: High-Speed Conceptual Layout of a pixel in a CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' See the text for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 4: An example of TCAD transient simulation during charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The text contains further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' node, increase the conversion gain of the pixel and reduce input-referred noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' At the same time, the electrical field between the TX1 gate and the floating diffusion node is reduced, which also reduces the dark current due to Gate-Induced-Drain-leakage (GIDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' A test chip is designed based on this sequential transfer gate pixel in a standard 180-nm process without any process modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Simulations show that the test chip can run at least 20 Mfps with less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='8 𝑒− input-referred noise, the lowest noise reported in the ultrafast burst-mode image sensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Noise emulation using neural network Noise is ubiquitous in imaging and especially in high-speed and ultrafast imaging, when the signal-to-noise ratio is limited in part by the source intensity and transients that may be induced in the electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Meanwhile, better understanding of the noise through modeling is useful for noise 4 口1e+03 #(e) Voltage (V) tx1OuterVoltage tx2OuterVoltage 613e- & 574e- tx3 OuterVoltage fd OuterVoltage rst OuterVoltage 110 IntegrWell(-5,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='5,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='3) eDensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='44163V IntegrWell(-5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='5,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='3) eDensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='2773V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='001 1e-05 1e-07 2e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='05e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='1e-07 timeUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang Figure 5: Electrostatic potential along the charge transfer path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 6: Comparison of experimental images (Top Row) and images produced using Contrastive Unpaired Translation (Bottom Row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Some examples of noisy images from inertial confinement fusion (ICF) experiments are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Further details may be found for example in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Here we describe the use of a generative adversarial network (GAN) to emulate image noise from the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Some examples of the synthetic images are given in Figure 6, which are qualitatively similar to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The work flow of image synthesis is summarized in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The first step is to produce noise-free synthetic radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Three dimensional (3D) models of ICF shells are generated using Legendre polynomials for the shell boundaries and constant densities for the shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' These shells consist of a 𝑆𝑖𝑂2 inner shell, an aluminum outer shell, and a foam between the two shells [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' These computer generated 3D shell models are projected to 2D images (or synthetic radiographs free of noise) using a ray-tracing algorithm implementing the python library TIGRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The second step is to ‘add’ noise to the synthetic radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The noise found in experimental radiographs does not follow a standard distribution such as a Gaussian function and therefore is difficult to simulate using traditional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The noise can instead be applied to the synthetically 5 Voltage(V) e C2(C%(nt87_snapn_0004_ps_0a[6 ca(C%(nt87_srapn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' T1 Red: TX3 On,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX2 On,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX1 On T2 Green: TX3 Off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX2 On,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX1 On T3 Blue: TX3 Off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX2 Off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX1 On T4 Cyan: Tx3 Off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX2 Off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TX1 Off Distance(um) TX3 TX2 TX1 FDUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang produced noise-free radiographs using a conditional generative adversarial network (cGAN) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Similar to traditional GANs, a cGAN consists of a generator network and a discriminator network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' however, rather than using a latent noise vector as the input of the generator, an image from experiment is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' We use contrastive unpaired translation (CUT) [10] to model the noise found in the experimental radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 7 also shows the process of training CUT using synthetically produced radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 7: The process of training CUT using synthetically produced radiographs and experimental images The CUT model consists of a residual convolutional network [11] with nine residual blocks for the generator network (𝐺), PatchGAN [12] for the discriminator network (𝐷) and a set of perceptrons (𝐻𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' By using feature vectors from the 𝑙th layer of the generator’s encoder 𝐺𝑙 𝑒𝑛𝑐, the perceptrons (𝐻𝑙) are used to classify image patches produced by network as patches from the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Classification loss helps to inform the generator through optimization of the patch contrastive loss, by preserving semantic features of the synthetic data such as the location of shell boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The noise content of the experimental images is transferred to the synthetic images by jointly optimizing the generator and discriminator through the least squares GAN loss (LSGAN) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The CUT model is trained using 200 synthetically produced images and 66 experimental images to generate the synthetic images shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Super-resolution using neural network We recently demonstrated sub-pixel resolution or ‘super-resolution’ using neural networks for post-processing of a boron-coated CCD (bCCD) pixelated images generated by neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Similar results have also been obtained for data from CMOS sensors, which will be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The detection principle for ultracold neutrons (UCNs) using a bCCD was discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' A scientific grade bCCD was used for UCN detection in our previous work [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The bCCD sensor was built by the Lawrence Berkeley National Laboratory (LBNL) [15] and has been extensively characterized by Fermilab for the Dark Energy Camera (DECam) project [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The detector is a 250 𝜇m thick, fully depleted, back-illuminated sensor fabriacated on high-resistivity silicon and has 8 million pixels (2k × 4k) with a pixel pitch of 15 × 15 𝜇m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' A thin 10B film up to 100 nm thick is deposited onto the transparent rear window of the bCCD camera to act as a conversion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 6 Synthetic Image Production Image Generation Noisy Object Parameters 3D Voxel TIGRE Synthetic Generator (Radii, Materials, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=') (Ray-Tracer) Image Synthetic Model (G) Data Losses Classification Generator Perceptrons Encoder Experimental Discriminator (H) (Genc) Image (D) Classification LSGAN Labels Loss LossUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang The UCN hit is captured through the nuclear reactions 10B (n, 𝛼0𝛾) 7Li (6%) and 10B (n, 𝛼1𝛾) 7Li (94%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The charged particles 𝛼, 7Li, and 𝛾-rays, penetrates the active silicon layer of the detector to generate electron-hole (e-h) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Influenced by the internal electric field, the generated holes will travel the full length of the active silicon layer to the potential wells near the poly electrode gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The collected charges are converted to digitized value to be readout by the camera to create an output image of the UCN hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='1 Allpix Squared High-statistics data samples produced with Monte Carlo simulations are required to train the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The Allpix Squared semiconductor simulation framework [17, 18] is used to gen- erate these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' It is an open source simulation tool that implements end-to-end simulations of particle detection from incident radiation to digitized detector output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The framework com- prises different algorithms for charge transport and front-end simulations as well as an interface to Geant4 [19] to describe the interaction of the incoming particle with the sensor material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Allpix Squared works on a first-principles basis, moving individual charge carriers or groups thereof along the electric field of the sensor using empirical mobility and recombination models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' This approach allows to replicate the sensor response of imaging devices given the detector parameters and electric field distributions in the sensing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Previous studies have demonstrated its capabilities of accurately describing the response of CMOS sensors to minimum ionizing particles [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The simulation is divided into several stages, each of which describes one component of the signal formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' In the first stage, the interaction of the incoming particle with the sensor material and the creation of electron-hole pairs is simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Subsequently, these charge carriers are propagated through the sensor in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The coupling to the front-end electronics is calculated in the third phase, and the front-end electronics and digitization is simulated in the fourth and last stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' For each of these stages, Allpix Squared can store the Monte Carlo truth information which allows to link detector output and initial particle and to trace the complete history of a detected pixel hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' This information can be exploited when training the neural network by providing both the true UCN position as well as the generated digital image obtained from the detector simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The built-in multithreading capabilities of Allpix Squared allow to scale the event generation and to simulate the large datasets required for training the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='2 bCCD Modeling in Allpix Squared While UCN hit images are acquired experimentally using a conversion layer and a silicon detector, the ground-truth UCN hit position is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' However, we can obtain synthetically generated UCN hit images and their corresponding ground-truth hit position by using the silicon detector framework Allpix Squared as summarized above [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' To accurately model the silicon detector physics, Allpix requires the detector to be fully characterized so that the output physics of Allpix Squared can well match the actual detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' One physics check we perform is comparing the experimental UCN hit images with Allpix’s synthetically generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 8 shows an example of matching an experimental hit with the closest generated synthetic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' We utilize a matching algorithm that computes the mean squared error (MSE) between the experimental and each synthetic image, and returns the synthetic image that 7 Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang results in the lowest MSE error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Recall that the generation of Allpix Squared hits is based on Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Therefore, the more synthetic hits that are generated, the higher the chance of generating a synthetic hit that better matches the experimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Another physics check is the verification of the captured UCN energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Due to the nuclear reaction between a neutron and the 10B film, the detector will capture one of four possible particle and energy combinations as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 9a plots the captured energy spectrum of the experimental hits, while Figure 9b plots the Allpix spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Both energy spectrum plots are reconstructed to properly center on the 1470 keV 𝛼 peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Under ideal conditions, the 7Li peak should naturally occur at 840 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' However, the 7Li peak is shifted about 70 keV which motivates the existence of a dead layer between the 10B film and fully depleted silicon layer due to the down- shift in the energy spectrum peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The Allpix spectrum was generated using the synthetic UCN hits while incorporating a dead layer into the bCCD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The dead layer for the bCCD is fully characterized by LBNL [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' With the dead layer modeled, the Allpix energy spectrum distribution and peaks well matches the experimental, which shows that the energy loss and charge creation within the detector is well captured by Allpix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Table 1: Detection probability (𝜔𝑖) and the produced reaction energy of the charged particles from the neutron capture process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Note that a dead layer exists between the 10B film and the fully depleted region of the Si sensor, which would reduce the actual energy captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Ion 𝜔𝑖 Energy (keV) 7Li 47% 840 7Li 3% 1020 𝛼 47% 1470 𝛼 3% 1780 Figure 8: We aim to match an experimental UCN hit with the best generated synthetic hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The matching algorithm computes the mean squared error (MSE) between the experimental image and a synthetic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The smaller the MSE, the closer the synthetic image is to the experimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The structural similarity index measure (SSIM) between the experimental and synthetic image is also computed, where a perfect match corresponds to SSIM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' In this example, synthetic image 5 best matches the experimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='3 Deep Learning for position super-resolution Machine learning techniques are very popular in recent years to learn a predictive model between input and output labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Deep learning is a special case of machine learning that is very 8 Experimental I Synthetic 1 2 3 4 5 MSE 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='59 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='58 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='76 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='96 SSIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='99Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang Figure 9: The captured UCN energy spectrum by (a) the bCCD and (b) Allpix Squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' (Note: The expected peaks are 7Li = 840 keV and 𝛼 =1470 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=') powerful in learning nonlinear predictive models by using neural networks [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' We aim to leverage deep learning to obtain a predictive model that maps from input UCN hit images to the ground-truth hit position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Deep learning typically requires a large dataset to attain accurate predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' We use Allpix Squared to generate a large synthetic dataset consisting of 60,000 images and their corresponding ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Note that, in addition to the ground-truth hit position, other types of ground-truth information from the simulation history or prior simulation knowledge can be included in the output labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The experimental UCN data is not used to train the neural network as the ground-truth labels are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Using the synthetic data, we propose to train a fully connected neural network (FCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 10a shows the overview of an arbitrary FCNN model with three hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' In the FCNN architecture, the 2-D input images are first flattened into a 1-D vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The flattened layer is then followed by three hidden layers with 124, 125, and 124 9 6000 a 5000 4000 3000 2000 1000 500 000 500 2000 2500 energy (keV) 2500 b 2000 Count 1500 1000 500 +0 500 1000 1500 2000 2500 Energy (kev)Ultrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang hidden neurons, respectively, and the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' While not shown in Figure 10a, dropout layers are also included in the FCNN during the training and testing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Dropout layers are popularly used in neural networks to help mitigate over-fitting issues as well as uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' We use the Pytorch library to train the FCNN to obtain a predictive mapping between the input UCN hit images and the ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The trained model can then be used to make predictions on the hit position of input UCN hit images, both synthetic and experimental images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 10b shows an example FCNN prediction on the entry point position for a synthetic hit image with sub-pixel resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' It is important to note that the accuracy of the predictive model for this arbitrary FCNN can be improved by tuning the network architecture, number of hidden layers and neurons, and other neural network parameters including the number of training epochs and learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Figure 10: (a) Overview of a FCNN model with three hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The input images of size 14 × 14 pixels are flattened into a 1-D vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The output of the neural network is a 1-D vector of size 𝑛, which denotes the number of ground-truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' (b) The ground-truth labels in this example include the (𝑥, 𝑦) position for the hit entry point, where the red ‘x’ denotes the actual entry point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The blue kernel density estimation (KDE) plot shows the FCNN prediction, which obtains sub-pixel position resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Summary A 20 Mfps CMOS image sensor design is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' TCAD simulations showed that the test chip can run at least 20 Mfps with less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content='8 𝑒− input-referred noise, the lowest noise reported in the ultra-fast burst-mode image sensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' By using neural networks for post data processing, we demonstrated noise emulation and super position resolution at a fraction of pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' The combination of novel CMOS pixel designs and data-enabled image post-processing provide a promising path towards ultrafast multi-modal radiographic imaging and tomography applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' SL and ZW wish to thank Drs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Don Groom and Steve Holland, both from Lawrence Berkeley National Laboratory, for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' This work is supported in part by the LANL LDRD, C3, and ICF programs under the Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 89233218CNA000001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' Zhu’s group at UT Austin would like to acknowledge the NSF support through the Award 1802319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 10 1x125 1x124 1x196 X ActualEntryPoint b 14x14 Input Output Flatten 3 Hidden LayersUltrafast CMOS image sensors and data-enabled super-resolution for multimodal radiographic imaging and tomography Zhehui Wang References [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FKT4oBgHgl3EQfoS5C/content/2301.11865v1.pdf'} +page_content=' 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b/c9FST4oBgHgl3EQfEDj7/content/2301.13713v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:06cbf03f4600943ae26c2a8da8cbcb886555b96a0a21f4dff1735ba667a2a0a8 +size 4966302 diff --git a/cdFJT4oBgHgl3EQfRCzr/content/tmp_files/2301.11494v1.pdf.txt b/cdFJT4oBgHgl3EQfRCzr/content/tmp_files/2301.11494v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2bbe09d6a314d968449afc0cdd59ef566228a05 --- /dev/null +++ b/cdFJT4oBgHgl3EQfRCzr/content/tmp_files/2301.11494v1.pdf.txt @@ -0,0 +1,2312 @@ +Published as a conference paper at ICLR 2023 +LEARNING VORTEX DYNAMICS FOR FLUID +INFERENCE AND PREDICTION +Yitong Deng +Dartmouth College +Stanford University +Hong-Xing Yu +Stanford University +Jiajun Wu +Stanford University +Bo Zhu +Dartmouth College +ABSTRACT +We propose a novel machine learning method based on differentiable vortex par- +ticles to infer and predict fluid dynamics from a single video. The key design of +our system is a particle-based latent space to encapsulate the hidden, Lagrangian +vortical evolution underpinning the observable, Eulerian flow phenomena. We de- +vise a novel differentiable vortex particle system in conjunction with their learn- +able, vortex-to-velocity dynamics mapping to effectively capture and represent +the complex flow features in a reduced space. We further design an end-to-end +training pipeline to directly learn and synthesize simulators from data, that can +reliably deliver future video rollouts based on limited observation. The value of +our method is twofold: first, our learned simulator enables the inference of hidden +physics quantities (e.g. velocity field) purely from visual observation, to be used +for motion analysis; secondly, it also supports future prediction, constructing the +input video’s sequel along with its future dynamics evolution. We demonstrate +our method’s efficacy by comparing quantitatively and qualitatively with a range +of existing methods on both synthetic and real-world videos, displaying improved +data correspondence, visual plausibility, and physical integrity.1 +1 +INTRODUCTION +As small as thin soap films, and as large as atmospheric eddies observable from outer space, fluid +systems can exhibit intricate dynamic features on different mediums and scales. However, despite +the inspiring recent progress, to effectively represent these flow features, identify the underlying +dynamics system, and predict the future evolution, remains an open problem for scientific machine +learning, due to the noisy data, imperfect modeling, and unavailable, hidden physics quantities. +Here, we identify three fundamental challenges that currently hinder the success of such endeavors. +First, flow features are difficult to represent. Traditional methods learn the fluid dynamics by storing +velocity fields either using regularly-spaced grids or smooth neural networks. These approaches +have demonstrated promising results for fluid phenomena that are relatively damped and laminar +(e.g. Chu et al., 2022), but for fluid systems that can exhibit turbulent features on varying scales, +these methods fall short due to the problem’s curse of dimensionality (high-resolution space and +time), local non-smoothness, and hidden constraints. As a result, more compact and structured +representation spaces and data structures are called for. +Secondly, hidden flow dynamics is hard to learn.. Fluid systems as prescribed by the Navier-Stokes +equations tightly couple multiple physical quantities (i.e. velocity, pressure, and density), and yet, +only the density information can be accessibly measured. Due to the system’s complexity, ambigu- +ity, and non-linearity, directly learning the underlying dynamics from the observable density space +is infeasible; and successful learning is usually contingent on velocity or pressure supervision, a +requirement that distances these methods from deployment in real-world scenarios. +Exciting recent progress has been made in hidden dynamics inference by PDE-based frameworks +such as (Raissi et al., 2020), which uncover the underlying physics variables solely from density +observations. However, this type of methods encounter the third fundamental challenge, which is +1We +invite +the +readers +to +view +the +video +results +via +an +anonymized +link: +learning-vortex-dynamics.github.io +1 +arXiv:2301.11494v1 [cs.LG] 27 Jan 2023 + +Published as a conference paper at ICLR 2023 +Observed real video +Synthesized future prediction +Figure 1: Our goal is to learn vortex dynamics for fluid prediction. The 3 frames on the left are observed from +a real video recording a soap film on a circular metal rim, where the red ink is spreading. The 3 frames on the +right are future prediction results produced by our method. +that performing future prediction is difficult. As we will demonstrate, although strong results are ob- +tained for interpolating inside the observation window provided by the training data, these methods +render themselves unsuited for extrapolating into the future, profoundly limiting their usage. +In this paper, we propose a novel fluid learning method to tackle the three aforementioned challenges +in a unified framework. In particular, harnessing the physics insights developed for the vortex meth- +ods in the computational fluid dynamics (CFD) literature, we design a novel, data-driven Lagrangian +vortex system to serve as a compact and structured latent representation of the flow dynamics. We +learn the complex dynamic system in the high-dimensional image space by learning a surrogate, +low-dimensional model on the latent vortex space, and use a physics-based, learnable mechanism: +the vortex-to-velocity dynamics module, to decode the latent space dynamics back to the image +space. Leveraging this advantageous representation, we design an end-to-end training pipeline that +learns from a single video containing only density information, to jointly perform accurate inference +of hidden physics quantities and robust long-term future predictions, as shown in Figure 1. +To examine the efficacy of our method, we compare our method’s performance on both motion in- +ference and future prediction against various state-of-the-art methods along with their extensions. +We conduct benchmark testing on synthetic videos generated using high-order numeric simulation +schemes as well as real-world videos in the wild. Evaluation is carried out both quantitatively +through exhaustive numerical analysis, and qualitatively by generating a range of realistic visual +effects. We compare the uncovered velocities both in terms of data correspondence and physical +integrity, and the predicted visual results in terms of both pixel-level and perceptual proximity. Re- +sults indicate that our proposed method provides enhanced abilities on both fronts, inferring hidden +quantities at higher accuracy, and predicting future evolution with higher plausibility. +In summary, the main technical contributions of our framework align with the three challenges +we have addressed regarding flow representation, dynamics learning, and simulator synthesis. (1) +We devise a novel fluid dynamics representation with differentiable vortex particles, to drastically +reduce the learning problem’s dimensionality on complex flow fields. Motivated by the vortex meth- +ods in CFD, we establish the vorticity-carrying fluid particles as a new type of learning primitive to +transform the existing PDE-constrained optimization problem to a particle ODE trajectory learning +problem. (2) We design a novel particle-to-field paradigm for learning the Lagrangian vortex dy- +namics. Instead of learning the interaction among particles (e.g. Sanchez-Gonzalez et al., 2020), +our model learns the continuous vortex-to-velocity induction mapping to naturally connect the vor- +tex particle dynamics in the latent space and the fluid phenomena captured in the image space. (3) +We develop an end-to-end differentiable pipeline composed of two network models to synthesize +data-driven simulators based on single, short RGB videos. +2 +RELATED WORK +Hidden Dynamics Inference. The problem of inferring dynamical systems based on noisy or incom- +plete observations has been addressed using a variety of techniques, including symbolic regression +(Bongard & Lipson, 2007; Schmidt & Lipson, 2009), dynamic mode decomposition (Schmid, 2010; +Kutz et al., 2016), sparse regression (Brunton et al., 2016; Rudy et al., 2017), Gaussian process re- +gression (Raissi et al., 2017; Raissi & Karniadakis, 2018), and neural networks (Raissi et al., 2019; +Yang et al., 2020; Jin et al., 2021; Chu et al., 2022). Among these inspiring advancements, the “hid- +den fluid mechanics” (HFM) method proposed in Raissi et al. (2020) is particularly noteworthy, as +it uncovers the continuous solutions of fluid flow using only images (the transport of smoke or ink). +Data-driven Simulation. Recently, growing interests are cast on learning numerical simulators ac- +cording to data supervision, which has shown promise to reduce computation time (Ladick`y et al., +2 + +Published as a conference paper at ICLR 2023 +2015; Guo et al., 2016; Wiewel et al., 2019; Pfaff et al., 2020; Sanchez-Gonzalez et al., 2020; Tomp- +son et al., 2017), increase simulation realism (Chu & Thuerey, 2017; Xie et al., 2018), enable stylized +control (Kim et al., 2020), estimate dynamic quantities such as viscosity and energy (Chang et al., +2016; Battaglia et al., 2016; Ummenhofer et al., 2019), and facilitate the training of control policies +(Sanchez-Gonzalez et al., 2018; Li et al., 2018). Akin to Watters et al. (2017), our system takes im- +ages as inputs and performs dynamics simulation on a low-dimensional latent space; but our method +learns purely from the input video and performs future rollout in the image space. Our method is +also related to Guan et al. (2022), which infers Lagrangian fluid simulation from observed images. +We propose sparse neural vortices as our representations while they use dense material points. +Vortex Methods. The underlying physical prior incorporated in our machine learning system is +rooted in the family of vortex methods that are rigorously derived, analyzed, and tested in the com- +putational fluid dynamics (CFD) (Leonard, 1980; Perlman, 1985; Beale & Majda, 1985; Winck- +elmans & Leonard, 1993; Mimeau & Mortazavi, 2021) and computer graphics (CG) community +(Selle et al., 2005; Park & Kim, 2005; Weißmann & Pinkall, 2010; Brochu et al., 2012). Xiong et al. +(2020) is pioneering for combining the Discrete Vortex Method with neural networks, but its pr +oposed method relies on a large set of ground truth velocity sequences, whereas our method learns +from single videos without needing the ground truth velocity. +3 +PHYSICAL MODEL +We consider the velocity-vorticity form of the Navier–Stokes equations (obtained by taking the curl +operator on both sides of the momentum equation, see Cottet et al. (2000) for details): +Dω +Dt = ∂ω +∂t + u · ∇ω = (ω · ∇)u + ν∇2ω + ∇ × b, +(1) +u = ∇ × φ, +∇2φ = −ω, +(2) +where ω denotes the vorticity, u the velocity, b the conservative body force, ν the kinematic viscos- +ity, and φ the streamfunction. If we ignore the viscosity and stretching terms (inviscid 2D flow), we +obtain Dω/Dt = 0, which directly conveys the Largangian conservative nature of vorticity (i.e. a +particle’s vorticity will not change during its advection). +If we assume the fluid domain has an open boundary, we can further obtain the vorticity-to-velocity +induction formula, which is derived by solving the the Poisson equation on φ using Green’s method +(also known as the Biot-Savart Law in fluid mechanics): +u(x) = +� +K(x − x′)ω(x′)dx′, +(3) +The kernel K exhibits a type-II singularity at 0 and causes numerical instabilities, therefore in CFD +practices, K is replaced by various mollified versions Kδ to improve the simulation accuracy (Beale +& Majda, 1985). We note that the mollified version Kδ is not unique, and can be customized and +tuned in different numerical schemes per human heuristics. Different types and parameters for the +mollification bring about significantly different simulation results. +Takeaways. The mathematical models above provide two central physical insights guiding the design +of our vortex-based learning framework: (1) The Lagrangian conservation of vorticity ω suggests +the suitablity of adopting Lagrangian data structures (e.g. particles as opposed to grids) to capture +the dynamics. Since the tracked variable ω remains temporally-invariant for each Lagrangian vortex, +the evolution of the continuous flow field is embodied fully by the movement of these vortices, which +significantly alleviates the difficulty in learning. (2) Equation 3 presents an induction mapping from +the vorticity ω, a Lagrangian quantity carried by particles, to the velocity u, an Eulerian variable that +can be queried continuously at an arbitrary location x. This lends the possibility for the Lagrangian +method to be used in conjunction with Eulerian data structures (e.g. a grid) for learning from the +widely available video data. Furthermore, such a mapping can benefit from data-driven learning, +as we can replace human heuristics by learning the mollified kernel Kδ (which is shared among all +vortex particles) to minimize the discrepancy between the induced and observed flow phenomena. +4 +METHOD +System Overview. Following the physics insight conveyed in Section 3, we design a learning system +whose workflow is illustrated in Figure 2. As shown on the top row, our system takes as input a +3 + +Published as a conference paper at ICLR 2023 +Eulerian +Integrator +Predicting The Future +Eulerian +Integrator +Analyzing The Past +Observed +RGB Image +Space +Intermediate +Velocity +Space +Learned +Vortex + Space +Dynamics +Module +Dynamics +Module +Dynamics +Module +Dynamics +Module +Lagrangian +Integrator +Trajectory +Module +Eulerian +Integrator +Eulerian +Integrator +Lagrangian +Integrator +Figure 2: Given an input RGB image sequence (top row), we learn the dynamic system of a low-dimensional +vortex space (bottom row), whose motion is decoded into the motion of the high-dimensional image space to +explain the observed phenomena. +single RGB video that captures the vortical flow phenomena. As shown on the bottom row, our +method learns and outputs a dynamic simulator — not on the image space itself, but on a latent +space consisting of discrete vortices. Learning the latent dynamics in the vortex space would only +be useful and feasible if we can tie it back to the image space, because it is the image space that we +want to perform future prediction on, and we have no ground truth values for the vortex particles to +begin with. The bridge to tie the vortex space with the image space is derived from Equation 3, which +supplies the core insight that there exists a learnable mapping from vortex particles to the continuous +velocity field at arbitrary positions. This mapping is modelled by our learned dynamics module D, +which gives rise to the intermediate velocity space, as shown in the middle row of Figure 2. +4.1 +DIFFERENTIABLE VORTEX PARTICLES +We track a collection V of n vortex particles, i.e. V := [V1, . . . , Vn]. We define each vortex Vi as +the 3-tuple (xi, ωi, δi), where x represents the position, ω the vortex strength, and δ the size. The +number of particles n is a hyperparameter which we set to 16 for all our results. Further discussions +and experiments regarding the choice of n can be found in Appendix D. We also note that, since +we are concerned with 2D inviscid incompressible flow, the size δ of a vortex does not change in +time due to incompressibility, and the vortex strength ω does not change in time due to Kelvin’s +circulation theorem (see Hald (1979) for a thorough discussion). +Learning Particle Trajectory. As shown in Figure 3, we learn a particle trajectory module: a query +function T such that Vt = T (t), which predicts the configuration of all the vortices at any time +t ∈ [0, tE] where tE represents the end time of the input video. As described above, predicting Vt +boils down to determining two time-invariant components: (1) [ω1, . . . , ωn], (2) [δ1, . . . , δn], and one +time-varying component: [(x1)t, . . . , (xn)t]. For the two time-invariant components, we introduce +two trainable n × 1 vectors ∆ and Ω to represent δ and ω respectively, such that [ω1, . . . , ωn] = +sin(Ω) and [δ1, . . . , δn] = sigmoid(∆) + ϵ. The vortex size ∆ and strength Ω are optimized to fit +the motion depicted by the input RGB video. For the time-varying component, we use a network +N1(t) to encode N1(t) = [(x1)t, . . . , (xn)t], and the particle velocities dN1 +dt can be extracted using +automatic differentiation. We note that learning the full particle trajectory, rather than the initial +particle configuration, allows the aggregation of dynamics information throughout the input video +for better inference and prediction. We provide further discussion on this design in Appendix F. +Trajectory Initialization. As discussed above, the trajectory T has three learnable components: ∆, +Ω and N1. We initialize ∆ and Ω as zero vectors, which gives δi = 0.5 + ϵ and ωi = 0 for all +i. Conceptually, these vortices are initialized as large blobs with no vortex strength, which learn to +alter their sizes and grow their strengths to better recreate the eddies seen in the video. The initial +positions [(x1)0, . . . , (xn)0] are regularly spaced points to populate the entire domain. We initialize +4 + +Published as a conference paper at ICLR 2023 +Figure 3: We encapsulate the motion of a continuous field by the motion of discrete particles. The blue trajec- +tory is encoded by a neural network N1, corresponding to the input video; while the red trajectory is unrolled +using our learned dynamics module and a numeric integrator, corresponding to the future prediction. +the 16 particles to lie at grid centers of a 4 × 4 grid. To do so, we simply pretrain N1 so that N1(0) +evaluates to the grid centers. The details regarding pretraining is given in Appendix A. +Learning the Vorticity-to-Velocity Mapping. The vorticity-to-velocity mapping is performed by our +dynamics module, which predicts the velocity u given arbitrary query point x and the collection of +vortices V = [(x1, ω1, δ1), . . . , (xn, ωn, δn)]. Following the physics insight conveyed in Section 3, +D embodies the integration: +u(x) = +� +Kδ(x − x′)ω(x′)dx′, +(4) +which replace the kernel K by a learnable Kδ : Rd → Rd mapping, with d representing the spatial +dimension. Rather than directly using a neural network to model this Rd → Rd mapping, we further +incorporate physical insights by analyzing the structure of Kδ. As derived in Beale & Majda (1985), +the kernel Kδ for 2-dimensional flow exhibits the following form: +Kδ(z) = +1 +2πrM(r, δ)R 2 +π (z), r = |z| +(5) +where R 2 +π is the 90◦ rotation applying which to z computes the unit direction of the cross product +of z and the out-of-plane vector ω; and M(r, δ) is the human heuristic term that varies by choice. +Hence, we opt to replace +1 +2πrM(r, δ) by a R2 → R neural network function N2(r, δ) so that: +u(x) = +� +N2(|x − x′|, δi)R 2 +π (x − x′)ω(x′)dx′ +(6) +≈ +n +� +i=1 +N2(|x − xi|, δi)R 2 +π (x − xi)ωi = D(V). +(7) +Learning this induction kernel N2(r, δ) instead of using heuristics-based kernels allows for more +accurate fluid learning and prediction from input videos. We discuss more on this in Appendix E. +4.2 +END-TO-END TRAINING +As previously mentioned, the dynamics on the latent vortex space is bridged to the evolution of +the image space through the differentiable, dynamic module D. Hence, we can optimize the vor- +tex representation Vt = T (t) at time t using images as supervision. First, we select m frames: +[It, . . . , It+m] from the video. Then, we compute ut = D(Vt). After that, (ut, It) is fed into an +integrator on the Eulerian grid to predict ˜It+1. Simultaneously, (ut, Vt) is fed into an integrator on +Lagrangian particles to predict ˜Vt+1. The process is then repeated, using ˜It+1 in place of It and ˜Vt+1 +in place of Vt, to generate ˜It+2 and ˜Vt+2, and so on. Eventually, we would obtain [˜It+1, . . . , ˜It+m], +which are the predicted future outcome starting at time t. We optimize T and D jointly by minimiz- +ing its difference between [˜It+1, . . . , ˜It+m] and [It+1, . . . , It+m] in and end-to-end fashion. +By picking different values of t in each iteration to cover [0, tE], we optimize T and D to fit the +input video. There remains one more caveat — that the trajectories in T are not enforced to be +consistent with D, because each frame of Vt is optimized individually. +In other words, if we +evaluate the particle velocities [ ˙x1, ..., +˙xn] = +dN1 +dt +as prescribed by T , it should coincide with +5 + +Published as a conference paper at ICLR 2023 +Ours +HFM Extrap. +HFM + UNet +ER + UNet +Ours +HFM Extrap. HFM + UNet +ER + UNet +Figure 4: Applied to real-world videos, our Lagrangian based method can create realistic future predictions +over long periods of time compared to existing methods (and their extensions). +Velocity +Stream- +lines +Velocity +Residue +Ground Truth +HFM +E-R +UNet +Ours +Figure 5: Hidden motion inference compared with existing methods on a synthetic video. Our method uncovers +the underlying velocity field with higher accuracy. +[D(V)(x1), . . . , D(V)(xn)], which as prescribed by D. Hence, in training, another loss is com- +puted between dT +dt and [D(V)(x1), . . . , D(V)(xn)] to align the vortex trajectory and the predicted +velocity. +Deployment. After successful training, the learned system allows us to perform two important tasks. +First, using our continuous query function T (t), we are able to interpolate for V(t), which then +uncovers the hidden velocity field u = D(Vt) at arbitrary precision, which provides the same func- +tionality as Raissi & Karniadakis (2018), but using vorticity instead of pressure as the secondary +variable. Moreover, with the dynamics module D, we can perform future prediction to unroll the +input video, a feature unsupported by previous methods. As shown in Figure 4, since our method is +forward-simulating by nature, it can provide more realistic and robust future prediction than existing +methods and their extensions. Further implementation details of our method, including hyperparam- +eters, network architectures, training schemes and computational costs can be found in Appendix A. +5 +EXPERIMENTS +We evaluate our method’s ability to perform motion inference and future prediction on both synthetic +and real videos, comparing against existing methods. +6 + +250 +24D +150 +14D +50 +0 + +50 +140 +150 +24D +250 +X250 +20D +150 +100 +50 +1iD +150 +2i0 +250250 +240 +150 +50 +50 +100 +150 +21D +250 +x250 +24D +150 +140 +50 +01 +50 +1iD +150 +240 +250 +x250 +240 +150 +140 +50 +01 +0 +50 +150 +210 +250 +xT +T0.7 +0.6 +240 +0.5 +150 +to- +0.3 +100 +0.2 +50 +o- +O.D +0 +50 +150 +250 +x250 +LD +150 +0.6 +140 +0.4 +50 +0.2 +0.D +I +50 +140 +150 +20D +250 +X250 +24D +150 +50 +0 + +0 +50 +14D +150 +24D +250 +X250 - +150 +100 +50 +50 +150 +24D +250 +x250 +24D +150 +50 +0 +- +50 +140 +150 +24D +250 +X250 - +240 +150 +140 +50 +0+ +0 +50 +140 +150 +24D +250 +x250 +24D +150 +140 +50 +0 +o +50 +1iD +150 +240 +250 +x250 +24D +150 +50 +0- +0 +50 +140 +150 +21D +250250 - +150 +100 +50 +0 +50 +150 +210 +250 +x250 - +150 +100 +50 +50 +150 +24D +250 +xPublished as a conference paper at ICLR 2023 +Future +Prediction +Errors +Motion +Inference +Errors +Figure 6: Error analysis on a synthetic dataset. The top row plots the inference errors of velocity, vorticity, and +compressibility. The bottom row plots the future prediction errors, which consider both the dynamic error in +the velocity and the perceptual error of the generated image sequence. +Baselines. For motion inference, we compare our method against Raissi & Karniadakis (2018) +(HFM) and Zhang et al. (2022) (ER). We reimplement the HFM method as prescribed, making only +the modification that instead of using only a single concentration variable c and its inverse d as spec- +ified by (Raissi & Karniadakis, 2018), we create three (c, d) pairs for each of the RGB channel for +the support of colored videos. The E-R method is evaluated using the published pretrained models. +We further compare against an ablated version of our proposed method, termed “UNet”, which es- +sentially replaces the Lagrangian components of the system with a UNet architecture (Ronneberger +et al., 2015), a classic method for conducting field-to-field mapping. The UNet baseline takes two +images It and It+1 and predicts a velocity field ut+1 to predict It+2 using the same Eulerian in- +tegrator as our method. For future prediction, there do not exist previous methods that perform in +the same setting, so we extend the inference methods in a few ways to support future prediction +in a logical and straightforward manner. First, since HFM offers a query function parameterized +by t, we test its future prediction behavior by simply extrapolating with t > tE; this is referred +to as “HFM extp.”. Since both Raissi & Karniadakis (2018) and Zhang et al. (2022) uncovers the +time-varying velocity field, we use a UNet to learn the evolution from ut to ut+1, and use this ve- +locity update mechanism to perform future prediction, the two baselines thus obtained are referred +to as “HFM+UNet” and “ER+UNet” respectively. The ablated version “UNet” does support future +prediction intrinsically. +5.1 +SYNTHETIC VIDEO +The synthetic video for vortical flow is generated using the Discrete Vortex Method with a first-order +Gaussian mollifying kernel M(δ). The high-fidelity BFECC advection scheme with Runge-Kutta-3 +time integration is deployed. The simulation advects a background grid of 256 × 256, with a time +step dt = 0.01 to create 300 simulation videos. Only the first 100 frames will be seen to train all +methods, and future predictions are tested and examined on the latter 200 frames. +Motion Inference. The results for the uncovering of hidden dynamic variables are illustrated in +Figure 5 and Figure 6. Shown in Figure 5 are the velocities uncovered by all 4 methods against +the ground truth, at frame 55 of a synthetic video with 100 frames. The velocity is visualized in the +form of colors (top row) as well as streamlines (middle row), while the velocity residue, measured in +end-point error(EPE), is depicted in the bottom row. It can be seen that HFM, UNet, and our method +achieve agreeing results, and matches the ground truth values to high accuracy. On the bottom row, +it can be seen that while both HFM and UNet provide sensible results, our method generates the +inference velocity that best matches the unseen ground truth. +The inference results over the full 100 frames at the top of Figure 6. We evaluate the velocity with +four metrics: the average end-point error (AEPE), average angular error (AAE), vorticity RMSE and +compressibility RMSE. From all 4 metrics, it can be seen that our method outperforms the baselines +consistently. The time-averaged data for all four metrics are shown on the left of Table 1, which +deems our method favorable for all metrics used. +Future Prediction. In Figure 7, we visually compare the future prediction results from frame 100 to +frame 299 done using our method and the 4 benchmarks, against the ground truth. It can be observed +7 + +Velocity AEPE over time +21 +(uaua)zBo) +2 +23 +24 +E-R + UNet +24 +HFM + UNet +HFM extrap. +2- +UNet +125 +150 +175 +20D +225 +250 +275 +30D +frameCompressibility eor over time +2 +2 +logz(error) +2' +E-R + UNet +HFM + UNet +HFMextrap. +24 +UNet +125 +150 +175 +240 +225 +250 +275 +310 +frameImage RMSE emor over time +2-2 +23 +24 +21 +2- +E-R + UNet +HFM + UNet +HFM extrap. +UNet +210 +140 +125 +150 +175 +20D +225 +250 +275 +30D +frameVelocity AEPE over time +211 +2-2 +(aua) +2-3 +logz1 +21 +E-R +2 +HFM +UNet +2-? +Ours +0 +24 +140 +rameVelocity AAE aver time +21 +20 +2-1 +(error) +2 +213 +21+ +E-R +HFM +21 +UNet +s.ino +21 +41 +81 +140 +rameVorticity errar over time +23 +22 +21 +E-R +HFM +UNet +Ours +0 +24 +4I +64 +140 +fameCompressibility eror over time +21 +21 +2-1 +27 +2-3 +E-R +HFM +24 +UNet +ours +区 +140 +frameE-R + UNet +HFM + UNet +HFM extrap. +UNet +OursPublished as a conference paper at ICLR 2023 +Ours +UNet +Ground Truth +HFM Extrap. +HFM + UNet +ER + UNet +t = 1.00 +t = 2.32 +t = 1.66 +t = 2.98 +Figure 7: Future predicting capacities of our method compared to benchmarks. Our method accurately predicts +the unseen, future sequence that’s twice as long as the seen sequence. +Time-averaged Inference Errors +Time-averaged Prediction Errors +AEPE +AAE +Vort. +Div. +VGG +RMSE +AEPE +AAE +Vort. +Div. +E-R +0.505 +1.393 +8.470 +2.319 ++UNet +4.346 +0.205 +0.631 +1.424 +12.84 +6.580 +HFM +0.100 +0.212 +3.949 +0.202 ++UNet +4.258 +0.205 +0.720 +1.062 +36.73 +10.41 +Extp. +4.080 +0.285 +0.541 +1.464 +7.761 +4.315 +UNet +0.048 +0.100 +1.799 +1.145 +4.530 +0.211 +0.424 +1.159 +7.334 +3.017 +Ours +0.020 +0.041 +0.976 +0.053 +2.010 +0.080 +0.048 +0.096 +1.621 +0.043 +Table 1: Error analysis of benchmark testing on a synthetic dataset. +that the sequence generated by our method best matches with the ground truth video, capturing the +vortical flow structures, while the other baselines either quickly diffuse or generates unnatural, hard- +edged patterns. Numerical analysis confirms these visual observations. We compare the unrolled +200 frames both in terms of velocity and visual similarity. The velocity analysis inherits the same +4 metrics, and the visual similarity is simultaneously gauged using the pixel-level RMSE and the +VGG perceptual loss (Johnson et al.). The time-averaged results of all 6 metrics are documented in +the right of Table 1, and 4 are plotted in Figure 6. It can be concluded that our method outperforms +the baselines. +5.2 +REAL VIDEO +A similar numerical analysis is carried out on a real video published on YouTube, as shown in +Figure 8. The video has 150 frames: the first 100 frames will be used for training, while the latter +50 will be used for testing. Since the ground truth velocities for the real video are nonexistent, we +will only analyze the future predicting performance. For all methods, we perform future prediction +for 150 frames, among these, the first 50 frames can be compared with the testing videos, and the +latter 100 frames will be compared qualitatively. We note that, since only part of the video is fluid +(within the circular rim), we pre-generate a signed distance field for all methods, so that only the +fluid regions are considered, and the same no-slip boundary condition will be placed for all unroll +methods (except for HFM extp. which requires no advection). +The numerical analysis for the first 50 predicted frames are documented and plotted in Table 2 and +Figure 9. We compare all methods against the baseline using the VGG perceptual loss for visual +loss, and compare the velocity divergence, which should be close to the theoretical value of 0. It +can be seen that in all 3 metrics our method prevails. For prediction results that surpass the duration +8 + +SPublished as a conference paper at ICLR 2023 +Ours +UNet +Ground Truth +HFM Extrap. +HFM + UNet +ER + UNet +t = 1.00 +t = 1.98 +t = 1.49 +t = 2.47 +Figure 8: Future predicting capacities of our method compared to benchmarks on a real video sequence. Our +method generates a predicted sequence that best matches the input video within its duration and remains visually +plausible way beyond its duration. +Figure 9: Plots corresponding to Figure 8. +VGG +(avg.) +VGG +(final) +Div. +(avg) +E-R +2.095 +2.205 +2.046 +HFM+UNet +2.151 +2.231 +0.940 +HFM Extp. +2.980 +3.271 +1.922 +UNet +2.111 +2.088 +1.447 +Ours +2.093 +2.045 +0.318 +Table 2: Data corresponding to Figure 8. +of the real video, qualitative observations can be made in that our method preserves the vortical +structure, generating smooth visualizations over the entire horizon, while other methods end up +yielding glitchy patterns. +We perform additional quantitative benchmark testings in Appendix B against a differentiable grid- +based simulator on real and synthetic videos; and in Appendix C against 4 baselines on another +synthetic video featuring different visual and dynamics distributions. +6 +CONCLUSION & LIMITATIONS +In this work, we propose a novel data-driven system to perform fluid hidden dynamics inference and +future prediction from single RGB videos, leveraging a novel, vortex latent space. The success of +our method in synthetic and real data, both qualitatively and quantitatively, suggests the potential for +embedding Lagrangian structures for fluid learning. Our method has several limitations. First, our +vortex model is currently limited to 2D inviscid flows. Extending to 3D, viscous flow is an exciting +direction, which can be enabled by allowing vortex strengths and sizes to evolve in time (Mimeau & +Mortazavi, 2021). Secondly, our vortex evolution did not take into account the boundary conditions +in a physically-based manner, hence it cannot accurately predict flow details around a solid bound- +ary. Incorporating learning-based boundary modeling may be an interesting exploration. Thirdly, +scaling our method to handle turbulence with multi-scale vortices remains to be explored. We con- +sider two additional directions for future work. 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Learning to estimate and +refine fluid motion with physical dynamics. arXiv preprint arXiv:2206.10480, 2022. +12 + +Published as a conference paper at ICLR 2023 +A +IMPLEMENTATION DETAILS +In this section, we describe the implementation details of our proposed method. +Integrators. As described above and illustrated in Figure 2, our system embeds two differentiable +integrators in the loop. The Eulerian integrator is implemented using the Back and Forth Error +Compensation and Correction (BFECC) (Kim et al., 2005) method for backtracking, and the 3rd +order Runge-Kutta method for time-stepping. The Lagrangian integrator is implemented using the +Forward Euler method. +Network N1. +The network N1 adopts a series of 3 residue blocks with increasing width +[64, 128, 256], whose architecture is similar to He et al. (2016) but with convolution layers replaced +by linear layers with sine activation functions. The frequency factor ω0 discussed in Sitzmann et al. +(2020) is set to 1. +Network N2. The network N2(r, δ) is structured as follows. First, the input r is scaled by the input δ +as ¯r = r · η +δ , where η is a hyperparameter selected to be 0.1, which corresponds to the characteristic +length scale of vortices. Then, ¯r is transformed into ˆr as ˆr = ¯r0.3, which is a reparametrization that +stretches the value ¯r near 0, which exploits the insight that velocity varies more aggressively when +near a vortex. The value ˆr is then fed through 4 residue blocks (same as N1) but with a fixed width +of 40. The output from these residue blocks would be scaled by multiplying with η +δ . The scaled +output is N2(r, δ), which is then used for velocity computation according to 7. +Training Details. Both the image loss and the velocity alignment loss are MSE, and the velocity +alignment loss has an extra scaling factor of 0.001. We use Adam optimizer with β1 = 0.9, β2 = +0.999 and learning rate = [0.001, 0.0003, 0.005, 0.005] for N1, N2, Ω and ∆ respectively. We use +a step learning rate scheduler and set the learning rate to decay to 0.1 of the original value at 20% +of the total training iterations. We use a batch size of 4, so for each iteration, 4 starting times are +picked uniformly randomly among [0, 1, . . . , tE] for evaluation. The sliding-window size m is set +to 3. +Pretraining N1. We pretrain N1 for 10000 iterations with 2 objectives: (1) for all t ∈ [0, tE], +N1(t) = [(x1)t, . . . , (xn)t] coincide with the centers of a 4×4 grid, (2) for all t ∈ [0, tE], dN1 +dt = 0, +so that these particles are initially stationary. We use MSE for the positional and velocity losses, and +the other training specifications are the same as described above. +Computational Performance. Running on a laptop with Nvidia RTX 3070 Ti and Intel Core i7- +12700H, our model takes around 0.4s per training iteration, and around 30000 iterations to converge +(for a 256 × 256 video with 100 frames). For inference, each advance step costs around 0.035s. +B +COMPARISON WITH DIFFERENTIABLE FLUID SIMULATION +We compare our method qualitatively and quantitatively against a standard, grid-based differentiable +fluid simulator (referred to as Diff-Sim) on both synthetic and real videos. This baseline method is +a differentiable implementation of the method proposed by Fedkiw et al. (2001), which is a clas- +sic, widely-adopted numerical method for simulating vortical fluids. The method is designed to +solve the 2D Euler equations for inviscid fluid, hence it can in theory recreate the inviscid fluid phe- +nomena represented by any video if provided with the appropriate initial conditions and simulation +parameters. +Therefore, in this experiment, we make use of its differentiable nature to optimize (1) the initial grid +velocities (a 256 × 256 × 2 tensor), and (2) the vorticity confinement strength, which is a scalar +value, with the objective of minimizing the discrepancy between the simulated results and the input +video. The loss computation between the simulated image sequence and the ground truth is the +same as in our method. We note that the idea of optimizing initial conditions using differentiable +fluid simulation to fit specific still frames has been demonstrated in Hu et al. (2020). However, their +task is notably simpler than ours, since they only require the simulated image to match a target frame +at the end of the simulation, while our goal is to match the underlying motion of the entire video, +and dynamically unroll into the future. +Comparison on a synthetic video. We start by comparing both methods on a synthetic video, which +yields a visual comparison which can be found in Figure 10. We observe that our method suc- +cessfully learns the dynamics represented in the video: the generated video and velocities closely +13 + +Published as a conference paper at ICLR 2023 +t = 0.00 +t = 3.00 +Observed Past +Predicted Future +Differentiable +Fluid +Simulation +Ours +Ground +Truth +t = 0.00 +t = 3.00 +Inferred Past Velocity +Predicted Future Velocity +Differentiable +Fluid +Simulation +Ours +Ground +Truth +Figure 10: Visual comparison between a differentiable grid-based simulator and ours on a synthetic video. The +upper half displays the simulated image, while the lower half displays the underlying velocity, whose color +wheel is depicted. On the bottom row of each half is the ground truth sequence, which has 300 frames. The +first 100 frames are available for both methods to learn from, while the latter 200 frames are unseen during +training. +Synthetic Test Errors +Real Test Errors +Figure 11: Error plots of the comparison between Diff-Sim and ours on a synthetic dataset. +resembles the groundtruth even in the unseen frames. Diff-Sim, on the other hand, shows a weak +resemblance with the ground truth for the seen frames, yet fails to capture the individual eddies in +the video. Consequently, it fails to predict the future dynamics. Diff-Sim’s lack of correspondence +to the dynamics of the ground truth is also made evident in Figure 13. The result clearly suggests +that our method has better learned the dynamics evolution. This performance discrepancy is nu- +merically supported by the errors shown on the left panel of Table 3 and plotted on the left panel +of Figure 11, both showing that our method yields reduced image-level and velocity-level errors +compared to Diff-Sim. +14 + +T +TDiff-Sim +OursDiff-Sim +OursDiff-Sim +OursDiff-Sim +OursPublished as a conference paper at ICLR 2023 +t = 0.00 +t = 1.55 +Observed Past +Predicted Future +Differentiable +Fluid +Simulation +Ours +Ground Truth +Figure 12: Comparison between Diff-Sim and ours on a real video. +Synthetic Video Errors +Real Video Errors +AEPE +AAE +Vort. +RMSE +VGG +VGG (avg.) +VGG (final) +Diff-Sim (Grid) +0.469 +0.953 +24.43 +0.157 +26043 +15076 +18792 +Ours +0.041 +0.081 +1.482 +0.055 +2171.4 +7846.2 +11081 +Table 3: Error comparison between Diff-Sim and ours on synthetic and real videos. +Comparison on a real video. We then use the same experimental set up to perform learning on a +real video, which is depicted in Figure 12. We observe that on the real video, the same behavioral +patterns for both systems seen on the synthetic one have carried over. For the results generated by +Diff-Sim (top row), we can see that the overall, large-scale motion (the large eddy moving towards +bottom-left) is correctly learned. Nevertheless, all the smaller vortices are gone and the entire image +quickly diffuses as the simulation goes on. This can be attributed to the numerical diffusion issues +innate to grid-based simulations, as well as the lack of embedded fluid structures. In comparison, our +method well-preserves the vortical movements due to its built-in structure, and produces a plausible +future rollout extending beyond the duration of the original video. Although both systems are unable +to perfectly model the exact dynamics that governs this real-world video (due to unmodelled factors +such as fluid viscosity, air friction, and 3-dimensional forces), our proposed method does a better +job in retaining the vortical patterns and energetic flows thanks to its vorticity-based formulation and +the Lagrangian-Eulerian design, as can be observed in the middle row. The advantage of our system +over Diff-Sim on the real video is numerically supported, as can be found on the right panel of +Table 3 and Figure 11. Since we do not have the ground-truth velocities for real videos, we compare +the VGG perceptual loss between the simulated sequence of both methods and the real video, which +demonstrates quantitatively that our generated results better resembles the input video than that of +its counterpart. +C +ADDITIONAL BENCHMARK TESTING +As depicted in Figure 14, to further illustrate our method’s advantage and generalizability, we have +conducted an additional set of numerical tests on another synthetic video, and compare our method’s +performance with 4 benchmarks in terms of both velocity inference quality and future prediction +quality. The ground truth data is generated using a significantly different background image (sharp +color tiles vs. smooth color gradients), and a different velocity kernel (second-order Gaussian kernel +vs. first-order Gaussian kernel). The experimental setup is otherwise the same as the one presented +in the main text (in Figure 7), with the same compared benchmarks. +15 + +N/APublished as a conference paper at ICLR 2023 +Color Plot +Streamline Plot +Quiver Plot +Residue +Ground +Truth +Ours +Differentiable +Fluid +Simulation +Figure 13: Comparing the quality of velocity inference of our method and Diff-Sim. We show the predicted +velocity of frame 200 in three different forms (color, streamline and quiver plots) in additional to the residue +(end-point error) compared to the ground truth. +Ours +UNet +Ground Truth +HFM Extrap. +HFM + UNet +ER + UNet +t = 0.60 +t = 1.4 +t = 1.0 +t = 1.8 +Figure 14: Future prediction results: our method compared to baselines on a synthetic video. +The comparison of the velocity inference quality can be found in Figure 16 and the top panel +Figure 15. Figure 16 depicts the uncoverred velocities of frame 40 (among the 60 input frames) by +all 4 methods compared to the ground truth. The top row depicts the respective velocities in colors +with the color wheel supplied; the middle row depicts the velocities in streamlines; and the bottom +row depicts the velocity residue compared to the ground truth, measured in end-point error (EPE). +As with the results in 5, we can see that HFM, UNet and our method can all infer the underlying +velocity field with high precision, whereas E-R yields a visibly noisier approximation. As seen on +the bottom row, the inference performance between UNet and Ours are very close, but our method +takes the slight edge with an average error (AEPE) of 0.0143 as compared to the error of 0.0215 +16 + +T +TPublished as a conference paper at ICLR 2023 +Future +Prediction +Errors +Motion +Inference +Errors +Figure 15: Error analysis on a synthetic dataset. The top row plots the inference errors of velocity, vorticity, +and compressibility. The bottom row plots the future prediction errors, which consider both the dynamic error +in the velocity and the perceptual error of the generated image sequence. +Velocity +Stream- +lines +Velocity +Residue +Ground Truth +HFM +E-R +UNet +Ours +Figure 16: Comparison to baselines on velocity inference on a synthetic video. Our method recovers the +underlying velocity field with higher accuracy. +yielded by UNet. The advantage of our method is not unique to the specific frame selected. As +plotted on the top row of Figure 15, it can be seen that our method (red) consistently yields the +lowest velocity-inference error throughout the 60 input frames, in terms of the average end-point +error (AEPE), average angular error (AAE), vorticity RMSE and compressibility RMSE. The time- +averaged errors of these metrics are documented in Table 4, which again shows that our method +yields the best estimations. +Future prediction. We also compare the future prediction results with the baselines. In Figure 14, +we show a visual comparison of all 5 methods against the ground truth. It highlights the close re- +semblance of our generated sequence with the ground truth, which is twice as long as the sequence +used for training. Compared to the baselines, our method yields the best match to the ground truth +video, capturing the accurate vortical flow structures. HFM+UNet, ER+UNet, and UNet can gen- +erate reasonable future prediction up to t = 1.0 (for 40 frames). For t > 1.0, these sequences start +to distort in different ways, due to their lack of physical structures and constraints. The direct ex- +trapolation of HFM yields the least plausible results, quickly degrading to noise. We compare these +sequences quantitatively using the 4 velocity-based metrics, along with 2 image-based metrics: the +pixel-level RMSE and the VGG perceptual loss. Four of these time-dependent errors are plotted +in the bottom row of Figure 15, with their time-averaged counterparts documented on the right of +Table 4. In summary, we observe that our method outperforms the existing baselines for this video +both quantitatively and qualitatively. +17 + +E-R + UNet +HFM + UNet +HFM extrap. +UNet +OursE-R +HFM +UNet +SInoE-R +HFM +UNet +OursE-R +HFM +UNet +OursE-R +HFM +UNet +oursE-R + UNet +HFM + UNet +HFM extrap. +UNet +OursE-R + UNet +HFM + UNet +HFM extrap. +UNet +OursE-R + UNet +HFM + UNet +HFM extrap. +UNet +OursT +TPublished as a conference paper at ICLR 2023 +Time-averaged Inference Errors +Time-averaged Prediction Errors +AEPE +AAE +Vort. +Div. +VGG +RMSE +AEPE +AAE +Vort. +Div. +E-R +0.229 +0.805 +4.380 +1.750 ++UNet +8138.8 +0.178 +0.272 +1.115 +4.694 +2.504 +HFM +0.038 +0.097 +3.001 +0.533 ++UNet +9389.5 +0.146 +0.201 +0.715 +10.58 +3.199 +Extp. +40967 +0.166 +0.293 +1.221 +4.862 +3.152 +UNet +0.026 +0.101 +1.013 +0.895 +7721.3 +0.170 +0.330 +1.141 +5.462 +1.496 +Ours +0.015 +0.046 +0.480 +0.015 +2045.0 +0.097 +0.057 +0.173 +1.547 +0.013 +Table 4: Time-averaged errors of our method compared to various baselines on a synthetic video. +D +NUMBER OF VORTEX PARTICLES +In our proposed method, we use n vortex particles to learn the fluid dynamics. However, we note +that vortices are not intrinsic to fluid phenomena, but are rather imposed constructs to allow fluids +to be better understood conceptually and modeled numerically. Thus, the number of vortices n is +fundamentally a hyperparameter that does not admit a uniquely-correct value. +With this in mind, we let ˆn denote the minimum number of particles that can be used to model +the fluid system to an acceptable accuracy. This natural number ˆn surely exists since it has been +proven that vortex particle methods converges to the exact solution of 2D Euler’s Equations (Beale +& Majda, 1985; Hald, 1979). We are mostly concerned with the cases where n > ˆn, which means +the deployed degrees of freedom (DoFs) are higher than that of the fluid system. In the following, +we show that our method can spontaneously prune the redundant vortices and thus it is robust to +a reasonable range of n > ˆn. In Figure 17, we show the results of learning the same underlying +motion with 4, 9, and 64 vortex particles. In Figure 18, we show the underlying velocity and vorticity +fields. +Spontaneous pruning of redundant DoFs. As shown on the top row of Figure 17, the ground truth +is generated with 4 vortices, so it is safe to assume that ˆn = 4. Learning with 4 vortices (as shown +on the second row) represents the case where n = ˆn. Comparing the first row with the second +row, we can see that there is a one-to-one correspondence between the ground-truth vortices and the +learned vortices, with each learned vortex assuming the role of one individual ground-truth vortex +(obtaining the same vorticity and initial position). +When we have 9 vortices (third row), there are more vortex particles than ground-truth. In this +case, two interesting phenomena occur to spontaneously prune these redundant particles: degenera- +tion and clustering. First, some particles degenerate themselves by reducing its strength to 0 or by +moving farther away from the domain. We can observe both mechanisms taking place on the two +lingering particles on the top part of the third row. They both have low strength (evident from their +turquoise color) and are peripheral to the domain. Secondly, particles would aggregate to simulate a +single particle with greater strength. Since the velocity computation is done with distance-weighted +summation (as in Equation 7), if multiple particles coincide at the same location, they effectively +act as one single particle with their vorticities summed together. This phenomenon can be observed +on the lower half of the images in the second and third row. Both of these mechanisms enable the +system to spontaneously prune redundant vortices. In the last row, we show that our method is robust +to even 64 vortices. +Figure 19 helps to illustrate this spontaneous pruning mechanism by observing different snapshots +of the training process. Shown on the left are the vortex particles’ behaviors soon after the training +has begun. It is particularly noticeable that, on the bottom row, the 64 particles are scattered in the +fluid domain, and the learned result appears quite different from the ground truth. Moving from +left to right, these particles become more and more clustered on the flow regions, with much fewer +particles wandering around; and the end result can approximate the ground truth much better. +Finally, we note that n < ˆn is still challenging to resolve as the system is over-constrained. Never- +theless, we empirically find that n = 16 is sufficient for all the real and synthetic videos we consider +in our experiments. +18 + +Published as a conference paper at ICLR 2023 +Ground +Truth +(4 Vortices) +Learned +with +4 Vortices +Learned +with +9 Vortices +Learned +with +64 Vortices +Observed Past +Predicted Future +Figure 17: The same underlying motion as learned with different numbers of vortex particles. The ground truth +sequence has 100 frames, the first 30 frames are provided during training, and the latter 70 frames are predicted. +Vorticity Field +at Frame 80 +Velocity Field +at Frame 80 +Ground Truth +(4 Vortices) +Learned with +4 Vortices +Learned with +9 Vortices +Learned with +64 Vortices +Figure 18: Different number of vortices can learn similar underlying dynamics. +E +ABLATION: LEARNABLE VELOCITY KERNEL +In traditional vortex simulation applications in Computer Graphics or Computational Fluid Dynam- +ics, the velocity kernel is hand-selected (typically from Gaussian kernels of different orders) with +a uniform support radius (size). Such approaches are designed to perform forward simulation, yet +they are limited when used for backward inference tasks, i.e. to reconstruct input videos. In our +method, we address this issue by learning neural kernels with learnable sizes. By leveraging data- +driven techniques, we can reconstruct and predict fluid flows that are not only visually pleasing, but +also resemble the particular dynamics traits depicted in the input video. +19 + +120 +1.00 +0.75 +100 +.0.50 +B0 +0.25 +0.00 +60 +0.25 +40 +-0.50 +20 +-0.75 +1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +0.00 +60 +0.25 +40 +-0.50 +20 +0.75 +1.00 +o. +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +0 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100- +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +0 +20 +40 +60 +100 +120120 +1.00 +0.75 +100 +0.50 +80 - +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +0.00 +0.25 +40 +~0.50 +20 +0.75 +1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +20 +Ot +60 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +0.00 +60 +0.25 +40 +-0.50 +20 +0.75 +-1.00 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +:0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +B0 +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +. +0 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +BO +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +DZ +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +~0.50 +20 +0.75 +1.00 +20 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +-0.75 +1.00 +O: +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100- +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +0: +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100: +0.50 +BO +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +-0.75 +1.00 +20 +60 +80 +100 +120T +T120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +-0.50 +20 +0.75 +1.00 +0- +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100- +0.50 +80- +0.25 +.0.00 +60- +0.25 +40 +0.50 +20 +0.75 +1.00 +04 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +B0- +0.25 +0.00 +60 +0.25 +40 +20 +~0.75 +1.00 +to +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80- +0.25 +-0.00 +60 +0.25 +40 +0.50 +20 +0.75 +1.00 +20 +40 +60 +80 +100 +120Published as a conference paper at ICLR 2023 +Learning +with +4 Vortices +Learning +with +9 Vortices +Learning +with +64 Vortices +Ground Truth +(4 Vortices) +Training Starts +Training Ends +Figure 19: The training evolution using different number of particles. +t = 0.00 +t = 2.15 +Observed Past +Predicted Future +With +Learnable +Kernel +Ground +Truth +Without +Learnable +Kernel +Figure 20: Ablation study: reconstruction and prediction on a real video with learnable velocity kernels (our +full method) and without learnable velocity kernels. +In Figure 20, we show an ablation study on learning velocity kernel. We reconstruct and predict a +real-world video using our method and an ablated version in which the learnable kernel is replaced +with a hard-coded first-order Gaussian kernel with uniform size. The ground truth, shown on the +bottom row, has 126 frames revealed (for training) and 62 frames hidden (for testing). In the middle +row, we learn to fit the video with our learnable kernel enabled. In the top row, we learn to do the +same with the learnable kernel disabled. It can be observed that the middle row well-captures the +characteristic smoothness of the flow, and simulates a image sequence that resembles the ground +truth. The ablated version (top row) can also learn the correct overall motion (clockwise rotation), +but it induces various smaller eddies and wrinkles uncharacteristic of the input video. +Extending to unseen frames, our method can continue to retain the overall structure of the eddies, +while the ablated version (without learnable kernel) drives the pattern to disintegrate and evolve +20 + +120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +-0.50 +20 +0.75 +-1.00 +0 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +0.50 +20 +-0.75 +1.00 +O: +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +-0.75 +-1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +-0.75 +-1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +-0.75 +-1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +-0.75 +-1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +0.75 +-1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +0.75 +-1.00 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +09 +0.25 +40 +-0.50 +20 +0.75 +-1.00 +0 +0 +20 +40 +60 +80 +100 +120120 +1.00 +0.75 +100 +0.50 +80 +0.25 +0.00 +60 +0.25 +40 +-0.50 +20 +0.75 +-1.00 +0 +0 +20 +40 +60 +80 +100 +120N/ASPublished as a conference paper at ICLR 2023 +Figure 21: Time-dependent losses corresponding to Figure 20. +VGG +(avg.) +VGG +(final) +RMSE +(avg) +Ours +w/o +learnable +kernels +9698.7 +11240 +0.183 +Ours +7365.6 +10027 +0.180 +Table 5: Data corresponding to Figure 21. +t = 0.00 +t = 1.80 +Without +Trajectory +Learning +Ours +Ground +Truth +t = 0.00 +t = 1.80 +Without +Trajectory +Learning +Ours +Ground +Truth +Figure 22: We compare our method against its ablated version which does not feature trajectory learning. On +the top depicts the simulated images, and on the bottom depicts the simulated velocities. The results by both +approaches are compared to the ground truth. +into various folds and wrinkles that do not resemble the dynamics characteristics in the real video. +We further show quantitative results plotted in Figure 21 and documented in Table 5. In summary, +learning velocity kernels allows for better reconstruction and prediction of fluid dynamics in the +input video. +F +ABLATION: TRAJECTORY LEARNING +In our approach, we learn the full trajectory of vortex particles for the input video. An alternative +to “learning initial condition through trajectory” is to learn the initial condition directly. However, +we find that the former option is more computationally tractable and effective, if we want to fully +exploit the input video. To see this, suppose we have 100 training frames in the video, and the +goal is to infer the initial condition at frame-1. If we directly optimize the initial condition using +the last frame, we need to simulate from frame-1 all the way to frame-100, compute the loss and +21 + +Without Learnable +With LeanableWithout Learnable +With LearmablePublished as a conference paper at ICLR 2023 +Velocity Errors +Image Errors +AEPE +AAE +Vort. +Div. +RMSE +VGG +Ours (Ablated) +0.257 +0.753 +4.689 +0.054 +0.170 +6759.4 +Ours +0.043 +0.131 +1.180 +0.014 +0.081 +1805.9 +Table 6: Time-averaged velocity-level and image-level errors by our method and its ablated version. +Figure 23: Time-dependent velocity-level and image-level errors by our method and its ablated version. +backpropagate. Unrolling such a long sequence for each training iteration (1) takes a long time, (2) +leads to noisy gradients, and (3) is practically infeasible due to memory constraints. On the other +hand, learning through the whole trajectory allows us to address these challenges by using a smaller +sliding window in time (e.g., simulating only 3 frames at a time) and aggregating the dynamics +information throughout the whole video. In Figure 22, we show a comparison of both methods in +action. +In Figure 22 top, we show the reconstruction and prediction results for both our full method and +an ablation version where we directly learn the initial condition. Note that the ablation version +can only unroll the first 13 frames (and thus it is learned using only the first 13 frames) due to the +same memory constraint. In Figure 22 bottom, we show the velocity corresponding to the top. We +observe that our method and its ablated version can approximate the ground truth reasonably well +at the beginning of simulation (the left three images). However, the ablated version starts to distort +significantly in terms of both the advected image and the underlying velocity. This observation is +in correspondence with the numerical evidence, as plotted in Figure 23 and documented in Table 6, +which shows that our full method consistently outperforms its ablated counterpart among all metrics. +We conjecture that the underlying reasons to this performance discrepancy is threefold: first, the +ablated version can only learn from the beginning of the fluid dynamics which provides limited +information to correctly infer the initial conditions. Secondly, only learning the initial condition +is more susceptible to accumulated error than our full method. Thirdly, using a limited number of +frames makes it harder to learn an appropriate velocity kernel. In summary, our observations suggest +that learning the full trajectory is computationally more tractable and effective compared to learning +the initial conditions only. +22 + +W/O TrajectoryW/O Trajectory +OursW/O Trajectory +Oursw/O Trajectory +OursW/O Trajectory +sInoW/O Trajectory +Ours \ No newline at end of file diff --git a/cdFJT4oBgHgl3EQfRCzr/content/tmp_files/load_file.txt b/cdFJT4oBgHgl3EQfRCzr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f55d33af82c3427020355972753ec931f2e8b325 --- /dev/null +++ b/cdFJT4oBgHgl3EQfRCzr/content/tmp_files/load_file.txt @@ -0,0 +1,1346 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf,len=1345 +page_content='Published as a conference paper at ICLR 2023 LEARNING VORTEX DYNAMICS FOR FLUID INFERENCE AND PREDICTION Yitong Deng Dartmouth College Stanford University Hong-Xing Yu Stanford University Jiajun Wu Stanford University Bo Zhu Dartmouth College ABSTRACT We propose a novel machine learning method based on differentiable vortex par- ticles to infer and predict fluid dynamics from a single video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The key design of our system is a particle-based latent space to encapsulate the hidden, Lagrangian vortical evolution underpinning the observable, Eulerian flow phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We de- vise a novel differentiable vortex particle system in conjunction with their learn- able, vortex-to-velocity dynamics mapping to effectively capture and represent the complex flow features in a reduced space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We further design an end-to-end training pipeline to directly learn and synthesize simulators from data, that can reliably deliver future video rollouts based on limited observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The value of our method is twofold: first, our learned simulator enables the inference of hidden physics quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' velocity field) purely from visual observation, to be used for motion analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' secondly, it also supports future prediction, constructing the input video’s sequel along with its future dynamics evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We demonstrate our method’s efficacy by comparing quantitatively and qualitatively with a range of existing methods on both synthetic and real-world videos, displaying improved data correspondence, visual plausibility, and physical integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='1 1 INTRODUCTION As small as thin soap films, and as large as atmospheric eddies observable from outer space, fluid systems can exhibit intricate dynamic features on different mediums and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' However, despite the inspiring recent progress, to effectively represent these flow features, identify the underlying dynamics system, and predict the future evolution, remains an open problem for scientific machine learning, due to the noisy data, imperfect modeling, and unavailable, hidden physics quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Here, we identify three fundamental challenges that currently hinder the success of such endeavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, flow features are difficult to represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Traditional methods learn the fluid dynamics by storing velocity fields either using regularly-spaced grids or smooth neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' These approaches have demonstrated promising results for fluid phenomena that are relatively damped and laminar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2022), but for fluid systems that can exhibit turbulent features on varying scales, these methods fall short due to the problem’s curse of dimensionality (high-resolution space and time), local non-smoothness, and hidden constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As a result, more compact and structured representation spaces and data structures are called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Secondly, hidden flow dynamics is hard to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='. Fluid systems as prescribed by the Navier-Stokes equations tightly couple multiple physical quantities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' velocity, pressure, and density), and yet, only the density information can be accessibly measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Due to the system’s complexity, ambigu- ity, and non-linearity, directly learning the underlying dynamics from the observable density space is infeasible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' and successful learning is usually contingent on velocity or pressure supervision, a requirement that distances these methods from deployment in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Exciting recent progress has been made in hidden dynamics inference by PDE-based frameworks such as (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2020), which uncover the underlying physics variables solely from density observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' However, this type of methods encounter the third fundamental challenge, which is 1We invite the readers to view the video results via an anonymized link: learning-vortex-dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='io 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='11494v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='LG] 27 Jan 2023 Published as a conference paper at ICLR 2023 Observed real video Synthesized future prediction Figure 1: Our goal is to learn vortex dynamics for fluid prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The 3 frames on the left are observed from a real video recording a soap film on a circular metal rim, where the red ink is spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The 3 frames on the right are future prediction results produced by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' that performing future prediction is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As we will demonstrate, although strong results are ob- tained for interpolating inside the observation window provided by the training data, these methods render themselves unsuited for extrapolating into the future, profoundly limiting their usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In this paper, we propose a novel fluid learning method to tackle the three aforementioned challenges in a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In particular, harnessing the physics insights developed for the vortex meth- ods in the computational fluid dynamics (CFD) literature, we design a novel, data-driven Lagrangian vortex system to serve as a compact and structured latent representation of the flow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We learn the complex dynamic system in the high-dimensional image space by learning a surrogate, low-dimensional model on the latent vortex space, and use a physics-based, learnable mechanism: the vortex-to-velocity dynamics module, to decode the latent space dynamics back to the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Leveraging this advantageous representation, we design an end-to-end training pipeline that learns from a single video containing only density information, to jointly perform accurate inference of hidden physics quantities and robust long-term future predictions, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' To examine the efficacy of our method, we compare our method’s performance on both motion in- ference and future prediction against various state-of-the-art methods along with their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We conduct benchmark testing on synthetic videos generated using high-order numeric simulation schemes as well as real-world videos in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Evaluation is carried out both quantitatively through exhaustive numerical analysis, and qualitatively by generating a range of realistic visual effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We compare the uncovered velocities both in terms of data correspondence and physical integrity, and the predicted visual results in terms of both pixel-level and perceptual proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Re- sults indicate that our proposed method provides enhanced abilities on both fronts, inferring hidden quantities at higher accuracy, and predicting future evolution with higher plausibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In summary, the main technical contributions of our framework align with the three challenges we have addressed regarding flow representation, dynamics learning, and simulator synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (1) We devise a novel fluid dynamics representation with differentiable vortex particles, to drastically reduce the learning problem’s dimensionality on complex flow fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Motivated by the vortex meth- ods in CFD, we establish the vorticity-carrying fluid particles as a new type of learning primitive to transform the existing PDE-constrained optimization problem to a particle ODE trajectory learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2) We design a novel particle-to-field paradigm for learning the Lagrangian vortex dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Instead of learning the interaction among particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2020), our model learns the continuous vortex-to-velocity induction mapping to naturally connect the vor- tex particle dynamics in the latent space and the fluid phenomena captured in the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (3) We develop an end-to-end differentiable pipeline composed of two network models to synthesize data-driven simulators based on single, short RGB videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 2 RELATED WORK Hidden Dynamics Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The problem of inferring dynamical systems based on noisy or incom- plete observations has been addressed using a variety of techniques, including symbolic regression (Bongard & Lipson, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Schmidt & Lipson, 2009), dynamic mode decomposition (Schmid, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Kutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2016), sparse regression (Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Rudy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2017), Gaussian process re- gression (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Raissi & Karniadakis, 2018), and neural networks (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Among these inspiring advancements, the “hid- den fluid mechanics” (HFM) method proposed in Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2020) is particularly noteworthy, as it uncovers the continuous solutions of fluid flow using only images (the transport of smoke or ink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Data-driven Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Recently, growing interests are cast on learning numerical simulators ac- cording to data supervision, which has shown promise to reduce computation time (Ladick`y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2 Published as a conference paper at ICLR 2023 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Wiewel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Pfaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Tomp- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2017), increase simulation realism (Chu & Thuerey, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2018), enable stylized control (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2020), estimate dynamic quantities such as viscosity and energy (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Ummenhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2019), and facilitate the training of control policies (Sanchez-Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Akin to Watters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2017), our system takes im- ages as inputs and performs dynamics simulation on a low-dimensional latent space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' but our method learns purely from the input video and performs future rollout in the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Our method is also related to Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2022), which infers Lagrangian fluid simulation from observed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We propose sparse neural vortices as our representations while they use dense material points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Vortex Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The underlying physical prior incorporated in our machine learning system is rooted in the family of vortex methods that are rigorously derived, analyzed, and tested in the com- putational fluid dynamics (CFD) (Leonard, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Perlman, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Beale & Majda, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Winck- elmans & Leonard, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Mimeau & Mortazavi, 2021) and computer graphics (CG) community (Selle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Park & Kim, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Weißmann & Pinkall, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Brochu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2020) is pioneering for combining the Discrete Vortex Method with neural networks, but its pr oposed method relies on a large set of ground truth velocity sequences, whereas our method learns from single videos without needing the ground truth velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 3 PHYSICAL MODEL We consider the velocity-vorticity form of the Navier–Stokes equations (obtained by taking the curl operator on both sides of the momentum equation, see Cottet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2000) for details): Dω Dt = ∂ω ∂t + u · ∇ω = (ω · ∇)u + ν∇2ω + ∇ × b, (1) u = ∇ × φ, ∇2φ = −ω, (2) where ω denotes the vorticity, u the velocity, b the conservative body force, ν the kinematic viscos- ity, and φ the streamfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' If we ignore the viscosity and stretching terms (inviscid 2D flow), we obtain Dω/Dt = 0, which directly conveys the Largangian conservative nature of vorticity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' a particle’s vorticity will not change during its advection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' If we assume the fluid domain has an open boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' we can further obtain the vorticity-to-velocity induction formula,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' which is derived by solving the the Poisson equation on φ using Green’s method (also known as the Biot-Savart Law in fluid mechanics): u(x) = � K(x − x′)ω(x′)dx′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (3) The kernel K exhibits a type-II singularity at 0 and causes numerical instabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' therefore in CFD practices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' K is replaced by various mollified versions Kδ to improve the simulation accuracy (Beale & Majda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We note that the mollified version Kδ is not unique, and can be customized and tuned in different numerical schemes per human heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Different types and parameters for the mollification bring about significantly different simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Takeaways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The mathematical models above provide two central physical insights guiding the design of our vortex-based learning framework: (1) The Lagrangian conservation of vorticity ω suggests the suitablity of adopting Lagrangian data structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' particles as opposed to grids) to capture the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Since the tracked variable ω remains temporally-invariant for each Lagrangian vortex, the evolution of the continuous flow field is embodied fully by the movement of these vortices, which significantly alleviates the difficulty in learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2) Equation 3 presents an induction mapping from the vorticity ω, a Lagrangian quantity carried by particles, to the velocity u, an Eulerian variable that can be queried continuously at an arbitrary location x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This lends the possibility for the Lagrangian method to be used in conjunction with Eulerian data structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' a grid) for learning from the widely available video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Furthermore, such a mapping can benefit from data-driven learning, as we can replace human heuristics by learning the mollified kernel Kδ (which is shared among all vortex particles) to minimize the discrepancy between the induced and observed flow phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 4 METHOD System Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Following the physics insight conveyed in Section 3, we design a learning system whose workflow is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As shown on the top row,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' our system takes as input a 3 Published as a conference paper at ICLR 2023 Eulerian Integrator Predicting The Future Eulerian Integrator Analyzing The Past Observed RGB Image Space Intermediate Velocity Space Learned Vortex Space Dynamics Module Dynamics Module Dynamics Module Dynamics Module Lagrangian Integrator Trajectory Module Eulerian Integrator Eulerian Integrator Lagrangian Integrator Figure 2: Given an input RGB image sequence (top row),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' we learn the dynamic system of a low-dimensional vortex space (bottom row),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' whose motion is decoded into the motion of the high-dimensional image space to explain the observed phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' single RGB video that captures the vortical flow phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As shown on the bottom row, our method learns and outputs a dynamic simulator — not on the image space itself, but on a latent space consisting of discrete vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Learning the latent dynamics in the vortex space would only be useful and feasible if we can tie it back to the image space, because it is the image space that we want to perform future prediction on, and we have no ground truth values for the vortex particles to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The bridge to tie the vortex space with the image space is derived from Equation 3, which supplies the core insight that there exists a learnable mapping from vortex particles to the continuous velocity field at arbitrary positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This mapping is modelled by our learned dynamics module D, which gives rise to the intermediate velocity space, as shown in the middle row of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='1 DIFFERENTIABLE VORTEX PARTICLES We track a collection V of n vortex particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' V := [V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , Vn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We define each vortex Vi as the 3-tuple (xi, ωi, δi), where x represents the position, ω the vortex strength, and δ the size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The number of particles n is a hyperparameter which we set to 16 for all our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Further discussions and experiments regarding the choice of n can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We also note that, since we are concerned with 2D inviscid incompressible flow, the size δ of a vortex does not change in time due to incompressibility, and the vortex strength ω does not change in time due to Kelvin’s circulation theorem (see Hald (1979) for a thorough discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Learning Particle Trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As shown in Figure 3, we learn a particle trajectory module: a query function T such that Vt = T (t), which predicts the configuration of all the vortices at any time t ∈ [0, tE] where tE represents the end time of the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As described above, predicting Vt boils down to determining two time-invariant components: (1) [ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , ωn], (2) [δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , δn], and one time-varying component: [(x1)t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , (xn)t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For the two time-invariant components, we introduce two trainable n × 1 vectors ∆ and Ω to represent δ and ω respectively, such that [ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , ωn] = sin(Ω) and [δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , δn] = sigmoid(∆) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The vortex size ∆ and strength Ω are optimized to fit the motion depicted by the input RGB video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For the time-varying component, we use a network N1(t) to encode N1(t) = [(x1)t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , (xn)t], and the particle velocities dN1 dt can be extracted using automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We note that learning the full particle trajectory, rather than the initial particle configuration, allows the aggregation of dynamics information throughout the input video for better inference and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We provide further discussion on this design in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Trajectory Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As discussed above, the trajectory T has three learnable components: ∆, Ω and N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We initialize ∆ and Ω as zero vectors, which gives δi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='5 + ϵ and ωi = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Conceptually, these vortices are initialized as large blobs with no vortex strength, which learn to alter their sizes and grow their strengths to better recreate the eddies seen in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The initial positions [(x1)0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , (xn)0] are regularly spaced points to populate the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We initialize 4 Published as a conference paper at ICLR 2023 Figure 3: We encapsulate the motion of a continuous field by the motion of discrete particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The blue trajec- tory is encoded by a neural network N1, corresponding to the input video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' while the red trajectory is unrolled using our learned dynamics module and a numeric integrator, corresponding to the future prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' the 16 particles to lie at grid centers of a 4 × 4 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' To do so, we simply pretrain N1 so that N1(0) evaluates to the grid centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The details regarding pretraining is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Learning the Vorticity-to-Velocity Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The vorticity-to-velocity mapping is performed by our dynamics module, which predicts the velocity u given arbitrary query point x and the collection of vortices V = [(x1, ω1, δ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , (xn, ωn, δn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Following the physics insight conveyed in Section 3, D embodies the integration: u(x) = � Kδ(x − x′)ω(x′)dx′, (4) which replace the kernel K by a learnable Kδ : Rd → Rd mapping, with d representing the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Rather than directly using a neural network to model this Rd → Rd mapping, we further incorporate physical insights by analyzing the structure of Kδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As derived in Beale & Majda (1985), the kernel Kδ for 2-dimensional flow exhibits the following form: Kδ(z) = 1 2πrM(r, δ)R 2 π (z), r = |z| (5) where R 2 π is the 90◦ rotation applying which to z computes the unit direction of the cross product of z and the out-of-plane vector ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' and M(r, δ) is the human heuristic term that varies by choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Hence, we opt to replace 1 2πrM(r, δ) by a R2 → R neural network function N2(r, δ) so that: u(x) = � N2(|x − x′|, δi)R 2 π (x − x′)ω(x′)dx′ (6) ≈ n � i=1 N2(|x − xi|, δi)R 2 π (x − xi)ωi = D(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (7) Learning this induction kernel N2(r, δ) instead of using heuristics-based kernels allows for more accurate fluid learning and prediction from input videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We discuss more on this in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='2 END-TO-END TRAINING As previously mentioned, the dynamics on the latent vortex space is bridged to the evolution of the image space through the differentiable, dynamic module D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Hence, we can optimize the vor- tex representation Vt = T (t) at time t using images as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, we select m frames: [It, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , It+m] from the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Then, we compute ut = D(Vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' After that, (ut, It) is fed into an integrator on the Eulerian grid to predict ˜It+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Simultaneously, (ut, Vt) is fed into an integrator on Lagrangian particles to predict ˜Vt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The process is then repeated, using ˜It+1 in place of It and ˜Vt+1 in place of Vt, to generate ˜It+2 and ˜Vt+2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Eventually, we would obtain [˜It+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , ˜It+m], which are the predicted future outcome starting at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We optimize T and D jointly by minimiz- ing its difference between [˜It+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , ˜It+m] and [It+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , It+m] in and end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' By picking different values of t in each iteration to cover [0, tE], we optimize T and D to fit the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' There remains one more caveat — that the trajectories in T are not enforced to be consistent with D, because each frame of Vt is optimized individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In other words, if we evaluate the particle velocities [ ˙x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', ˙xn] = dN1 dt as prescribed by T , it should coincide with 5 Published as a conference paper at ICLR 2023 Ours HFM Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' HFM + UNet ER + UNet Ours HFM Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' HFM + UNet ER + UNet Figure 4: Applied to real-world videos, our Lagrangian based method can create realistic future predictions over long periods of time compared to existing methods (and their extensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Velocity Stream- lines Velocity Residue Ground Truth HFM E-R UNet Ours Figure 5: Hidden motion inference compared with existing methods on a synthetic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Our method uncovers the underlying velocity field with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' [D(V)(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , D(V)(xn)], which as prescribed by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Hence, in training, another loss is com- puted between dT dt and [D(V)(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , D(V)(xn)] to align the vortex trajectory and the predicted velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' After successful training, the learned system allows us to perform two important tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, using our continuous query function T (t), we are able to interpolate for V(t), which then uncovers the hidden velocity field u = D(Vt) at arbitrary precision, which provides the same func- tionality as Raissi & Karniadakis (2018), but using vorticity instead of pressure as the secondary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Moreover, with the dynamics module D, we can perform future prediction to unroll the input video, a feature unsupported by previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As shown in Figure 4, since our method is forward-simulating by nature, it can provide more realistic and robust future prediction than existing methods and their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Further implementation details of our method, including hyperparam- eters, network architectures, training schemes and computational costs can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 5 EXPERIMENTS We evaluate our method’s ability to perform motion inference and future prediction on both synthetic and real videos, comparing against existing methods.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='24D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='xPublished as a conference paper at ICLR 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Future ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Errors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Motion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Errors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='Figure 6: Error analysis on a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The top row plots the inference errors of velocity, vorticity, and compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The bottom row plots the future prediction errors, which consider both the dynamic error in the velocity and the perceptual error of the generated image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For motion inference, we compare our method against Raissi & Karniadakis (2018) (HFM) and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2022) (ER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We reimplement the HFM method as prescribed, making only the modification that instead of using only a single concentration variable c and its inverse d as spec- ified by (Raissi & Karniadakis, 2018), we create three (c, d) pairs for each of the RGB channel for the support of colored videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The E-R method is evaluated using the published pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We further compare against an ablated version of our proposed method, termed “UNet”, which es- sentially replaces the Lagrangian components of the system with a UNet architecture (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2015), a classic method for conducting field-to-field mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The UNet baseline takes two images It and It+1 and predicts a velocity field ut+1 to predict It+2 using the same Eulerian in- tegrator as our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For future prediction, there do not exist previous methods that perform in the same setting, so we extend the inference methods in a few ways to support future prediction in a logical and straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, since HFM offers a query function parameterized by t, we test its future prediction behavior by simply extrapolating with t > tE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' this is referred to as “HFM extp.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Since both Raissi & Karniadakis (2018) and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2022) uncovers the time-varying velocity field, we use a UNet to learn the evolution from ut to ut+1, and use this ve- locity update mechanism to perform future prediction, the two baselines thus obtained are referred to as “HFM+UNet” and “ER+UNet” respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The ablated version “UNet” does support future prediction intrinsically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='1 SYNTHETIC VIDEO The synthetic video for vortical flow is generated using the Discrete Vortex Method with a first-order Gaussian mollifying kernel M(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The high-fidelity BFECC advection scheme with Runge-Kutta-3 time integration is deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The simulation advects a background grid of 256 × 256, with a time step dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='01 to create 300 simulation videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Only the first 100 frames will be seen to train all methods, and future predictions are tested and examined on the latter 200 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Motion Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The results for the uncovering of hidden dynamic variables are illustrated in Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Shown in Figure 5 are the velocities uncovered by all 4 methods against the ground truth, at frame 55 of a synthetic video with 100 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The velocity is visualized in the form of colors (top row) as well as streamlines (middle row), while the velocity residue, measured in end-point error(EPE), is depicted in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It can be seen that HFM, UNet, and our method achieve agreeing results, and matches the ground truth values to high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' On the bottom row, it can be seen that while both HFM and UNet provide sensible results, our method generates the inference velocity that best matches the unseen ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The inference results over the full 100 frames at the top of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We evaluate the velocity with four metrics: the average end-point error (AEPE), average angular error (AAE), vorticity RMSE and compressibility RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' From all 4 metrics, it can be seen that our method outperforms the baselines consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The time-averaged data for all four metrics are shown on the left of Table 1, which deems our method favorable for all metrics used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Future Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 7, we visually compare the future prediction results from frame 100 to frame 299 done using our method and the 4 benchmarks, against the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It can be observed 7 Velocity AEPE over time 21 (uaua)zBo) 2 23 24 E-R + UNet 24 HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=" 2- UNet 125 150 175 20D 225 250 275 30D frameCompressibility eor over time 2 2 logz(error) 2' E-R + UNet HFM + UNet HFMextrap." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 24 UNet 125 150 175 240 225 250 275 310 frameImage RMSE emor over time 2-2 23 24 21 2- E-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet 210 140 125 150 175 20D 225 250 275 30D frameVelocity AEPE over time 211 2-2 (aua) 2-3 logz1 21 E-R 2 HFM UNet 2-?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Ours 0 24 140 rameVelocity AAE aver time 21 20 2-1 (error) 2 213 21+ E-R HFM 21 UNet s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='ino 21 41 81 140 rameVorticity errar over time 23 22 21 E-R HFM UNet Ours 0 24 4I 64 140 fameCompressibility eror over time 21 21 2-1 27 2-3 E-R HFM 24 UNet ours 区 140 frameE-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursPublished as a conference paper at ICLR 2023 Ours UNet Ground Truth HFM Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' HFM + UNet ER + UNet t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='32 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='66 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='98 Figure 7: Future predicting capacities of our method compared to benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Our method accurately predicts the unseen, future sequence that’s twice as long as the seen sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Time-averaged Inference Errors Time-averaged Prediction Errors AEPE AAE Vort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' VGG RMSE AEPE AAE Vort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' E-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='505 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='393 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='470 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='319 +UNet 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='631 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='424 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='580 HFM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='212 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='424 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='159 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='334 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='017 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='053 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='043 Table 1: Error analysis of benchmark testing on a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' that the sequence generated by our method best matches with the ground truth video, capturing the vortical flow structures, while the other baselines either quickly diffuse or generates unnatural, hard- edged patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Numerical analysis confirms these visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We compare the unrolled 200 frames both in terms of velocity and visual similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The velocity analysis inherits the same 4 metrics, and the visual similarity is simultaneously gauged using the pixel-level RMSE and the VGG perceptual loss (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The time-averaged results of all 6 metrics are documented in the right of Table 1, and 4 are plotted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It can be concluded that our method outperforms the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='2 REAL VIDEO A similar numerical analysis is carried out on a real video published on YouTube, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The video has 150 frames: the first 100 frames will be used for training, while the latter 50 will be used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Since the ground truth velocities for the real video are nonexistent, we will only analyze the future predicting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For all methods, we perform future prediction for 150 frames, among these, the first 50 frames can be compared with the testing videos, and the latter 100 frames will be compared qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We note that, since only part of the video is fluid (within the circular rim), we pre-generate a signed distance field for all methods, so that only the fluid regions are considered, and the same no-slip boundary condition will be placed for all unroll methods (except for HFM extp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' which requires no advection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The numerical analysis for the first 50 predicted frames are documented and plotted in Table 2 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We compare all methods against the baseline using the VGG perceptual loss for visual loss, and compare the velocity divergence, which should be close to the theoretical value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It can be seen that in all 3 metrics our method prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For prediction results that surpass the duration 8 SPublished as a conference paper at ICLR 2023 Ours UNet Ground Truth HFM Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' HFM + UNet ER + UNet t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='98 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='49 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='47 Figure 8: Future predicting capacities of our method compared to benchmarks on a real video sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Our method generates a predicted sequence that best matches the input video within its duration and remains visually plausible way beyond its duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Figure 9: Plots corresponding to Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' VGG (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=') VGG (final) Div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (avg) E-R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='095 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='205 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='046 HFM+UNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='151 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='940 HFM Extp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='980 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='271 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='922 UNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='111 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='088 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='447 Ours 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='093 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='318 Table 2: Data corresponding to Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' of the real video, qualitative observations can be made in that our method preserves the vortical structure, generating smooth visualizations over the entire horizon, while other methods end up yielding glitchy patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We perform additional quantitative benchmark testings in Appendix B against a differentiable grid- based simulator on real and synthetic videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' and in Appendix C against 4 baselines on another synthetic video featuring different visual and dynamics distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 6 CONCLUSION & LIMITATIONS In this work, we propose a novel data-driven system to perform fluid hidden dynamics inference and future prediction from single RGB videos, leveraging a novel, vortex latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The success of our method in synthetic and real data, both qualitatively and quantitatively, suggests the potential for embedding Lagrangian structures for fluid learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Our method has several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, our vortex model is currently limited to 2D inviscid flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Extending to 3D, viscous flow is an exciting direction, which can be enabled by allowing vortex strengths and sizes to evolve in time (Mimeau & Mortazavi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Secondly, our vortex evolution did not take into account the boundary conditions in a physically-based manner, hence it cannot accurately predict flow details around a solid bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Incorporating learning-based boundary modeling may be an interesting exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Thirdly, scaling our method to handle turbulence with multi-scale vortices remains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We con- sider two additional directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, we plan to explore the numerical accuracy of our neural vortex representation to improve the current vortex particle methods for scientific com- puting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Secondly, we plan to combine our differentiable simulator with neural rendering methods to synthesize visually appealing simulations from 3D videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 9 N/AE-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursE-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursPublished as a conference paper at ICLR 2023 REPRODUCIBILITY STATEMENT To ensure reproducibility of our work, we will release the training and testing code, as well as the data to reproduce our results upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' REFERENCES Peter Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Interaction networks for learning about objects, relations and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Advances in neural information processing systems, 29, 2016.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Physics-informed generative adversarial networks for stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' SIAM Journal on Scientific Computing, 42(1): A292–A317, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Mingrui Zhang, Jianhong Wang, James Tlhomole, and Matthew D Piggott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Learning to estimate and refine fluid motion with physical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='10480, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 12 Published as a conference paper at ICLR 2023 A IMPLEMENTATION DETAILS In this section, we describe the implementation details of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As described above and illustrated in Figure 2, our system embeds two differentiable integrators in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The Eulerian integrator is implemented using the Back and Forth Error Compensation and Correction (BFECC) (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', 2005) method for backtracking, and the 3rd order Runge-Kutta method for time-stepping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The Lagrangian integrator is implemented using the Forward Euler method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Network N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The network N1 adopts a series of 3 residue blocks with increasing width [64, 128, 256], whose architecture is similar to He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2016) but with convolution layers replaced by linear layers with sine activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The frequency factor ω0 discussed in Sitzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2020) is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Network N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The network N2(r, δ) is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, the input r is scaled by the input δ as ¯r = r · η δ , where η is a hyperparameter selected to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='1, which corresponds to the characteristic length scale of vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Then, ¯r is transformed into ˆr as ˆr = ¯r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='3, which is a reparametrization that stretches the value ¯r near 0, which exploits the insight that velocity varies more aggressively when near a vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The value ˆr is then fed through 4 residue blocks (same as N1) but with a fixed width of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The output from these residue blocks would be scaled by multiplying with η δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The scaled output is N2(r, δ), which is then used for velocity computation according to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Training Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Both the image loss and the velocity alignment loss are MSE, and the velocity alignment loss has an extra scaling factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We use Adam optimizer with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='999 and learning rate = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='005] for N1, N2, Ω and ∆ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We use a step learning rate scheduler and set the learning rate to decay to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='1 of the original value at 20% of the total training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We use a batch size of 4, so for each iteration, 4 starting times are picked uniformly randomly among [0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , tE] for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The sliding-window size m is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Pretraining N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We pretrain N1 for 10000 iterations with 2 objectives: (1) for all t ∈ [0, tE], N1(t) = [(x1)t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' , (xn)t] coincide with the centers of a 4×4 grid, (2) for all t ∈ [0, tE], dN1 dt = 0, so that these particles are initially stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We use MSE for the positional and velocity losses, and the other training specifications are the same as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Computational Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Running on a laptop with Nvidia RTX 3070 Ti and Intel Core i7- 12700H, our model takes around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='4s per training iteration, and around 30000 iterations to converge (for a 256 × 256 video with 100 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For inference, each advance step costs around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='035s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' B COMPARISON WITH DIFFERENTIABLE FLUID SIMULATION We compare our method qualitatively and quantitatively against a standard, grid-based differentiable fluid simulator (referred to as Diff-Sim) on both synthetic and real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This baseline method is a differentiable implementation of the method proposed by Fedkiw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2001), which is a clas- sic, widely-adopted numerical method for simulating vortical fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The method is designed to solve the 2D Euler equations for inviscid fluid, hence it can in theory recreate the inviscid fluid phe- nomena represented by any video if provided with the appropriate initial conditions and simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Therefore, in this experiment, we make use of its differentiable nature to optimize (1) the initial grid velocities (a 256 × 256 × 2 tensor), and (2) the vorticity confinement strength, which is a scalar value, with the objective of minimizing the discrepancy between the simulated results and the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The loss computation between the simulated image sequence and the ground truth is the same as in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We note that the idea of optimizing initial conditions using differentiable fluid simulation to fit specific still frames has been demonstrated in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' However, their task is notably simpler than ours, since they only require the simulated image to match a target frame at the end of the simulation, while our goal is to match the underlying motion of the entire video, and dynamically unroll into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Comparison on a synthetic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We start by comparing both methods on a synthetic video, which yields a visual comparison which can be found in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We observe that our method suc- cessfully learns the dynamics represented in the video: the generated video and velocities closely 13 Published as a conference paper at ICLR 2023 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 Observed Past Predicted Future Differentiable Fluid Simulation Ours Ground Truth t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 Inferred Past Velocity Predicted Future Velocity Differentiable Fluid Simulation Ours Ground Truth Figure 10: Visual comparison between a differentiable grid-based simulator and ours on a synthetic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The upper half displays the simulated image, while the lower half displays the underlying velocity, whose color wheel is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' On the bottom row of each half is the ground truth sequence, which has 300 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The first 100 frames are available for both methods to learn from, while the latter 200 frames are unseen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Synthetic Test Errors Real Test Errors Figure 11: Error plots of the comparison between Diff-Sim and ours on a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' resembles the groundtruth even in the unseen frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Diff-Sim, on the other hand, shows a weak resemblance with the ground truth for the seen frames, yet fails to capture the individual eddies in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Consequently, it fails to predict the future dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Diff-Sim’s lack of correspondence to the dynamics of the ground truth is also made evident in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The result clearly suggests that our method has better learned the dynamics evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This performance discrepancy is nu- merically supported by the errors shown on the left panel of Table 3 and plotted on the left panel of Figure 11, both showing that our method yields reduced image-level and velocity-level errors compared to Diff-Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 14 T TDiff-Sim OursDiff-Sim OursDiff-Sim OursDiff-Sim OursPublished as a conference paper at ICLR 2023 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='55 Observed Past Predicted Future Differentiable Fluid Simulation Ours Ground Truth Figure 12: Comparison between Diff-Sim and ours on a real video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Synthetic Video Errors Real Video Errors AEPE AAE Vort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' RMSE VGG VGG (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=') VGG (final) Diff-Sim (Grid) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='953 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='157 26043 15076 18792 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='081 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='055 2171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='4 7846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='2 11081 Table 3: Error comparison between Diff-Sim and ours on synthetic and real videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Comparison on a real video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We then use the same experimental set up to perform learning on a real video, which is depicted in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We observe that on the real video, the same behavioral patterns for both systems seen on the synthetic one have carried over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For the results generated by Diff-Sim (top row), we can see that the overall, large-scale motion (the large eddy moving towards bottom-left) is correctly learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Nevertheless, all the smaller vortices are gone and the entire image quickly diffuses as the simulation goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This can be attributed to the numerical diffusion issues innate to grid-based simulations, as well as the lack of embedded fluid structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In comparison, our method well-preserves the vortical movements due to its built-in structure, and produces a plausible future rollout extending beyond the duration of the original video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Although both systems are unable to perfectly model the exact dynamics that governs this real-world video (due to unmodelled factors such as fluid viscosity, air friction, and 3-dimensional forces), our proposed method does a better job in retaining the vortical patterns and energetic flows thanks to its vorticity-based formulation and the Lagrangian-Eulerian design, as can be observed in the middle row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The advantage of our system over Diff-Sim on the real video is numerically supported, as can be found on the right panel of Table 3 and Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Since we do not have the ground-truth velocities for real videos, we compare the VGG perceptual loss between the simulated sequence of both methods and the real video, which demonstrates quantitatively that our generated results better resembles the input video than that of its counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' C ADDITIONAL BENCHMARK TESTING As depicted in Figure 14, to further illustrate our method’s advantage and generalizability, we have conducted an additional set of numerical tests on another synthetic video, and compare our method’s performance with 4 benchmarks in terms of both velocity inference quality and future prediction quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The ground truth data is generated using a significantly different background image (sharp color tiles vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' smooth color gradients), and a different velocity kernel (second-order Gaussian kernel vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' first-order Gaussian kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The experimental setup is otherwise the same as the one presented in the main text (in Figure 7), with the same compared benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 15 N/APublished as a conference paper at ICLR 2023 Color Plot Streamline Plot Quiver Plot Residue Ground Truth Ours Differentiable Fluid Simulation Figure 13: Comparing the quality of velocity inference of our method and Diff-Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We show the predicted velocity of frame 200 in three different forms (color, streamline and quiver plots) in additional to the residue (end-point error) compared to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Ours UNet Ground Truth HFM Extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' HFM + UNet ER + UNet t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='60 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='4 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='8 Figure 14: Future prediction results: our method compared to baselines on a synthetic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The comparison of the velocity inference quality can be found in Figure 16 and the top panel Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Figure 16 depicts the uncoverred velocities of frame 40 (among the 60 input frames) by all 4 methods compared to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The top row depicts the respective velocities in colors with the color wheel supplied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' the middle row depicts the velocities in streamlines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' and the bottom row depicts the velocity residue compared to the ground truth, measured in end-point error (EPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As with the results in 5, we can see that HFM, UNet and our method can all infer the underlying velocity field with high precision, whereas E-R yields a visibly noisier approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As seen on the bottom row, the inference performance between UNet and Ours are very close, but our method takes the slight edge with an average error (AEPE) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0143 as compared to the error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0215 16 T TPublished as a conference paper at ICLR 2023 Future Prediction Errors Motion Inference Errors Figure 15: Error analysis on a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The top row plots the inference errors of velocity, vorticity, and compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The bottom row plots the future prediction errors, which consider both the dynamic error in the velocity and the perceptual error of the generated image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Velocity Stream- lines Velocity Residue Ground Truth HFM E-R UNet Ours Figure 16: Comparison to baselines on velocity inference on a synthetic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Our method recovers the underlying velocity field with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' yielded by UNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The advantage of our method is not unique to the specific frame selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As plotted on the top row of Figure 15, it can be seen that our method (red) consistently yields the lowest velocity-inference error throughout the 60 input frames, in terms of the average end-point error (AEPE), average angular error (AAE), vorticity RMSE and compressibility RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The time- averaged errors of these metrics are documented in Table 4, which again shows that our method yields the best estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Future prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We also compare the future prediction results with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 14, we show a visual comparison of all 5 methods against the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It highlights the close re- semblance of our generated sequence with the ground truth, which is twice as long as the sequence used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Compared to the baselines, our method yields the best match to the ground truth video, capturing the accurate vortical flow structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' HFM+UNet, ER+UNet, and UNet can gen- erate reasonable future prediction up to t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0 (for 40 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' For t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0, these sequences start to distort in different ways, due to their lack of physical structures and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The direct ex- trapolation of HFM yields the least plausible results, quickly degrading to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We compare these sequences quantitatively using the 4 velocity-based metrics, along with 2 image-based metrics: the pixel-level RMSE and the VGG perceptual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Four of these time-dependent errors are plotted in the bottom row of Figure 15, with their time-averaged counterparts documented on the right of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In summary, we observe that our method outperforms the existing baselines for this video both quantitatively and qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 17 E-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursE-R HFM UNet SInoE-R HFM UNet OursE-R HFM UNet OursE-R HFM UNet oursE-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursE-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursE-R + UNet HFM + UNet HFM extrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' UNet OursT TPublished as a conference paper at ICLR 2023 Time-averaged Inference Errors Time-averaged Prediction Errors AEPE AAE Vort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' VGG RMSE AEPE AAE Vort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' E-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='805 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='380 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='750 +UNet 8138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='272 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='115 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='694 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='504 HFM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='097 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='533 +UNet 9389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='715 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='199 Extp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 40967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='293 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='221 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='862 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='152 UNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='895 7721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='330 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='141 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='462 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='496 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='015 2045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='173 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='013 Table 4: Time-averaged errors of our method compared to various baselines on a synthetic video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' D NUMBER OF VORTEX PARTICLES In our proposed method, we use n vortex particles to learn the fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' However, we note that vortices are not intrinsic to fluid phenomena, but are rather imposed constructs to allow fluids to be better understood conceptually and modeled numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Thus, the number of vortices n is fundamentally a hyperparameter that does not admit a uniquely-correct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' With this in mind, we let ˆn denote the minimum number of particles that can be used to model the fluid system to an acceptable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This natural number ˆn surely exists since it has been proven that vortex particle methods converges to the exact solution of 2D Euler’s Equations (Beale & Majda, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Hald, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We are mostly concerned with the cases where n > ˆn, which means the deployed degrees of freedom (DoFs) are higher than that of the fluid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In the following, we show that our method can spontaneously prune the redundant vortices and thus it is robust to a reasonable range of n > ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 17, we show the results of learning the same underlying motion with 4, 9, and 64 vortex particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 18, we show the underlying velocity and vorticity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Spontaneous pruning of redundant DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' As shown on the top row of Figure 17, the ground truth is generated with 4 vortices, so it is safe to assume that ˆn = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Learning with 4 vortices (as shown on the second row) represents the case where n = ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Comparing the first row with the second row, we can see that there is a one-to-one correspondence between the ground-truth vortices and the learned vortices, with each learned vortex assuming the role of one individual ground-truth vortex (obtaining the same vorticity and initial position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' When we have 9 vortices (third row), there are more vortex particles than ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In this case, two interesting phenomena occur to spontaneously prune these redundant particles: degenera- tion and clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' First, some particles degenerate themselves by reducing its strength to 0 or by moving farther away from the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We can observe both mechanisms taking place on the two lingering particles on the top part of the third row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' They both have low strength (evident from their turquoise color) and are peripheral to the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Secondly, particles would aggregate to simulate a single particle with greater strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Since the velocity computation is done with distance-weighted summation (as in Equation 7), if multiple particles coincide at the same location, they effectively act as one single particle with their vorticities summed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This phenomenon can be observed on the lower half of the images in the second and third row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Both of these mechanisms enable the system to spontaneously prune redundant vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In the last row, we show that our method is robust to even 64 vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Figure 19 helps to illustrate this spontaneous pruning mechanism by observing different snapshots of the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Shown on the left are the vortex particles’ behaviors soon after the training has begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It is particularly noticeable that, on the bottom row, the 64 particles are scattered in the fluid domain, and the learned result appears quite different from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Moving from left to right, these particles become more and more clustered on the flow regions, with much fewer particles wandering around;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' and the end result can approximate the ground truth much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Finally, we note that n < ˆn is still challenging to resolve as the system is over-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Never- theless, we empirically find that n = 16 is sufficient for all the real and synthetic videos we consider in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 18 Published as a conference paper at ICLR 2023 Ground Truth (4 Vortices) Learned with 4 Vortices Learned with 9 Vortices Learned with 64 Vortices Observed Past Predicted Future Figure 17: The same underlying motion as learned with different numbers of vortex particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The ground truth sequence has 100 frames, the first 30 frames are provided during training, and the latter 70 frames are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Vorticity Field at Frame 80 Velocity Field at Frame 80 Ground Truth (4 Vortices) Learned with 4 Vortices Learned with 9 Vortices Learned with 64 Vortices Figure 18: Different number of vortices can learn similar underlying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' E ABLATION: LEARNABLE VELOCITY KERNEL In traditional vortex simulation applications in Computer Graphics or Computational Fluid Dynam- ics, the velocity kernel is hand-selected (typically from Gaussian kernels of different orders) with a uniform support radius (size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Such approaches are designed to perform forward simulation, yet they are limited when used for backward inference tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' to reconstruct input videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In our method, we address this issue by learning neural kernels with learnable sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' By leveraging data- driven techniques, we can reconstruct and predict fluid flows that are not only visually pleasing, but also resemble the particular dynamics traits depicted in the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 19 120 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 20 40 60 80 100 120Published as a conference paper at ICLR 2023 Learning with 4 Vortices Learning with 9 Vortices Learning with 64 Vortices Ground Truth (4 Vortices) Training Starts Training Ends Figure 19: The training evolution using different number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='15 Observed Past Predicted Future With Learnable Kernel Ground Truth Without Learnable Kernel Figure 20: Ablation study: reconstruction and prediction on a real video with learnable velocity kernels (our full method) and without learnable velocity kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 20, we show an ablation study on learning velocity kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We reconstruct and predict a real-world video using our method and an ablated version in which the learnable kernel is replaced with a hard-coded first-order Gaussian kernel with uniform size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The ground truth, shown on the bottom row, has 126 frames revealed (for training) and 62 frames hidden (for testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In the middle row, we learn to fit the video with our learnable kernel enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In the top row, we learn to do the same with the learnable kernel disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' It can be observed that the middle row well-captures the characteristic smoothness of the flow, and simulates a image sequence that resembles the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The ablated version (top row) can also learn the correct overall motion (clockwise rotation), but it induces various smaller eddies and wrinkles uncharacteristic of the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Extending to unseen frames, our method can continue to retain the overall structure of the eddies, while the ablated version (without learnable kernel) drives the pattern to disintegrate and evolve 20 120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='75 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='50 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='25 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='25 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='50 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 0 0 20 40 60 80 100 120N/ASPublished as a conference paper at ICLR 2023 Figure 21: Time-dependent losses corresponding to Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' VGG (avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=') VGG (final) RMSE (avg) Ours w/o learnable kernels 9698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='7 11240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='183 Ours 7365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='6 10027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='180 Table 5: Data corresponding to Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='80 Without Trajectory Learning Ours Ground Truth t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='00 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='80 Without Trajectory Learning Ours Ground Truth Figure 22: We compare our method against its ablated version which does not feature trajectory learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' On the top depicts the simulated images, and on the bottom depicts the simulated velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' The results by both approaches are compared to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' into various folds and wrinkles that do not resemble the dynamics characteristics in the real video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We further show quantitative results plotted in Figure 21 and documented in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In summary, learning velocity kernels allows for better reconstruction and prediction of fluid dynamics in the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' F ABLATION: TRAJECTORY LEARNING In our approach, we learn the full trajectory of vortex particles for the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' An alternative to “learning initial condition through trajectory” is to learn the initial condition directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' However, we find that the former option is more computationally tractable and effective, if we want to fully exploit the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' To see this, suppose we have 100 training frames in the video, and the goal is to infer the initial condition at frame-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' If we directly optimize the initial condition using the last frame, we need to simulate from frame-1 all the way to frame-100, compute the loss and 21 Without Learnable With LeanableWithout Learnable With LearmablePublished as a conference paper at ICLR 2023 Velocity Errors Image Errors AEPE AAE Vort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' RMSE VGG Ours (Ablated) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='753 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='170 6759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='4 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='131 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='081 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='9 Table 6: Time-averaged velocity-level and image-level errors by our method and its ablated version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Figure 23: Time-dependent velocity-level and image-level errors by our method and its ablated version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' backpropagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Unrolling such a long sequence for each training iteration (1) takes a long time, (2) leads to noisy gradients, and (3) is practically infeasible due to memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' On the other hand, learning through the whole trajectory allows us to address these challenges by using a smaller sliding window in time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=', simulating only 3 frames at a time) and aggregating the dynamics information throughout the whole video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 22, we show a comparison of both methods in action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 22 top, we show the reconstruction and prediction results for both our full method and an ablation version where we directly learn the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Note that the ablation version can only unroll the first 13 frames (and thus it is learned using only the first 13 frames) due to the same memory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In Figure 22 bottom, we show the velocity corresponding to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We observe that our method and its ablated version can approximate the ground truth reasonably well at the beginning of simulation (the left three images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' However, the ablated version starts to distort significantly in terms of both the advected image and the underlying velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' This observation is in correspondence with the numerical evidence, as plotted in Figure 23 and documented in Table 6, which shows that our full method consistently outperforms its ablated counterpart among all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' We conjecture that the underlying reasons to this performance discrepancy is threefold: first, the ablated version can only learn from the beginning of the fluid dynamics which provides limited information to correctly infer the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Secondly, only learning the initial condition is more susceptible to accumulated error than our full method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' Thirdly, using a limited number of frames makes it harder to learn an appropriate velocity kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' In summary, our observations suggest that learning the full trajectory is computationally more tractable and effective compared to learning the initial conditions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} +page_content=' 22 W/O TrajectoryW/O Trajectory OursW/O Trajectory Oursw/O Trajectory OursW/O Trajectory sInoW/O Trajectory Ours' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFJT4oBgHgl3EQfRCzr/content/2301.11494v1.pdf'} diff --git a/ctA0T4oBgHgl3EQfGv-w/content/tmp_files/2301.02052v1.pdf.txt b/ctA0T4oBgHgl3EQfGv-w/content/tmp_files/2301.02052v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..314e0c4e1e7e018fb5afc7d9ac70440920aebec2 --- /dev/null +++ b/ctA0T4oBgHgl3EQfGv-w/content/tmp_files/2301.02052v1.pdf.txt @@ -0,0 +1,2406 @@ +Relaxing Instrument Exclusion with Common +Confounders +Christian Tien ∗ +January 6, 2023 +Abstract +Instruments can be used to identify causal effects in the presence of unobserved con- +founding, under the famous relevance and exclusion assumptions. +As exclusion is +difficult to justify and to some degree untestable, it often invites criticism in applica- +tions. Hoping to alleviate this problem, we propose a novel identification approach, +which relaxes traditional IV exclusion to exclusion conditional on some unobserved +common confounders. We assume there exist some relevant proxies for the unobserved +common confounders. Unlike typical proxies, our proxies can have a direct effect on +the endogenous regressor and the outcome. +We provide point identification results +with a linearly separable outcome model in the disturbance, and alternatively with +strict monotonicity in the first stage. Using this novel method with NLS97 data, we +demonstrate the insignificant role of ability bias compared to general selection bias in +the economic returns to education problem. Beyond economics, the approach is just as +relevant in health treatment evaluation with an unobserved underlying health status, +or a psychological study where character traits are unobserved common confounders. +Keywords: +Causal Inference, Unobserved Confounding, Instrumental Variables, Control Function, +Proximal Learning +∗ct493@cam.ac.uk; Faculty of Economics, University of Cambridge +arXiv:2301.02052v1 [econ.EM] 5 Jan 2023 + +1 +Introduction +Unobserved confounding complicates the identification of a causal effect of a regressor of +interest on an outcome. Despite the endogeneity of a regressor of interest, instrumental vari- +able (IV) approaches can identify their causal effect if the famous relevance and exclusion +assumptions hold for the instruments. These assumptions are strong and often invite criti- +cism of IV estimates in practice. We propose a novel approach to relax exclusion, in favour +of exclusion conditional on an unobserved common confounder, for which some relevant +variables are observed. +Relaxing exclusion (or exogeneity) is only possible when it is replaced by other strong +assumptions. One way to identify causal effects without instrument exclusion is from resid- +ual distributions, not variation in the explanatory variables [Heckman, 1979, Millimet and +Tchernis, 2013]. Very specific forms of heteroskedasticity across the first stage and outcome +model can also be used to establish identification without an exclusion restriction [Klein +and Vella, 2010, Lewbel, 2012]. Others have suggested to use irrelevant variation in instru- +ments to test for the exclusion of the relevant variation in the instruments [D’Haultfœuille +et al., 2021]. A similar idea is followed when integrated conditional moments use nonlinear +mean-dependence of endogenous variables on instruments, such that the instruments may +violate the exclusion restrictions in pre-specified parametric ways. Despite recent advances +in estimation with integrated conditional moments [Tsyawo, 2021], the strong identifying +assumptions render all these approaches difficult to justify in applications. Our solution +differs significantly from these approaches as it only uses a relaxed exclusion restriction and +variation in explanatory variables to identify the causal effect of interest. +From the perspective of IV, we allow for some endogeneity in the instruments. That +endogeneity originates from unobserved common confounders. +We call these unobserved +confounders common, because there are some observed variables, which are relevant for +them. These observed variables are called proxies. In other words, we assume there are some +unobserved variables that explain all correlation (association) between the instruments and +these proxies. This assumption is testable. Then, we need to argue for the exclusion of the +instruments, but only conditional on the unobserved common confounders (and observable +variables). +In general, this is a strong relaxation of instrument exclusion conditional on +observed variables only. +Another way to understand our proposal is as a solution to measurement error in observed +confounders for IV. Residual bias is a well-known problem when confounders are measured +with error. Proximal learning [Cui et al., 2020] is a solution to the problem, where observed +variables measure all unobserved confounders with some error. In proximal learning, the +1 + +proxies for the unobserved confounder may either be causes of the treatment or outcome +variable. These proxies must be sufficiently relevant (e.g. complete) for all unobserved con- +founders. +Separately developed from the proximal learning approach, a control function +solution exists with identical conditional independence assumptions and mismeasured con- +founders [Nagasawa, 2018]. Our solution is different, as we do not assume the existence of +measurements for all confounders. Instead, we use instruments and assume that measure- +ments exist for all confounders, conditional on which the instruments would be exogenous. +In this sense, our solution can be understood as IV with mismeasured confounders. +As it is standard in the control function literature, our approach will identify average +causal (structural) quantities of interest. Unless the outcome model is fully linearly separa- +ble in the treatment and disturbance [Newey et al., 1999], where in our case the disturbance +includes the effect of the common confounder, we identify those average causal (structural) +quantities of interest that integrate out the unobservables without dependence on the treat- +ment using a control function [Imbens and Newey, 2009]. Our identification approach is +most similar to recent advances in nonlinear panels [Liu et al., 2021]. In panel data, unob- +served fixed effects are common to the same variables across time, in a similar way as the +unobserved common confounders are common to the instruments and proxies in our setup. +In Liu et al. [2021], identification stems from a parametric dimension reduction of the effect +of observed variables on the outcome, and an index sufficiency assumption that renders the +observed variables independent from the fixed effects conditional on an index of the observed +variables. In our approach, identification stems from the existence of more instruments than +treatments, and an index sufficiency assumption that renders the instruments independent +from the unobserved common confounders conditional on an index of the instruments. Just +like Blundell and Powell [2004], Liu et al. [2021] do not explain how to derive this crucial +index function. One of our main contributions is the derivation of the index function, which +arises naturally in the common confounding setup. +A motivating example for our proposal is the returns to college education identification +problem. It features various biases, ability and selection, and we motivate pre-college test +scores as instruments exogenous to selection, while clearly endogenous to ability. proxies +are pre-college risky behaviour dummies, which appear to correlate negatively with ability. +With NLS97 data, we show that selection bias is the much more economically relevant bias +compared to ability bias in this problem. +2 + +2 +Setup +The treatment (action) A ∈ A is discrete or continuous with base measure µA of A ⊆ RdA. +Y ∈ Y ⊆ R is the one-dimensional outcome variable. Other important variables are the +instruments Z ∈ Z ∈ RdZ, the proxies W ∈ W ⊆ RdW , and the common confounders +U ∈ U ⊆ RdW . +Assumption 2.1 (Common Confounding IV Model). +1. SUTVA: Y = Y (A, Z). +2. Instruments +(a) Exclusion: Y (a, z) = Y (a) ⊥⊥ (A, Z) | U. +(b) Index sufficiency: For some τ ∈ L2(Z), where T := τ(Z), U ⊥⊥ Z | T. +(c) Relevance (completeness): For any g(A, T) ∈ L2(A, T), +E +�g(A, T)|Z +� = 0 only when g(A, T) = 0. +(2.1) +3. Proxies +(a) Exclusion: W ⊥⊥ Z | U. +(b) Relevance (completeness): For any g(U) ∈ L2(U), +E +�g(U)|W +� = 0 only when g(U) = 0. +(2.2) +Assumption 2.1.1 is the standard stable unit treatment value assumption (SUTVA), which +implies no interference across units. In assumption 2.1.2a we capture the key relaxation of +this model compared to standard IV. It states that the instruments are excluded, yet this +exclusion may be conditional on an unobserved (vector-valued) random variable U. This +is a significant relaxation of the standard exclusion restriction, which is possible only with +assumptions 2.1.2b, 2.1.2c, and 2.1.3. In assumption 2.1.2b, we introduce a control function +τ and a control variable T = τ(Z). Conditional on the control variable T, the instruments Z +are independent from the common confounders U. This assumption describing the existence +the control function τ is often called index sufficiency, where T is a (multiple) index of Z. In +assumption 2.1.2c, we require that conditional on the control variable T, the instruments Z +are complete for treatment A. This is a standard completeness condition. It simply means +that keeping the variation of Z described by T fixed, the instruments must remain sufficiently +relevant for A. In slightly different words, after conditioning on T, enough variation must be +left in the instruments Z to infer the effect of treatment A on outcome Y . As in standard +3 + +IV with observed confounders, this relevance requirement is typically testable. Assumption +2.1.3a states that the proxies W are independent from instruments Z conditional on the +common confounders U. The proxies W must also be complete for the unobserved common +confounders U, as stated in 2.1.3b. Again, completeness means the proxies W are sufficiently +relevant for U. +A different way to understand these assumptions is that the unobserved variable U, which +explains all association (correlation) between the proxies W and instruments Z, renders the +instruments exogenous when observed. In this sense, W can be (possibly quite poor) proxies +for what we consider the unobserved common confounder U, as long as they are sufficiently +relevant. Conditional on W, the instruments Z are still endogenous. Common confounders U +are never observed, and W could be quite poor proxies for it. Yet, we prove that conditioning +on a control function T, which makes the instruments Z and proxies W independent, restores +the exclusion of instruments Z which holds conditional on the unobserved U. +3 +Learning the Confounding Structure +In this section, we describe the main idea of the paper. Using only observable information, we +find a control variable T, conditional on which the instruments Z are independent from the +unobserved common confounders U. We then explain what may be considered the optimal +control variable T. +3.1 +Learning a Control Function +The control function τ ∈ L2(Z), described in lemma 3.0.1, generates the control variable T. +This control variable renders the instruments Z independent from the unobserved common +confounders U. Logically, if the instruments Z and the proxies W are independent condi- +tional on U, it follows that any such control variable T also renders Z and W independent +conditional on T. +Lemma 3.0.1. Assume W ⊥⊥ Z | U (2.1.3a). Take any τ ∈ L2(Z), where T := τ(Z), such +that U ⊥⊥ Z | T. Then, also W ⊥⊥ Z | T. +One possible such control variable is T = Z, yet this would leave no remaining variation +in Z to instrument for A conditional on T. Also, lemma 3.0.1 does not provide a way to +identify any control function τ apart from a function which captures the same information +as Z itself. For this purpose, we need lemma 3.0.2. In this lemma, we establish that any +T = τ(Z), conditional on which the instruments Z and proxies W are independent, also +renders Z conditionally independent from the unobserved common confounders U. +4 + +Lemma 3.0.2. Assume W ⊥⊥ Z | U (2.1.3a), and for any g(U) ∈ L2(U), E +�g(U)|W +� = 0 +only when g(U) = 0 (2.1.3b). Take any τ ∈ L2(Z), where T := τ(Z), such that W ⊥⊥ Z | T. +Then, also U ⊥⊥ Z | T. +Unlike lemma 3.0.1, the conclusion of lemma 3.0.2 is not obvious and requires the com- +pleteness of proxies W for the unobserved common confounders U. Again, completeness +means that the proxies W must be sufficiently relevant for U. If this were not the case, +it would be impossible to keep all variation in Z that is associated with U fixed, using a +control variable T derived only using information about the association of Z and W. We +can interpret U as all unobserved confounders that associate (correlate) Z and W. +Lemma 3.0.2 is an important result, because it allows the identification of a control +function that does not capture all variation in instruments Z. For any T that we identify +conditional on which instruments Z and proxies W are independent, the instrument exclusion +assumption can be relaxed to exclusion conditional on all unobservables U that associate +(correlate) Z and W (assumption 2.1.2a). The parallels to standard IV are quite clear: The +conditional exclusion assumption 2.1.2a is untestable, yet relaxed compared to standard IV. +The relevance requirement of Z for A conditional on T (assumption 2.1.2c) is testable, yet +stricter compared to standard IV. The requirement for relevance of Z for A conditional on +T implies that only a subset of control functions τ ∈ L2(Z), which leave enough relevant +variation in Z conditional on T, enable model identification under assumption 2.1: +T valid := +� +τ ∈ L2(Z) : +�W ⊥⊥ Z | τ(Z) +� and +� +E +�g(A, τ(Z)|Z +� = 0 only when g(A, τ(Z)) = 0 +�� +(3.1) +As both defining relevance conditions of this set T valid are testable, its non-emptiness is +testable as well. +3.2 +Optimal Control Function +Under assumption 2.1, the optimal control function τ ∗ ∈ T valid out of the set of valid control +functions captures the minimum feasible information in Z in a sense of minimising the +variance of the asymptotically unbiased estimator ˆJ of some causal estimand J. Figure 1 +illustrates schematically how the bias and variance of the IV estimator conditional on the +control variable T depend on the complexity of τ. +In figure 1, the complexity of T = τ(Z) on the x-axis increases from T = 0 on the +extreme left to T = Z on the extreme right. Moving further to the right on the x-axis +means that the control function captures more information in Z, starting with variation +5 + +Figure 1: Implied typical estimator properties with control functions τ of varying complexity +Notes: This figure illustrates some typical properties of an estimator ˆJ of the causal effect of a treatment A +on an outcome Y , using instruments Z while conditioning on a control function T = τ(Z). Moving to the +right on the x-axis, the control function captures more information in Z, starting with variation in Z which +correlates with the unobserved common confounders U. +In the left rectangle, the control function is too simple to render Z exogenous conditional on T. Hence, +the estimator is inconsistent, yet the degree of inconsistency decreases as the complexity of the control +function T increases. In the central rectangle, the control function captures enough information for Z to be +excluded conditional on T. So, the estimator is consistent. In the right rectangle, the control function is too +complex. Conditional on T, the instruments Z are no longer sufficiently relevant for A and the estimator +ˆJ asymptotically does not exist. In the first two rectangles, the asymptotic variance of the estimator ˆJ +increases with the complexity of the control function, because conditional on T, less information in Z is used +to infer the effect of A on Y . +in Z which correlates with the unobserved common confounders U. In the left rectangle, +the complexity of τ is low. T does not capture all information in Z that correlates with +U, so even conditional on T the instruments Z remain endogenous, and the estimator ˆJ is +inconsistent. However, as all information in Z is used for inference, the asymptotic variance +of the estimator ˆJ will be relatively small. As the complexity of τ increases towards the +right in the left quadrant, more information about the elements of Z which correlate with +U is captured in T. Increasing the complexity of τ increases the asymptotic variance of ˆJ +as less information in Z is used. Importantly, as this information corresponds to variation +in Z that correlates with U, inconsistency is being reduced. +In the central rectangle of figure 1, the complexity of τ is sufficient for Z to be exogenous +conditional on T. Hence, the estimator ˆJ is consistent. However, as τ increases in complexity, +we use less information in Z to infer the causal estimand J. Hence, inevitably the asymptotic +variance of the estimator ˆJ increases. Consequently, the optimal τ would be that of minimal +6 + +UKZIT +UIZIT +UIZIT +Z relevant +Z relevant +Z not relevant +plim(IJ JI) +avar(J) +T=0 +T=Z +T = T(Z) captures more information in Zcomplexity, such that Z is excluded conditional on T. In practice, we do not know but can +only estimate T valid, so the exact minimum complexity valid τ is unknown and estimated with +sampling error. However, even if a τ is chosen with slightly too little complexity, the resulting +inconsistency may still be small. The margin of sufficient complexity is at the border of the +left and central rectangle. When a τ with slightly too little complexity is chosen, a small +degree of inconsistency is incurred, but depending on the sample size possibly outweighed +in terms of mean squared error contribution by the associated standard deviation reduction. +This is an example of a small in-sample bias-variance tradeoff, while we otherwise focus on +identification to enable the construction of consistent estimators. +As the complexity of τ increases, at some point the instruments Z are no longer relevant +for treatment A conditional on T. Asymptotically, the estimator ˆJ no longer exists. This is +the case in the right rectangle of figure 1. In the extreme, τ is simply an identify function +and T = Z. No variation in Z remains to infer the effect of A on Y . However, even in less +extreme cases where there is some variation left in Z conditional on T, it may simply be +insufficient variation to be relevant for A. +3.3 +Specification Test +A straightforward way to ensure the sufficient complexity of some τ is to test W ⊥⊥ Z | T. +An alternative to this is a specification test, similar in spirit to specification testing in +overidentified IV models. Consider the two control functions τ1 and τ2, such that T1 = τ1(Z) +and T2 = τ2(Z). +Without loss of generality, let τ1 be less complex than τ2. +The null +hypothesis is the conditional exogeneity of Z given T1, +H0 : Z ⊥⊥ Y (a) | T1, with alternative Ha : Z ̸⊥⊥ Y (a) | T1. +Let ˆJ1 and ˆJ2 be the two causal estimators of estimand J based on τ1 and τ2. Suppose that +under both control functions, the instruments Z remain conditionally relevant for treatment +A, so that both estimators ˆJ1 and ˆJ2 have some probability limit. We also still assume that +conditional on U, the instruments Z are exogenous. The conditional exogeneity of Z given +U is assumed, because here we only test for the sufficient complexity of τ, i.e. Z ⊥⊥ U | T.1 +Under the null hypothesis H0, both estimators ˆJ1 and ˆJ2 converge to the true causal effect +J. However, the asymptotic variance of ˆJ2 with the more complex control function τ2 will be +larger than that of ˆJ1, as ˆJ2 uses less variation in Z than ˆJ2. Under the alternative Ha, the +estimators do not have the same probability limit. If τ2 still captures enough variation T2 +1To test whether some instruments Z are exogenous conditional on U, we can use a standard specification +test for different Z, if J is overidentified conditional on T. +7 + +in Z for the instruments Z to be conditionally exogenous, ˆJ2 still converges to J. ˆJ1 on the +other hand will no longer converge to J. Generally, there is no guarantee that T2 still renders +Z conditionally exogenous. In this case, ˆJ2 converges to some value other than J. However, +unless the additional variation that we condition on in T2 compared to T1 is exogenous due +to some particularly poor construction of τ2, ˆJ1 and ˆJ2 still have different probability limits. +A specification test using this logic is generally possible for the sufficient complexity of a +control function τ. +4 +Point Identification +Without further parametric restrictions on the outcome or first stage model, at most set +identification is possible. When the outcome model is linearly separable in the observables +and unobservables, we show how to point-identify the model part relating to the observ- +ables [Newey and Powell, 2003]. If instead the first stage is monotonous, a control function +approach can be used to point-identify average structural functions and thus causal effects +with a common support assumption (instead of completeness) [Imbens and Newey, 2009]. +We construct a control function for the endogenous variation in A while already keeping the +endogenous variation in Z fixed. +4.1 +Linearly separable outcome model +An outcome model with linear separability in the treatment and a disturbance is one special +case where point identification is possible. With a linearly separated disturbance ε, it is +straightforward to represent the exclusion of instrument Z as mean-independence conditional +on common confounders U. Assumption 4.1 fully describes this setting. +Assumption 4.1 (Linearly separable outcome model). There exists some function k0 ∈ +L2(A) such that +Y = Y (A) = k0(A) + ε, +E +�ε|Z, U +� = E +�ε|U +� . +(4.1) +The conditional moment describes the mean-independence of instruments Z conditional +on the unobserved common confounders U. +Theorem 4.1 (Identification in linearly separable model). Let assumptions 2.1.(1/2b/2c/3) +and 4.1 hold. There is a unique h ∈ L2(A, T) for which E +�Y |Z +� = E +�h(A, T)|Z +�, and it +satisfies h(A, T) = k0(A) + δT(T), where δT(T) = E +�ε|T +�. +8 + +Theorem 4.1 establishes point identification of the function k0 of the effect of the ob- +servable treatment A on outcome Y . While we do not make this explicit, k0 may also be +a function of the proxies W or other observed covariates X. +The linear separability in +combination with the completeness assumption leads to a straightforward identification in +the linearly separable model. Unlike in Tien [2022], identification of an average structural +function when there are interactions of the observables and unobservable U is much more +difficult in this model where the proxies W may also have a direct effect on treatment A. +We could have considered other model specifications or versions of completeness to es- +tablish identification [D’Haultfoeuille, 2011]. For now, we leave this exercise for future work. +4.2 +First stage monotonicity +If the outcome model is not linearly separable in treatment and disturbance, monotonicity +in the first stage reduced form is an alternative assumption to identify average causal (struc- +tural) effects [Imbens and Newey, 2009]. If the common confounders U were observed, there +would be a simple control function for the endogenous variation in A due to monotonicity. +Assumption 4.2 describes the necessary first stage reduced form monotonicity. +Assumption 4.2 (Monotonicity). +A = h(Z, η) +(4.2) +1. h(Z, η) is strictly monotonic in η with probability 1. +2. η is a continuously distributed scalar with a strictly increasing conditional CDF Fη|U +on the conditional support of η. +3. Z ⊥⊥ η | U. +Assumption 4.2.1 describes the strict montonicity of A in the disturbance η. This dis- +turbance η is scalar and continuously distributed conditional on the unobservable U, with +a strictly increasing conditional CDF according to assumption 4.2.2. Jointly, these two as- +sumptions ensure that for any given Z, any A is associated with a unique η. The unobserved +confounders U may affect A, but only through their effect on η. This restriction keeps the +model monotonous in the unobservables to ensure point identification. +Finally, assump- +tion 4.2.3 requires full independence of instruments Z and η conditional on the common +confounders U. +The above setup does not immediately help with identification, because the common +confounders U are always unobserved. In lemma 4.1.1, we establish a few useful facts about +9 + +the conditional distribution of the scalar disturbance η given T. Notably, this conditional +distribution Fη|T is also strictly increasing on the conditional support of η, and unsurprisingly +the instruments Z are independent from η conditional on T. +Lemma 4.1.1. +Fη|T := +� +U FA|Z,U(A, Z, u)fU|T(u, T) dµU(u) +is a strictly increasing CDF on the conditional support of η, and Z ⊥⊥ η | T. +The above lemma 4.1.1 implies that Fη|T(η) is a one-to-one function of η conditional on +T, just like Fη|U(η) conditional on U. This fact is useful, because it is no longer necessary +to condition on the unobservable U to identify the endogenous variation in A, which is η, +exactly. Instead, if we can identify Fη|T(η), η is held fixed as long as (Fη|T(η), T) is held +fixed. The remaining difficulty is to identify Fη|T(η). In this regard, theorem 4.2 states that +Fη|T(η) is equal to the conditional CDF of A given Z. This conditional CDF is defined as +VT in equation 4.3. +Theorem 4.2. Let +VT := FA|Z;T(A, Z). +(4.3) +Under assumption 4.2, VT = Fη|T(η), and +A ⊥⊥ Y (a) | (VT, T), for all a ∈ A. +(4.4) +Theorem 4.2 states that despite the unobservable common confounder U, there exist the +observable control functions VT and T conditional on which we retrieve unconfoundedness. +Specifically, we retrieve unconfoundedness because all variation in treatment A stems from +instruments Z once we condition on VT and T. Fortunately, conditional on T, the instruments +Z are fully exogenous. So far, our arguments have only been with respect to exogeneity, not +yet relevance. +To describe relevance, we use a common support assumption 4.3, with focus on a causal +effect of interest +J := +� +A Y (a)π(a) dµA(a). +This common support assumption requires the sufficient relevance of instruments Z for +treatment A, and sufficient variation in Z, both conditional on T. +In slightly different +10 + +words, after holding all variation in Z associated with U fixed through T, the variation in +Z must still be sufficiently rich and relevant for A. +Assumption 4.3 (Common Support). For all a ∈ A, where the contrast function is non- +zero (π(a) ̸= 0), the support of (VT, T) equals the support of (VT, T) conditional on A. +With the common support assusmption 4.3, average causal quantities J in our model +are identified under monotonicity (assumption 4.2). In theorem 4.3, we explicitly replace +the completeness assumption in assumption 2.1.2c by the common support assumption 4.3, +which is the correct relevance requirement with a control function. +Theorem 4.3 (Average causal quantity identification). Suppose assumption 2.1.(1/2a/2b/3) +[relaxed IV model], 4.2 [monotonicity], and 4.3 [common support] hold. Then, any J := +� +A Y (a)π(a) dµA(a) is identified as +J = +� +VT ,T +� +A E +�Y |A = a, (VT, T) = (vT, t) +� π(a) dµA(a) dFVT ,T(vT, t). +We simply integrate out the control functions (VT, T) without dependence on treatment +A to obtain the causal quantity of interest J. Typically, J will be some form of average +treatment effect. +Other functions of interest than the above (weighted) averages of potential outcomes, +e.g. quantile structural functions, are also identified as a consequence of theorem 4.2, but +require corresponding common support assumption which will differ from assumption 4.3. +5 +Linear Model +In this section, we explain identification in the common confounding model in linear terms. +Apart from the illustrative purpose of linear models, their tractability and ease of use make +them attractive. For the common confounding IV approach, the linear model provides useful +intuition regarding the relevance and exclusion assumptions. +First, we describe the model assumptions in linear form. The outcome variable Y ∈ R +is one-dimensional. For ease of notation, we let A ∈ R be one-dimensional too. All other +variables X are of some general dimensions dX, i.e. Z ∈ RdZ, W ∈ RdW , and U ∈ RdU. As +previously, instruments are called Z, proxies W, and the unobserved common confounders +U. +Y = Aβ + UγY + WυY + εY , +E +�εY |Z +� = 0 +(5.1) +A = Zζ + UγA + WυA + εA, +E +�εA|Z +� = 0 +(5.2) +11 + +Equation 5.1 simply states that Y is a linear function of A, U, W and a disturbance εY . +The disturbances εY are mean-independent from the instruments Z. With the conditional +moment equation, the model parameters would be identified under a conditonal relevance +requirement of Z for A if U could be observed. The parameter of interest in this model is +β, the effect of treatment A on Y . In equation 5.2, A is a linear function of Z, U, W, and +a disturbance εA. The dZ-dimensional vector of parameters ζ describes the marginal effect +of Z on A. Equation 5.2 for A describes the model’s first stage. The conditional relevance +requirement of Z for A would simply be ζ ̸= 0, if U were observed. With observable U, +the model would be sufficiently described at this point to point identify β. As the common +confounders U are never observed, the model requires further assumptions. +Z = UγZ + εZ, +E +�εZ|U, W +� = 0 +(5.3) +W = UγW + εW, +E +�εW|U, Z +� = 0 +(5.4) +Equations 5.3 and 5.4 imply that all correlation between Z and W stems from the unobserved +common confounders U. There is no direct effect from either on the other. If there were, +we could model this by increasing the dimension of U by the corresponding element of Z +or W until Z and W are uncorrelated conditional on U. Assumption 5.1 describes the rank +condition for the richness of W with respect to U. +Assumption 5.1 (Rank conditions for γW). +rank(γW) ≥ dU +(5.5) +This first rank condition 5.5 implies dW ≥ dU. For simplicity, suppose that rank(γW) = +dW = dU. Then, we can simply invert γW to write +E +�U|Z +� = E +�W|Z +� γ−1 +W . +The expected value of U given Z is proportional to the expected value of W given Z, as +long as the rank condition 5.5 for γW holds. With this result, we can write both E +�Y |Z +� +and E +�A|Z +� as functions of the random variables Z, E +�W|Z +�, and model parameters: +E +�Y |Z +� = Zζβ + E +�W|Z +� � +γ−1 +W γAβ + γ−1 +W γY + υAβ + υY +� +, +E +�A|Z +� = Zζ + E +�W|Z +� � +γ−1 +W γA + υA +� +. +From the above derivations, it follows that the parameter of interest β can be written in the +12 + +following ratio form: +β = +E +� +(Zζ) Y +��� E +�W|Z +�� +E +� +(Zζ) A +��� E +�W|Z +��. +(5.6) +As we found previously, the instruments Z are endogenous only due to the common con- +founders U. In the linear model, all endogeneity in Z stems from changes in E +�U|Z +�, as Z +varies. As discussed previously, E +�U|Z +� is held fixed with E +�W|Z +� as long as W is sufficiently +relevant for U as defined via the rank condition 5.5 in assumption 5.1. With the conditional +exclusion of Z established, the focus shifts to the conditional relevance of Z for A. E +�U|Z +� +is dU-dimensional, and thus is E +�W|Z +�. Given that the variation in E +�W|Z +� has dimension +dU, there is spare variation in Z to infer the causal effect J of A on Y , as long as dZ > dU. +After keeping the dU-dimensional variation in E +�W|Z +� fixed, the expected predicted values +of treatment A given instruments Z, E +� +Zζ| E +�W|Z +�� +, must be non-degenerate. The rank +condition for any dA is described in assumption 5.2. +Assumption 5.2 (Rank conditions for Zζ). +rank +� +E +� +(Zζ)A| E +�W|Z +��� += dA +(5.7) +Usually, this slightly involved rank condition can be understood more simply as +rank(ζ) − dU ≥ dA. +dU dimensions of variation in Z are lost by conditioning on E +�W|Z +�. The remaining variation +in Z after this conditioning step must be sufficiently relevant for A. +With its transparency, the linear model sheds light on the assumptions in IV with common +confounders. The same intuition for relevance and exclusion assumptions in the linear model +carries on to nonparametric models - specifically the idea of a pre-IV control function. In the +linear model, instrument variation is used conditional on the dU-dimensional control function +E +�W|Z +�, while in nonlinear settings the control function is a general τ ∈ L2(Z), which serves +the same purpose: To render the instruments Z conditionally independent from the proxies +W and hence from the unobserved common confounders U. Exclusion of the instruments Z +conditional on this control function is the desired consequence. +13 + +6 +Practical Guide +Straightforward testing and discussion of model assumptions is key in any application. In +this section, we provide a practical guide to identification with this approach. Each of the +four steps we describe has its own subsection. As in standard IV, the relevance assumptions +generally remain testable. +The conditional exclusion assumption is not testable up to a +specification test. +6.1 +Find T and test relevance of Z, W for U +In this step, we test the relevance of W for U (assumption 2.1.3), as well as the relevance of +Z for U. The latter is a necessary condition for the relevance of Z for A conditional on T +(assumption 2.1.2c), which is tested explicitly in subsection 6.2. +First, we find some T = τ(Z) such that Z ⊥⊥ W | T is satisfied. As long as some T = τ(Z) +leads to the conditional independence of Z and W, this control function τ ∈ L2(Z) also +renders Z and U conditionally independent. To test for the sufficient relevance of Z and W +with respect to U, we can use T. A sufficient condition for relevance of Z and W for U is that +both Z and W contain spare information conditional on T. This motivates the choice of a +valid τ ∈ T valid, which captures the least information about Z while ensuring the conditional +exclusion of Z conditional on T, as discussed in section 3. To simplify the argument, suppose +that our model is linear and the dimensions for (Z, T, W, U) are (dZ, dT, dW, dU). Often, +relevance of Z and W for U imply min{dZ, dW} ≥ dU. The minimum dimension of T to +ensure conditional independence of Z and W is dU, so we know that any T = τ(Z) such that +Z ⊥⊥ W | T must satisfy dT ≥ dU. If we can reject a test with the null hypothesis +H0 : min{dZ, dW} ≤ dT, and alternative Ha : min{dZ, dW} > dT, +this implies min{dZ, dW} > dU. Thus, there is a test whether Z and W are relevant for U. +However, min{dZ, dW} > dT is a sufficient, not a necessary condition for the relevance of Z +and W for U. Z and W are still relevant for U when min{dZ, dW} = dU. Unfortunately, +this hypothesis is not testable with unobserved U. So, how should an applied researcher +proceed when min{dZ, dW} = dT (which could mean min{dZ, dW} = dU)? Here, we need to +distinguish dZ = dT from dW = dT. +dZ = dT When T contains as much information as Z, there is no point in moving to step 2. +The instruments Z contain no variation conditional on T, so Z cannot be relevant for +treatment A conditional on T. +14 + +dW = dT We know for sure that dW ≤ dU. If dW = dU, the proxies W are exactly relevant for +U without spare information. If we are willing to assume dW = dU, we could move +forward to step 2. The variation in U associated with Z would still be held fixed with +T in this case. Yet, dW = dU is not testable. It may well be that dW < dU. Then, +the variation in U associated with Z is not held fixed with T. We can never test the +completeness of W for U when dW = dT. Accordingly, we do not generally suggest to +move to step 2 by relying on the assumption dW = dU, when dW = dT is observed. +6.2 +Test relevance of Z for A given T +In step 1, the relevance of Z, W for U was confirmed. Now, we test the relevance of the +instruments Z for treatment A conditional on T (assumption 2.1.2c). +Depending on the additional model assumptions, this is either a test of completeness +(2.1.2c), or common support (4.3). As any T = τ(Z) is simply the control function τ ∈ L2(Z) +applied to instruments Z, the test of relevance of Z for A given T is straightforward for any +given τ. It is as simple as a test of instrument relevance with observed confounders. Let +us return to a linear model. If all components of the instrument vector Z are correlated +with both U and A, the conditional relevance requirement simplifies to (dZ − dT) ≥ dA. +After conditioning on all variation in Z which correlates with U by holding T fixed, dZ − dT +dimensions of instruments Z remain to infer the causal effect of treatment A on outcome Y . +The remaining instrument variation of dimension dZ − dT is relevant for treatment A only if +the treatment’s dimension dA is smaller than or equal to dZ − dT. +If Z is found to be relevant for A given T, it implies that Z is relevant for both U and +A. Interpreting U as any source of unobserved variation which associates instruments Z and +proxies W, Z would have no variation conditional on T if they were not sufficiently relevant +for U (dZ < dU). So, if Z is relevant for A given T, it implies that Z already had to be +relevant for U. +6.3 +Exclusion of Z conditional on U +In step 1 and 2 we tested all relevance assumptions in this model. The conditional exclusion +assumption for Z conditional on U remains untestable. To be precise, Y ⊥⊥ Z | (A, U) +(assumption 2.1.2a) can only be justified on theoretic grounds, not observed data. In order +to justify conditional exclusion theoretically, T can be used to understand the unobserved +common confounders U. As T captures all variation in Z associated with U, T immediately +explains U in terms of its association with Z and W. From the association of T and W, we +can interpret U even better. For example: If subject-specific pre-college GPA measures are +15 + +used as instruments Z, and T turns out to capture average GPA, then U could be interpreted +as general ability. Suppose W contains dummies capturing whether someone has engaged in +risky behaviour, including drugs and illegal activity, while in high school. From theory and +empirical evidence we would expect high ability to lead to less risky behaviour. Thus, if T +is an average GPA, it would be expected to negatively correlated with W. +Once we used T to understand the variation reflected by the unobserved confounders +U, we can construct a theoretic argument with respect to the conditional exclusion of in- +struments Z. +In our example, the common confounder U reflected general ability. +The +conditional exclusion assumption reduces to whether conditional on general ability U, the +subject-specific pre-college GPA measures Z are excluded. This argument clearly depends +on the respective choice of treatment A and outcome Y . +As in standard IV, specification tests, which can be revealing about the exclusion of Z, +are possible if the model is overidentified. If a specification test suggests that different subset +of instruments Z conditional on T result in estimators with different probability limits, we +reject that all instruments Z are excluded conditional on T (unless the estimand is the local +average treatment effect which can vary across subpopulations). Necessary for any such test +is model overidentification for the causal effect J conditional on T. In a simple linear model, +overidentification would e.g. mean (dZ − dT) > dA. After keeping dT dimensions of Z fixed, +the instruments must still contain overidentifying information. +Ultimately, just as in standard IV, the conditional exclusion assumption for Z remains +largely untestable. Therefore, it is crucial to better understand U from the control variable +T. +6.4 +Estimation +In the final fourth step, use Z to instrument for treatment A conditional on control variable +T, to identify (and estimate) the structural function or average causal effect J of A on Y . +Having established that all necessary relevance and exclusion requirements hold, an estimator +ˆJ can be formulated for the causal effect of interest J. The form of this estimator depends +on the type of parametric model assumptions made. +7 +Example: Linear Returns to Education +Interested in the returns to education, we use data from the National Longitudinal Survey +of Youth 1997 [Bureau of Labor Statistics, 2019]. The variables of interest are introduced +below. +16 + +Y Household net worth at 35: continuous variable, in USD. +A BA degree: 1 if individual i obtained a BA degree, 0 otherwise. +Z Pre-college test results: subject-specific and overall GPA; ASVAB percentile. +W Risky behaviour dummies: whether i drank, smoked, or engaged in other behaviours +considered risky by the age of 17. +U Ability: Unmeasured intellectual capacity. +˜U Other biases: Selection on unobservables into obtaining a BA degree (at least in part +result of optimising individuals). +X Covariates: sex, college GPA, parental education/net worth, siblings, region, etc. +A review of the vast literature on returns to education is far beyond the scope of this +paper [Psacharopoulos and Patrinos, 2018]. Instead, we focus on estimation of a very specific +return to education: The causal effect of obtaining a bachelor’s degree A on household net +worth at 35. Even in a simple linear model like +Y = αY + Aβ + UγY + WυY + XηY + εY , +(7.1) +two distinct potential sources of confounding are easily identified via the unobservable com- +ponents of 7.1: +UγY Ability U likely has a positive effect on household net worth Y , by means of salary and +non-salary based net worth accumulation [Griliches, 1977]. The vector-valued linear +parameter γY captures this positive linear effect of ability on net worth. +εY The disturbance εY captures all variation in Y , which is jointly unexplained by (A, U, W, X). +This can be understood as individual-specific, heterogeneous characteristics, and chance. +Any correlation of A with either of these terms leads to biased estimates of β. +How does obtaining a BA degree A correlate with ability U and a general dis- +turbance εY ? +In this identification problem, selection bias in inherent. At least to some degree, individuals +choose whether to obtain a BA degree as a result of an optimisation problem of expected +17 + +utility subject to an information set I: +A = arg max +a∈{0,1} +� +E +�u(Y (a)) − c(a)|A = a, I +�� +, +(7.2) +where u : Y → R is a utility function for net worth with diminishing returns, and c : +{0, 1} → R a cost function for obtaining a BA degree A. Both utility and cost function can +be individual-specific. +For ease of illustration, suppose individuals are perfectly informed with I = (A, U, W, X, εY ). +Then, each individual chooses a ∈ {0, 1} to maximise the utility associated with potential +outcome Y (a) minus cost c(a). In this case, there is an easy decision rule to determine +optimal A: +A = arg max +a∈{0,1} +u(Y (a)) − c(a), += 1 +�u(Y (1)) − u(Y (0)) > c(1) − c(0) +� . +UγY Ceteris paribus, an increase in ability U equally increases Y (0) and Y (1) according to +model 7.1. Due to diminishing returns in the utility function u, u(Y (1)) − u(Y (0)) +decreases. However, also c(1) decreases as higher-ability individuals experience a lower +utility cost of obtaining a BA degree. The overall effect on the choice of A is ambiguous +and depends on the utility functions of the individual. +εY The effect of εY on the choice of A, on the other hand, is unambiguous. An increase +in εY reduces u(Y (1)) − u(Y (0)) due to the diminishing returns of u. Cost c, however, +is unaffected by εY . Hence, A inevitably negatively correlates with εY . +This logic regarding negative selection bias when treatment is chosen by utility-maximising +individuals is by no means novel [Heckman et al., 2006], or unique to the returns to education +identification problem. Negative selection bias is inherent to the treatment variable when +it is at least in part the result of optimising behaviour by utility-maximising heterogeneous +individuals. Novel in our approach is the ability to explicitly account for certain biases, in +this case ability bias, when proxies for them exist. Finding excluded instruments can be +much more straightforward when pertinent biases, like ability bias, have already been taken +care of. +In our identification approach, instruments Z are pre-college test results. These results +are strongly correlated with ability U. Yet, conditional on ability, and some other covariates, +pre-college test results contain random variation, which is excluded with respect to household +net worth Y (at age 35). Concurrently, even random variation in pre-college test results is +18 + +a strong predictor of obtaining a BA degree. Hence, instrument relevance likely holds. The +proxies W are dummies for whether an individual engaged in risky behaviours at high school +age. Among others, the risky behaviour dummies include drinking, smoking (marijuana), +selling drugs and stealing. Theory and empirical evidence suggest the correlation of low +intelligence and risky behaviour [Loeber et al., 2012]. +Therefore, ability U both causes +instruments Z and proxies W in our data. Ability U is the common confounder in this +causal question. Clearly, additional covariates are necessary to justify instrument exclusion. +These include sex, college GPA, parental education and net worth, the number of siblings, +region of residence, etc. +7.1 +Assumptions +The linear equivalent to the general common confounding IV model in assumption 2.1 is +described as assumption 7.1. Again, for ease of notation assume dA = 1, just as in this +returns to education identification problem. +Assumption 7.1 (Linear IV Model with Common Confounding). +1. Linear outcome model projection: +Y = αY + Aβ + UγY + WυY + XηY + εY +(7.3) +2. Instruments +(a) Exclusion: E +�εY (Z, U, W, X) +� = 000. +(b) Relevance: For the linear projection of A on (Z, U, W, X), +A = αA + Zζ + UγA + WυA + XηA + εA, E +�εA(Z, U, W, Z) +� = 000 +(7.4) +rank +� +E +� +(Zζ) A +�� T, X +�� += dA. +(7.5) +3. Proxies +(a) Exclusion: For the linear projection of W on (Z, U, X) and (Z, X), +W = αW + UγW + XηW + εW, +E +�εW(Z, U, X) +� = 000, +(7.6) +W = ˜αW + Z˜γW + X˜ηW + ˜εW, +E +�˜εW(Z, X) +� = 000. +(7.7) +with T := Z˜γW + X˜ηW. +19 + +(b) Relevance: rank(γW) ≥ dU +. +(7.8) +To simplify notation, let Z|X be the true residual of a projection of Z onto X. The +linearity of the outcome model implies that the covariance Cov +� +(Z|Xζ), Y +� +is +Cov +� +(Z|Xζ), Y +� += Cov +� +(Z|Xζ), A +� +β + Cov +� +(Z|Xζ), U +� +γY + Cov +� +(Z|Xζ), W +� +υY . +The above expression uses the uncorrelatedness of Z and εY in assumption 7.1.2a. If it were +not for the linear confounding from the unobserved common confounders U and proxies W, +Z would be excluded. Next, we demonstrate how to use the proxies W to keep Cov +�(Zζ)U +� +fixed. +Cov +� +(Z|Xζ), U +� += Cov +� +(Z|Xζ), W +� +γ⊺ +W +� +γWγ⊺ +W +�−1 +The inverse +� +γWγ⊺ +W +�−1 exists under assumption 7.1.3 that the rank of γW is at least dU. +Then, slightly rewriting Cov +�(Zζ)Y +� as +Cov +� +(Z|Xζ), Y +� += Var +� +Z|Xζ +� +β + Cov +� +(Z|Xζ), W +� +˜υW, +˜υY := υY + υAβ + γ⊺ +W +� +γWγ⊺ +W +�−1 (γY + γAβ) , +implies that any endogeneity of residualised instruments Z|X is controlled for by conditioning +on Z˜γW from the linear projection 7.7. To be precise, +Cov +� +(Z|Xζ), Y |Z˜γW +� += Var +� +Z|Xζ|Z˜γW +� +β + Cov +� +(Z|Xζ)W|Z˜γW +� +� +�� +� +=0 +˜υY . +The covariance of the first stage can be rewritten as +Cov +� +(Z|Xζ), A|Z˜γW +� += Var +� +Z|Xζ|Z˜γW +� ++ Cov +� +(Z|Xζ)W|Z˜γW +� +� +�� +� +=0 +˜υA. +Using both of these results, and one-dimensional treatment A to simplify notation, a simple +20 + +ratio form for the linear effect of A on outcome Y is +β = +Cov +�� +Z|Xζ +� +Y | Z˜γW +� +Cov +�� +Z|Xζ +� +A | Z˜γW +� = Cov +�(Zζ) Y | T, X +� +Cov +�(Zζ) A | T, X +�, with T = Z˜γW. +(7.9) +Hence, the estimator differs from standard IV based estimation only by also holding a linear +prediction T of W fixed as the partial predicted values Zζ for A change. Thus, the relevance +requirement 7.1.2b for the instruments Z is conditional on T and X. T can be represented +by a dU-dimensional linear function of Z, E +�U|Z, X +�, multiplied by γW. Hence, a simpler +way to understand the relevance requirement 7.1.2b is as +rank(ζ) ≥ (dU + dA) +(7.10) +A total of dU dimensions of variation in Z are typically needed to account for the dU- +dimensional confounding effect of U via E +�U|Z, X +�, while the remaining variation in Z still +needs to be relevant for A. Other than in trivial cases2, equation 7.10 describes this relevance +requirement satisfactorily as a rank condition on ζ, the partial linear projection effect of Z +on A conditional on (U, W, X). +7.2 +Find T and test relevance of Z, W for U +A valid control function is the linear prediction T = Z˜γW under assumption 7.1.3, meaning +that conditional on (T, X), instruments Z are still relevant for A. However, its OLS estimate +T = Zˆ˜γW generally is not a valid control function, because T and Z are perfectly correlated +due to sampling variation, unless dZ > dW. However, even when dZ > dW, the true ˜γW will +have rank dU ≤ dW, while its OLS estimate ˆ˜γW always has possibly larger than necessary +rank dW due to sampling variation. Ultimately, the estimate ˆ˜γW should at best have exactly +rank dU. A test is needed for the rank r0 of matrix ˜γW. If E +�U|Z, X +� = ZγZ + XγX, then +˜γW = γZγW. Sufficient for the rank condition in assumption 7.1.3 is r0 < min{dZ, dW}. This +condition means that an unobservable variable of smaller dimension than both W and Z can +explain all correlation between W and Z conditional on X. This unobserved variable is the +common confounder U. By the definition of U as the (minimum information) unobserved +variable which renders W and Z mean-independent, γZ has dU ≤ dZ linearly independent +rows (rank(γZ) = dU). As γW has dimensions dU × dW and dU ≤ dW, rank(γW) ≤ dU and +thus r0 = rank(γZγW) = rank(γW). +While r0 < dW suffices to confirm the relevance of W for U in assumption 7.1.3, r0 < dZ +2e.g. when Z contains perfectly collinear variation conditional on (U, W, X). +21 + +is necessary for Z to be relevant for treatment A in assumption 7.1.2b. A suitable test for +some r < min{dZ, dW} has null hypothesis +H0 : r0 ≤ r, and alternative Ha : r0 > r. +(7.11) +With the OLS estimator ˆ˜γW, we apply a bootstrap based test for its rank. First, write the +singular value decomposition as +˜γW = +P0 +dZ×dZ +Π0 +dZ×dW +Q⊺ +0 +dW ×dW +(7.12) +Then, let φr(A) := �mA +j=r+1 π2 +j(A) be the sum of squared singular values of A from the (r +1) +largest to the smallest singular value, which is the mA-th singular value, where mA is the +minimum across A’s number of rows and columns. Then, an equivalent test to 7.11 is a test +with null hypothesis +H0 : φr (˜γW) = 0, and alternative Ha : φr (˜γW) > 0. +(7.13) +The bootstrap procedure is as follows: +1. For each binary proxy Wj ∈ W, calculate the probability pj := Pr (Wj = 1) under H0 +as pj,0 = Logit +� +� +� +� +�Z P0,r +dZ×r +Π0,r +r×r +Q⊺ +0,r,j +r×1 ++ X ˜ηW,j +dX×1 +� +� β0 + α0 +� +� +�, where +(a) P0,r corresponds to the first r columns of P0, +(b) Q0,j corresponds to the first r entries of the j-th row of Q0 +(c) Π0,r corresponds to the r × r matrix of of the first r rows and columns of Π0, +(d) ˜ηW,j corresponds to j-th column of ˜ηW, +(e) β0 and α0 are univariate coefficients, which need to be estimated. +2. Draw 1000 new bootstrap samples b ∈ B of binary proxies as W b +0 using the n × dW +probability matrix (p0,0, p1,0, . . . , pdW ,0). +3. For each bootstrap sample b ∈ B: Calculate the sample projection coefficient ˆ˜γb +W,0 by +projecting W b +0 onto (Z, X) (all demeaned), and the sum of its smallest squared singular +values starting at the (r + 1) largest as φr,0,b := φr +�ˆ˜γb +W,0 +� +. +4. Obtain the p-value as 1 − +1 +|B| +� +b∈B 1 +� +φr,0,b < φr(˜γW) +� +. +22 + +Figure 2: Bootstrap based test for H0 : rank(˜γW) = r0 ≤ r +Notes: This figure illustrates the bootstrap distribution of the test statistic nφr +� +ˆ˜γW +� +in the test with null +hypothesis H0 : rank(˜γW ) = r0 ≤ r. The left figure depicts the test statistic distribution under r = 0, and +the left under r = 1. For r = 0, the p-value is zero. H0 : r0 = 0 is strongly rejected. For r = 1, the p-value +is 0.933. H0 : r0 ≤ 1 cannot be rejected at any meaningful level of significance. +In figure 2, the bootstrapped distributions of the test statistic nφr +�ˆ˜γW +� +are depicted +under two different null hypotheses: r0 = 0 and r0 ≤ 1. Non-rejection of the test is evidence +in favour of the low rank r0 of ˜γW. In the left diagram of figure 2, where the test concerns +r0 = 0, the p-value is at zero. The test provides strong evidence against r0 = 0, which +indicates some correlation between W and Z conditional on X. The right diagram of figure +2 depicts the test statistic bootstrap distribution for H0 : r0 ≤ 1, and provides strong +evidence against rejection. The associated p-value is 93.3%. Thus, we can conclude that +the rank of γW is at most one. In the NLS97 data, pre-college test results Z and risky +behaviour dummies have dimensions dZ = 7 and dW = 9. Thus, r0 ≤ 1 allows the conclusion +that the common confounder dimension is small: dU ≤ 1. Conditional on covariates X, all +covariance between Z and W is explained by a one-dimensional unobserved U. Successfully, +the proximal assumption 7.1.3 was tested. In addition, the necessary dZ > dU condition for +conditional instrument relevance (assumption 7.1.2b) was confirmed. +7.3 +Test relevance of Z for A given T +Despite satisfying the necessary dZ > dU condition for IV relevance (assumption 7.1.2b), +a proper test for the conditional relevance of Z for A given the control function T is still +missing. In this step, we first explain how to construct the here one-dimensional control +variable T after having conducted the tests in section 7.3. Then, we test for the conditional +relevance of instrument Z for treatment A given this control function T. +23 + +Ho : ro = 0 +Ho : ro < 1 +95% CV +95% CV +0.002 +0.004 +0.006 +0.008 +0.010 +0.001 +0.002 +0.003 +0.004 +0.005Given the statistical evidence in favour of dU ≤ 1, we construct the variable +T +N×1 := +Z +N×dZ +ˆP0,1 +dZ×1 +ˆΠ0,1 +1×1 +, +(7.14) +with the singular value decomposition of the OLS estimator ˆ˜γW = ˆP0 ˆΠ0 ˆQ⊺ +0. ˆP0,1 is the first +column of ˆP0, and ˆΠ0,1 is the top-left entry of ˆΠ0. Aside from sampling error, proxies W are +mean-independent from instruments Z conditional on (T, X). +With the control T now defined, we can use a bootstrap based test to confirm the relevance +of instruments Z for A conditional on (T, X). The null hypothesis can be formulated as +H0 : rank +� +E +�(Zζ)A|T, X +�� +< dA, with alternative Ha : rank +� +E +�(Zζ)A|T, X +�� += dA. +(7.15) +Importantly, under H0 the effect of Z on A (given X) would be fully described by a one- +dimensional T, as the dimension of U was found to be r0 ≤ 1 in section 7.2. When treatment +A is one-dimensional, a simple test for this null hypothesis compares the R2 of an unrestricted +(7.16) and restricted regression (7.17). +A = ˜αA,ur + Z ˜ζ + X˜ηA,ur + ˜εA,ur, +E +� +˜εA,ur|Z, X +� += 0, +(7.16) +A = ˜αA,r + T ˜γA + X˜ηA,r + ˜εA,r, +E +� +˜εA,r|T, X +� += 0. +(7.17) +Under H0, both regressions would predict A equally well, despite the dimension reduction +on Z in the second regression, 7.17. With the uncertainty in estimated T, we use a sim- +ple bootstrap-based test. +With 1000 bootstrap samples bt ∈ Bt, we obtain a bootstrap +distribution of R2 +r. Under H0, R2 +ur is (asymptotically) distributed as R2 +r. +Figure 3: Test for Relevance of Z for A given T (H0 : rank +� +E +�Zζ|T, X +�� +< dA) +Notes: This figure illustrates the bootstrap distribution of the restricted R2 +r in regression 7.17. The dimension +of T,dT = 1, is based on the test in section 7.3. +24 + +Bootstrap distribution of R2 +R2 +95% CV +0.18 +0.20 +0.22 +0.24 +0.26 +0.28 +0.30Figure 3 depicts the bootstrap distribution of R2 +r based on the restricted regression 7.17. +The control variable T is constructed for each bootstrap sample as described in 7.14. The +unrestricted R2 +ur based on the unrestricted regression 7.16 fits the data significantly better, +which indicates rejection of H0. The p-value is 0.023. There is predictive information in +Z for A, beyond that controlled for in (T, X). In other words, Z satisfies the conditional +instrument relevance requirement 7.1.2b. +7.4 +Exclusion of Z conditional on U +While both relevance assumptions 7.1.3 and 7.1.2b could be tested successfully, the exclusion +of instrument Z conditional on U in assumption 7.1.2a remains generally untestable. +To argue whether Z is exogenous conditional on U, it is worth asking: Which information +is being held fixed in T, and what does this imply about U? The linear construction of T +from Z is illustrated in table 1, where the instruments have been normalised to standard +deviation one. Z is mean-independent from W conditional on (T, X). T mostly consists of +an average of subject-specific pre-college GPA measures. In this sense, T closely measures +academic ability, as captured by pre-college GPA measures. Without the transcript GPA +and ASVAB percentile, the subject-specific GPA measures describe 94.5% of variation in T. +Despite the negative dependence of T on ASVAB percentile in its construction, T positively +correlates with ASVAB percentile unconditional on the GPA measures with a 0.31 correla- +tion coefficient. The interpretation of T and consequently U is pretty straightforward: It +positively reflects (academic) ability. +Table 1: Construction of T = Z ˆP0,1 ˆΠ0,1 +GPA +ASVAB +English +Math +SocSci +LifeSci +percentile +T +0.537 +0.216 +0.280 +0.214 +-0.262 +As U reflects (academic) ability, an increase in T is expected to result in a reduction of +risky behaviour [Loeber et al., 2012]. Indeed, a one standard deviation increase in T reduces +the probability of having engaged in risky behaviour by the age of 17 between 3% and +9%, as illustrated in table 2. All effects have strong statistical and economic significance. +Compared to the average probability of engaging in risky behaviour, the estimated effect +of a one standard deviation change in T is largest for some of the riskiest behaviour we +considered: selling drugs (-54%), running away (-46%), and attacking someone (-41%). +T captures the information we expected based on our suspicion about the unobserved +confounder ability. T closely reflects (academic) ability as measured by high-school GPA +25 + +Table 2: Effect of T on W +try +run +attack +sell +destroy +steal +steal +drink +smoke +marijuana +away +someone +drugs +property +< 50$ +> 50$ +Pr +65.5% +47.1% +29.8% +10.9% +19.0% +9.1% +32.0% +38.1% +8.0% +T +-7.9% +-9.5% +-9.6% +-5.0% +-7.8% +-4.9% +-3.5% +-4.7% +-3.2% +Notes: The table contains sample probabilities for engaging in risky behaviour by the age of 17 in the Pr +row. The estimated decrease in the probability of engaging in risky behaviour from a linear probability +model for a one standard deviation increase in T is noted in the row corresponding to T. +measures, which reduces the probability of engaging in risky behaviours during high-school. +Thus, we can conclude that the common confounder U contains the unobserved variable +ability. +Now, an argument is required for the conditional exogeneity of instruments Z given +unobserved ability U and observed covariates X: +Y = αY + Aβ + UγY + WυY + XηY + εY , +E +�εY | Z, X +� = 0. +While ability is the obvious confounder of the effect of pre-college GPA measures on net worth +later in life, there are other possible confounders. Among everyone who goes to college, those +with higher pre-college GPA are likely to also have a higher college GPA. Even conditional on +whether someone obtained a BA degree, a higher college GPA likely leads to higher earnings +later in life. Thus, college GPA is an important observed confounder. Family and individual +net worth at young age can affect pre-college GPA measures as more learning resources are +available. Their effect on net worth later in life is undeniable. Apart from net worth, other +family background characteristics likely affect both pre-college test scores and net worth later +in life. We include parental education, maternal age at first birth and the individual’s birth, +as well as the number of siblings to capture family background characteristics. Individual- +specific characteristics are other important confounders. +We include sex and citizenship +status based on birth as further covariates. Conditional on this rich set of covariates X, +and the unobserved variable ability U, there is no reason to believe that pre-college test +scores Z would affect or be correlated with post-college earnings through any other channel +than obtaining a BA degree A. Despite our best efforts in explaining U, and the provided +arguments in favour of assumption 7.1.2a, a test or conditional instrument exclusion is not +possible. +A specification test is not feasible, because in this example the model is not +overidentified. +26 + +7.5 +Estimation +Estimation of the fully linear model is now straightforward. As in Tien [2022], we call the +estimator an instrumented common confounding (ICC) estimator. +ˆβICC = +� +A⊺PZMT,XA +�−1 � +A⊺PZMT,XY +� +. +(7.18) +Here, PZ = Z (Z⊺Z)−1 Z⊺ is the projection matrix of Z, and MT,X = In − PT,X is the +annihilator matrix of (T, X). In table 3, the estimates of four major methods are compared: +ordinary least squares (OLS), instrumental variables (IV), proximal learning (PL), and the +here suggested ICC estimator. The row corresponding to T describes the estimated partial +effect of T (normalised to standard deviation one) on net worth Y (at 35) in the respective +regressions. T is only used in proximal learning and ICC, but derived from the covariation of +(Z, A) and W in negative control [Cui et al., 2020], as opposed to Z and W in our approach. +The row corresponding to A contains estimates for β, the causal effect of obtaining a BA +degree A on net worth Y (at 35). Their unit is US Dollar. +Table 3: Estimates with different estimators (Y in thousands (k)) +OLS +PL +IV +ICC +A +59.18*** +30.90*** +222.97*** +125.15** +(9.12) +(10.40) +(34.74) +(52.93) +T +27.76*** +16.05** +(4.82) +(7.37) +Notes: The table contains estimates and their standard errors (in paran- +theses) for β in the A row, and the linear parameter on T if used in the +method from four estimators: Ordinary Least Squares (OLS), Proximal +Learning (NC), Instrumental Variables (IV), and Instrumented Common +Confounding (ICC). Asterisks indicate significance at the 1% (***), 5% +(**) and 10% (*) level. +OLS estimates that obtaining a BA degree increases net worth at 35 by 59k$. +The +proximal learning estimator conditions on its own T, so implicitly anything fixed that covaries +(Z, A) and proxies W. +The proximal learning estimate at 31k$, is indeed economically +significantly smaller than the OLS estimate. +As hypothesised, this might indicate that +unobserved ability, which correlates (Z, A) and W, is a confounder which biases the estimated +effect of education on net worth upwards. In contrast, the IV estimate is much larger at +223k$. +The inherent negative selection bias may thus be quite large. +However, the IV +27 + +estimator ignores the strong correlation of the pre-college test score instruments Z with +ability U, which may lead to an accentuated ability bias compared to that in OLS. As the +estimator should be robust to ability bias, we condition on T and obtain the ICC estimator +at 125k$. Indeed, conditioning on T attenuates the estimate by the expected ability bias. +As relevance is not strongly satisfied for the instruments Z conditional on T in the ICC +estimator, the standard error is expectedly large for this method. Still, both general selection +bias and ability bias appear to be strong confounders in this difficult identification problem. +Quantitatively separating ability and general selection bias helps add the necessary cred- +ibility to IV, which misses under the original IV exclusion assumption. +8 +Conclusion +In this work, we relax instrument exclusion in the presence of mismeasured confounders. +Other observed variables, the proxies, must be relevant for the unobserved confounders, +which cause endogeneity in the instruments. The mild parametric index sufficiency assump- +tion is also required. Importantly, the proxies can be economically meaningful variables, +with their own effects on treatment and outcome. This method can be useful in various +causal identification problems with observational data, where the unobserved confounders +are otherwise unrestricted observed variables. The linear returns to education identification +problem illustrates how this method can identify causal effects when instrument exclusion, +as often in practice, is a strong and hardly testable assumption. +This paper established two point identification results. +When point identification is +impossible, this approach can still identify informative bounds on causal effects. This set +identification exercise is left to future work. Further, we have not demonstrated how to +construct estimators other than in the linear example. Uncertainty in the control function +estimation will be reflected in the performance of any estimator using this identification +approach. +The integration of this approach, which at best identifies averages of causal +effects across unobservables, with marginal treatment effects, is another remaining task. +28 + +References +Richard W Blundell and James L Powell. Endogeneity in semiparametric binary response +models. The Review of Economic Studies, 71(3):655–679, 2004. +U.S. Department of Labor Bureau of Labor Statistics. National longitudinal survey of youth +1997 cohort, 1997-2017 (rounds 1-18), 2019. +Yifan Cui, Hongming Pu, Xu Shi, Wang Miao, and Eric Tchetgen Tchetgen. Semiparametric +proximal causal inference. arXiv preprint arXiv:2011.08411, 2020. +Xavier D’Haultfoeuille. On the completeness condition in nonparametric instrumental prob- +lems. Econometric Theory, 27(3):460–471, 2011. +Xavier D’Haultfœuille, Stefan Hoderlein, and Yuya Sasaki. 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Treatment effect estimation with noisy conditioning variables. arXiv +preprint arXiv:1811.00667, 2018. +Whitney K Newey and James L Powell. Instrumental variable estimation of nonparametric +models. Econometrica, 71(5):1565–1578, 2003. +Whitney K Newey, James L Powell, and Francis Vella. Nonparametric estimation of trian- +gular simultaneous equations models. Econometrica, 67(3):565–603, 1999. +George Psacharopoulos and Harry Anthony Patrinos. Returns to investment in education: +a decennial review of the global literature. Education Economics, 26(5):445–458, 2018. +Christian Tien. Instrumented common confounding. 2022. +Emmanuel Selorm Tsyawo. +Feasible iv regression without excluded instruments. +arXiv +preprint arXiv:2103.09621, 2021. +30 + +9 +Proofs +Section 3 proofs +Proof of lemma 3.0.1. Let t ∈ T . +fW,Z|T(W, Z|T) = +� +U fW,Z|U,T(W, Z|u, T)fU|T(u, T) dµU(u) += +� +U fW|Z,U,T(W|Z, u, T)fZ|U,T(Z|u, T)fU|T(u, T) dµU(u) += +� +U fW|U(W|u)fZ|T(Z|T)fU|T(u, T) dµU(u) += fZ|T(Z|T) +� +U fW|U(W|u)fU|T(u, T) dµU(u) += fZ|T(Z|T)fW|T(W|T) +From line two to three we use W ⊥⊥ Z | U (assumption 2.1.3a) and U ⊥⊥ Z | T (by +construction of T). From line four to five we again use W ⊥⊥ τ(Z) | U (assumption 2.1.3a). +Proof of lemma 3.0.2. We write fW|Z(W, Z) in two separate ways using T and relate those. +1. +fW|Z(W, Z) = +� +U fW|U(W, u)fU|Z(u, Z) dµU(u) += +� +U fW|U(W|u)fU|T,Z(u, t, z) dµU(u) +The first line follows from W ⊥⊥ Z | U (assumption 2.1.3a). +2. +fW|Z(W, Z) = fW|T(W, T) += +� +U fW|U(W, u)fU|T(u, t) dµU(u) +The first line follows from the construction of T = τ(Z), τ ∈ L2(Z), such that W ⊥⊥ +Z | T. +By the equality of the expressions in steps 1 and 2 above, for all z ∈ Z, +� +U fW|U(W, u)fU|T,Z(u, t, z) dµU(u) = +� +U fW|U(W, u)fU|T(u, t) dµU(u), +� +U fU|T,Z(u, t, z)fW(W) +fU(u) fU|W(u, W) dµU(u) = +� +U fU|T(u, t)fW(W) +fU(u) fU|W(u, W) dµU(u). +31 + +Let gt,z(U) := +fU|T,Z(U,t,z) +fU(U) +and gt(U) := +fU|T (U,t) +fU(U) +for any z ∈ Z. Then, +E +� +gt,z(U)fW(W)|W +� += E +�gt(U)fW(W)|W +� +E +�� +gt,z(U) − gt(U) +����� W +� +fW(W) = 0 +By completeness of W for U (assumption 2.1.3b), the above only holds if +gt,z(U) = gt(U). +This implies +fU|T,Z(U, t, z) = fU|T(U, t), +and thus U ⊥⊥ Z | T for any T = τ(Z), τ ∈ L2(Z) such that W ⊥⊥ Z | T. +Section 4.1 proofs +Proof of Theorem 4.1. +E +�Y |Z +� = E +� +k(A) + ε +�� Z +� += E +�k(A)|Z +� + E +� +E +�ε|U, Z +� |Z +� += E +�k(A)|Z +� + E +� +E +�ε|U +� |Z +� += E +�k(A)|Z +� + E +� +E +�ε|U +� |T +� += E +�k(A)|Z +� + E +�ε|T +� +From line two to three we use the moment E +�ε|Z, U +� = E +�ε|U +� formulated in assumption +4.1. From line three to four we use U ⊥⊥ Z | T and T = τ(Z) (assumption 2.1.2b/2c). +Required for this step is the identifiability of T, given by lemma 3.0.1 subject to assumption +2.1.3. +Completeness of Z for A given T (assumption 2.1.2c) ensures that there is a unique +h ∈ L2(A, T) for which E +�Y |Z +� = E +�h(A, T)|Z +�, where +h(A, T) = k(A) + E +� +E +�ε|U +� |T +� += k(A) + E +�ε|T +� . +32 + +Section 4.2 proofs +Proof of lemma 4.1.1. +Fη|T(η) = +� +U Fη|u(η)fU|T(u, T) dµU(u), +∂Fη|T(e) +∂e +����� +e=η += +� +U +∂Fη|u(e) +∂e +����� +e=η +� +�� +� +>0 ∀u,η +fU|T(u, T) dµU(u) > 0, +so Fη|T(η) is strictly increasing on the conditional support of η. +Now prove Z ⊥⊥ η | T. +fZ,η|T = +� +U fZ,η|U,TfU|T dµU(u) += +� +U fη|Z,U,TfZ|U,TfU|T dµU(u) += +� +U fη|U,TfZ|TfU|T dµU(u) += fZ|T +� +U fη|U,TfU|T dµU(u) += fZ|Tfη|T +From second to third line we use Z ⊥⊥ η | U (assumption 4.2.3) and Z ⊥⊥ U | T (assumption +2.1.4). We have shown that Z ⊥⊥ η | T. +Proof of theorem 4.2. First, we derive a preliminary result as if U were observed. +FA|Z,U(a, z, u) = Pr +�A ≤ a|Z = z, U = u +� = Pr +�h(z, η) ≤ a|Z = z, U = u +� += Pr +� +η ≤ h−1(a, z)|Z = z, U = u +� += Pr +� +η ≤ h−1(a, z)|U = u +� += Fη|u +� +h−1(a, z) +� += Fη|u (η) . +From line one to two we use the invertibility of h(z, η) (assumption 4.2.(1/2)). From line +33 + +two to three we use Z ⊥⊥ η | U (assumption 4.2.3). Then, +VT := FA|Z(A, Z) = +� +U FA|Z,U(A, Z, u)fU|T(u, T) dµU(u) += +� +U Fη|U (η) fU|T(u, T) dµU(u) += Fη|T(η) +On line one we use Z ⊥⊥ U | T (assumption 2.1.2c). From line one to two, we use the +result derived at the beginning of this proof. The final line again follows from τ(Z) ⊥⊥ η | U +(assumption 4.2.3). Hence, +VT = FA|Z(A, Z) = Fη|T(η). +By lemma 4.1.1 (strictly increasing Fη|T on the support of η), (η, T) and (VT, T) = (Fη|T(η), T) +are associated with the same sigma algebra. +We show A ⊥⊥ Y (a) | (VT, T). +fY (a),A|VT ,T(Y (a), A|VT, T) += +� +U fY (a)|A,VT ,T,U(Y (a)|h(Z, η), VT, T, u) +� +�� +� +=fY (a)|VT ,T,u(Y (a)|VT ,T,u) +fA|T,VT ,U(h(Z, η)|VT, T, u) +� +�� +� +=fA|VT ,T (h(Z,η)|VT ,T) +fU|T(u|T) dµU(u) += fA|VT ,T(h(Z, η)|VT, T) +� +U fY (a)|VT ,T,u(Y (a)|VT, T, u)fU|T(u|T) dµU(u) +� +�� +� +=fY (a)|VT ,T (Y (a)|VT ,T) += fA|VT ,T(A|VT, T)fY (a)|VT ,T(Y (a)|VT, T) +=⇒ +A ⊥⊥ Y (a) | (VT, T) +34 + +Proof of theorem 4.3. J := E +�� +A Y (a)π(a) dµA(a) +� +is identified as +� +VT ,T +� +A E +�Y |A = a, (VT, T) = (vT, t) +� π(a) dµA(a) dFVT ,T(vT, t) +E +�� +A E +�Y (a)|A = a, VT, T +� π(a) dµA(a) +� +E +�� +A E +�g(A, ε)|A = a, VT, T +� π(a) dµA(a) +� +E +�� +A +� +E g(A, ε) dFε|A,VT ,T(ε, a, VT, T)π(a) dµA(a) +� +E +�� +A +� +E g(A, ε) dFε|VT ,T(ε, VT, T)π(a) dµA(a) +� +E +�� +E +� +A g(A, ε)π(a) dµA(a) dFε|VT ,T(ε, VT, T) +� +E +�� +A g(A, ε)π(a) dµA(a) +� +E +�� +A Y (a)π(a) dµA(a) +� += J. +We let Y (a) = g(a, ε) for some ε ∈ E and g ∈ L2(A, ε) from line two to three. +Then, +A ⊥⊥ Y (a) | (VT, T) from theorem 4.2 implies A ⊥⊥ ε | (VT, T), which we use from line four +to five. All other steps are algebra. +10 +Data Description +The sample consists of 1,983 individuals. +Y : Household net worth at 35 (Z9141400) +Household net worth was top-coded at 600,000$ and bottom-coded at -300,000$. 7.0% of +individuals were top-coded, 0.3% bottom-coded. +A: Bachelor’s degree obtained (Z9084400) +If there is a date of obtaining a bachelor’s degree (Z9084400 ≥ 0 or invalid skip −3), A = 1. +50.0% of individuals in the sample have obtained a BA degree. +Z: Pre-college ability measures +Instruments are credit-weighted high-school GPAs in English (R9872000), Math (R9872200), +Social Sciences (R9872300) and Life Sciences (R9872400), as well as the ASVAB percentile +in each individual’s respective age group. +35 + +Figure 4: Histogram for Y +Notes: Distribution of household net worth. +W: Pre-college risky behaviour +The proxies consist of risky behaviour dummies. They equal one when an individual has +engaged in behaviour considered ”risky” by age 17 or earlier if missing. +Table 4: Probability of engaging in risky behaviour W by 17 (in %) +mari- +run +attack +sell +destroy +steal +steal +drink +smoke +juana +away +someone +drugs +property +< 50$ +> 50$ +Pr +65.5 +47.1 +29.8 +10.9 +19.0 +9.1 +32.0 +38.1 +8.0 +X: Individual, family, and regional covariates +The proxies consist of risky behaviour dummies. They equal one when an individual has +engaged in behaviour considered ”risky” by age 17 or earlier if missing. +36 + +400 +300 - +200 - +00T +0 +300000 +200000 +100000 +0 +100000 +200000 +300000 +400000 +500000 +600000Figure 5: Histogram for high-school GPA +Notes: Distribution of household high-school GPAs. +Figure 6: Histogram for ASVAB percentile in 3-month age cohort +Notes: Distribution of ASVAB percentiles. +37 + +English +Math +300 +200 +100 +-0 +Social Sciences +Life Sciences +300 +200 +00T +0 - +0 +100 +200 +300 +400 +0 +00T +200 +300 +400140 +120 +00T +08 +60 +40 - +20 - +0 +0 +20 +40 +09 +08 +100Table 5: Correlation matrix of pre-college risky behaviour W (in %) +mari- +run +attack +sell +destroy +steal +steal +drink +smoke +juana +away +someone +drugs +property +< 50$ +> 50$ +drink +100 +49 +41 +18 +15 +21 +24 +28 +14 +smoke +49 +100 +49 +21 +20 +28 +27 +33 +17 +marijuana +41 +49 +100 +26 +24 +43 +29 +35 +21 +run away +18 +21 +26 +100 +27 +22 +20 +17 +23 +attack +15 +20 +24 +27 +100 +27 +28 +24 +18 +sell drugs +21 +28 +43 +22 +27 +100 +30 +27 +27 +destroy property +24 +27 +29 +20 +28 +30 +100 +39 +21 +steal < 50$ +28 +33 +35 +17 +24 +27 +39 +100 +29 +steal > 50$ +14 +17 +21 +23 +18 +27 +21 +29 +100 +Table 6: Covariates in first stage +variable +NLS97 basis +info +replace invalid by +hh net worth 1997 +R1204700 +35% quantile (8834.35) +male +R0536300 +46.6% +citizen birth non us +R1201300 +2.3% (not US-born) +citizen birth other +R1201300 +13.2% (not det.) +bio mom age first birth +R1200100 +median (23) +bio mom age subject birth +R1200100 +median (26) +bio mom educ +R1302500 +median (13) +bio dad educ +R1302400 +median (12) +relation parent figure +R1205300 +both biological parents (1) +parent religious +R1486900 +median (3) +n siblings +T6745900 +median (2) +region north central at 17 +R1200300 (i.a.) +27.8% +region south at 17 +R1200300 (i.a.) +36.3% +region west at 17 +R1200300 (i.a.) +21.6% +urban at 17 +R1217500 (i.a.) +69.8% +other non rural at 17 +R1217500 (i.a.) +0.3% +non english home +R0551900 +14.1% +0 +poor english interview +R2394903 +0.3% +0 +38 + +Table 7: Additional covariates in outcome model +variable +NLS97 basis +info +replace invalid by +net worth age 20 +Z9048900 +35% quantile (2737.7) +gpa college +B0004600 +400+ as 400 +region north central at 35 +U0001900 (i.a.) +24.5% +region south at 35 +U0001900 (i.a.) +38.7% +region west at 35 +U0001900 (i.a.) +23.2% +urban at 35 +U0015000 (i.a.) +81.9% +other non rural at 35 +U0015000 (i.a.) +0.4% +39 + +Table 8: OLS coefficients key variables +OLS +NC +IV +ICC +const +-32.36 +-30.21 +46.85 +13.52 +(37.41) +(36.21) +(43.38) +(45.10) +T +27.76*** +16.05** +(4.82) +(7.37) +A +59.18*** +30.90*** +222.97*** +125.15** +(9.12) +(10.40) +(34.74) +(52.93) +Notes: The table contains estimates and their standard errors (in paran- +theses) for β in the A row, and the linear parameter on T if used in the +method from four estimators: Ordinary Least Squares (OLS), Proximal +Learning (NC), Instrumental Variables (IV), and Instrumented Common +Confounding (ICC). Asterisks indicate significance at the 1% (***), 5% +(**) and 10% (*) level. +40 + +Table 9: OLS coefficients individual- and family-covariates +OLS +NC +IV +ICC +hh net worth 1997 +0.15*** +0.16*** +0.12*** +0.14*** +(0.03) +(0.03) +(0.03) +(0.03) +net worth age 20 +0.35*** +0.36*** +0.35*** +0.36*** +(0.09) +(0.08) +(0.09) +(0.08) +gpa college +16.00*** +11.58*** +-6.01 +2.86 +(3.44) +(4.00) +(6.19) +(6.94) +male +30.19*** +24.33*** +40.32*** +32.06*** +(8.39) +(7.94) +(9.14) +(9.05) +n siblings +-6.29*** +-6.78*** +-6.08** +-6.47*** +(1.73) +(2.16) +(2.37) +(2.21) +citizen birth non us +-5.50 +1.61 +-16.48 +-7.42 +(27.65) +(28.22) +(30.87) +(29.11) +citizen birth other +34.53*** +37.24*** +29.66** +33.18** +(13.24) +(12.88) +(14.09) +(13.27) +bio mom age first birth +2.28** +2.60** +1.11 +1.80 +(1.00) +(1.03) +(1.15) +(1.13) +bio mom age subject birth +-0.08 +-0.06 +-0.01 +0.00 +(0.55) +(0.57) +(0.62) +(0.58) +bio mom educ +2.91 +3.21* +-0.16 +1.53 +(1.95) +(1.77) +(2.04) +(2.03) +bio dad educ +1.84 +3.22* +-2.52 +0.38 +(1.84) +(1.71) +(2.06) +(2.32) +relation parent figure +-8.81*** +-10.29*** +-5.41** +-7.78*** +(2.22) +(2.44) +(2.75) +(2.82) +parent religious +-0.03 +-0.03 +-0.04 +-0.03 +(0.03) +(0.03) +(0.03) +(0.03) +non english home +-10.64 +-12.37 +-9.47 +-11.23 +(13.01) +(13.20) +(14.40) +(13.43) +poor english interview +75.41 +56.96 +78.91 +65.72 +(112.29) +(78.33) +(85.58) +(79.79) +Notes: The table contains estimates and their standard errors (in parantheses) for slope coeffi- +cients on individual and family covariates using four estimators: Ordinary Least Squares (OLS), +Proximal Learning (NC), Instrumental Variables (IV), and Instrumented Common Confounding +(ICC). Asterisks indicate significance at the 1% (***), 5% (**) and 10% (*) level. +41 + +Table 10: Slope coefficients regional covariates +OLS +NC +IV +ICC +urban 17 +1.08 +-1.48 +5.09 +1.59 +(10.04) +(9.68) +(10.64) +(10.01) +urban 35 +-30.42*** +-28.21*** +-38.53*** +-32.79*** +(11.48) +(10.89) +(12.00) +(11.47) +other non rural 17 +24.13 +19.02 +42.62 +30.34 +(23.31) +(24.07) +(26.55) +(25.25) +other non rural 35 +-92.49 +-82.51 +-48.20 +-66.35 +(72.08) +(66.52) +(73.14) +(68.57) +region north central 17 +43.59* +42.51** +38.27* +40.92** +(22.91) +(20.27) +(22.13) +(20.68) +region north central 35 +-28.57 +-27.57 +-11.03 +-19.45 +(24.64) +(21.07) +(23.28) +(22.11) +region south 17 +16.45 +14.74 +8.67 +11.96 +(18.82) +(18.22) +(19.92) +(18.66) +region south 35 +-22.18 +-20.87 +-1.67 +-11.54 +(19.98) +(18.40) +(20.49) +(19.65) +region west 17 +27.39 +25.91 +24.62 +25.23 +(22.03) +(19.97) +(21.79) +(20.34) +region west 35 +16.58 +15.37 +34.70 +24.20 +(22.98) +(20.06) +(22.19) +(21.22) +Notes: The table contains estimates and their standard errors (in parantheses) for +slope coefficients on regional covariates using four estimators: Ordinary Least Squares +(OLS), Proximal Learning (NC), Instrumental Variables (IV), and Instrumented +Common Confounding (ICC). Asterisks indicate significance at the 1% (***), 5% +(**) and 10% (*) level. +42 + +Table 11: Slope coefficients risky behaviour proxies +OLS +IV +ever drank 17 +14.44 +12.56 +(10.10) +(10.63) +ever smoked 17 +-2.40 +7.39 +(10.01) +(10.90) +ever marijuana 17 +-9.03 +-0.38 +(10.90) +(12.10) +ever ran away 17 +-8.52 +-12.39 +(12.91) +(14.74) +ever attack 17 +-1.71 +15.39 +(10.46) +(12.54) +ever sell drugs 17 +-22.44 +-19.73 +(15.87) +(17.27) +ever destroy property 17 +3.86 +2.57 +(9.89) +(10.59) +ever steal bit 17 +-1.29 +-4.15 +(9.50) +(10.30) +ever steal lot 17 +-20.98 +-11.98 +(15.45) +(17.12) +Notes: The table contains estimates and their standard errors +(in parantheses) for slope coefficients on risky behaviour proxies +using four estimators: Ordinary Least Squares (OLS), Proximal +Learning (NC), Instrumental Variables (IV), and Instrumented +Common Confounding (ICC). Asterisks indicate significance at +the 1% (***), 5% (**) and 10% (*) level. +43 + diff --git a/ctA0T4oBgHgl3EQfGv-w/content/tmp_files/load_file.txt b/ctA0T4oBgHgl3EQfGv-w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e2c4c8e80b4732cc1edad196eb016f7929cd657 --- /dev/null +++ b/ctA0T4oBgHgl3EQfGv-w/content/tmp_files/load_file.txt @@ -0,0 +1,1491 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf,len=1490 +page_content='Relaxing Instrument Exclusion with Common Confounders Christian Tien ∗ January 6, 2023 Abstract Instruments can be used to identify causal effects in the presence of unobserved con- founding, under the famous relevance and exclusion assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As exclusion is difficult to justify and to some degree untestable, it often invites criticism in applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hoping to alleviate this problem, we propose a novel identification approach, which relaxes traditional IV exclusion to exclusion conditional on some unobserved common confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We assume there exist some relevant proxies for the unobserved common confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Unlike typical proxies, our proxies can have a direct effect on the endogenous regressor and the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We provide point identification results with a linearly separable outcome model in the disturbance, and alternatively with strict monotonicity in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Using this novel method with NLS97 data, we demonstrate the insignificant role of ability bias compared to general selection bias in the economic returns to education problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Beyond economics, the approach is just as relevant in health treatment evaluation with an unobserved underlying health status, or a psychological study where character traits are unobserved common confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Keywords: Causal Inference, Unobserved Confounding, Instrumental Variables, Control Function, Proximal Learning ∗ct493@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Faculty of Economics, University of Cambridge arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='02052v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='EM] 5 Jan 2023 1 Introduction Unobserved confounding complicates the identification of a causal effect of a regressor of interest on an outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Despite the endogeneity of a regressor of interest, instrumental vari- able (IV) approaches can identify their causal effect if the famous relevance and exclusion assumptions hold for the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' These assumptions are strong and often invite criti- cism of IV estimates in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We propose a novel approach to relax exclusion, in favour of exclusion conditional on an unobserved common confounder, for which some relevant variables are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Relaxing exclusion (or exogeneity) is only possible when it is replaced by other strong assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' One way to identify causal effects without instrument exclusion is from resid- ual distributions, not variation in the explanatory variables [Heckman, 1979, Millimet and Tchernis, 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Very specific forms of heteroskedasticity across the first stage and outcome model can also be used to establish identification without an exclusion restriction [Klein and Vella, 2010, Lewbel, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Others have suggested to use irrelevant variation in instru- ments to test for the exclusion of the relevant variation in the instruments [D’Haultfœuille et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A similar idea is followed when integrated conditional moments use nonlinear mean-dependence of endogenous variables on instruments, such that the instruments may violate the exclusion restrictions in pre-specified parametric ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Despite recent advances in estimation with integrated conditional moments [Tsyawo, 2021], the strong identifying assumptions render all these approaches difficult to justify in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Our solution differs significantly from these approaches as it only uses a relaxed exclusion restriction and variation in explanatory variables to identify the causal effect of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From the perspective of IV, we allow for some endogeneity in the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' That endogeneity originates from unobserved common confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We call these unobserved confounders common, because there are some observed variables, which are relevant for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' These observed variables are called proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In other words, we assume there are some unobserved variables that explain all correlation (association) between the instruments and these proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This assumption is testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, we need to argue for the exclusion of the instruments, but only conditional on the unobserved common confounders (and observable variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In general, this is a strong relaxation of instrument exclusion conditional on observed variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Another way to understand our proposal is as a solution to measurement error in observed confounders for IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Residual bias is a well-known problem when confounders are measured with error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Proximal learning [Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2020] is a solution to the problem, where observed variables measure all unobserved confounders with some error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In proximal learning, the 1 proxies for the unobserved confounder may either be causes of the treatment or outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' These proxies must be sufficiently relevant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' complete) for all unobserved con- founders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Separately developed from the proximal learning approach, a control function solution exists with identical conditional independence assumptions and mismeasured con- founders [Nagasawa, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Our solution is different, as we do not assume the existence of measurements for all confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instead, we use instruments and assume that measure- ments exist for all confounders, conditional on which the instruments would be exogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this sense, our solution can be understood as IV with mismeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As it is standard in the control function literature, our approach will identify average causal (structural) quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Unless the outcome model is fully linearly separa- ble in the treatment and disturbance [Newey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 1999], where in our case the disturbance includes the effect of the common confounder, we identify those average causal (structural) quantities of interest that integrate out the unobservables without dependence on the treat- ment using a control function [Imbens and Newey, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Our identification approach is most similar to recent advances in nonlinear panels [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In panel data, unob- served fixed effects are common to the same variables across time, in a similar way as the unobserved common confounders are common to the instruments and proxies in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' [2021], identification stems from a parametric dimension reduction of the effect of observed variables on the outcome, and an index sufficiency assumption that renders the observed variables independent from the fixed effects conditional on an index of the observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In our approach, identification stems from the existence of more instruments than treatments, and an index sufficiency assumption that renders the instruments independent from the unobserved common confounders conditional on an index of the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Just like Blundell and Powell [2004], Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' [2021] do not explain how to derive this crucial index function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' One of our main contributions is the derivation of the index function, which arises naturally in the common confounding setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A motivating example for our proposal is the returns to college education identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' It features various biases, ability and selection, and we motivate pre-college test scores as instruments exogenous to selection, while clearly endogenous to ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' proxies are pre-college risky behaviour dummies, which appear to correlate negatively with ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With NLS97 data, we show that selection bias is the much more economically relevant bias compared to ability bias in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 2 2 Setup The treatment (action) A ∈ A is discrete or continuous with base measure µA of A ⊆ RdA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Y ∈ Y ⊆ R is the one-dimensional outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Other important variables are the instruments Z ∈ Z ∈ RdZ, the proxies W ∈ W ⊆ RdW , and the common confounders U ∈ U ⊆ RdW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 (Common Confounding IV Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' SUTVA: Y = Y (A, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instruments (a) Exclusion: Y (a, z) = Y (a) ⊥⊥ (A, Z) | U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (b) Index sufficiency: For some τ ∈ L2(Z), where T := τ(Z), U ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (c) Relevance (completeness): For any g(A, T) ∈ L2(A, T), E �g(A, T)|Z � = 0 only when g(A, T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Proxies (a) Exclusion: W ⊥⊥ Z | U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (b) Relevance (completeness): For any g(U) ∈ L2(U), E �g(U)|W � = 0 only when g(U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2) Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 is the standard stable unit treatment value assumption (SUTVA), which implies no interference across units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a we capture the key relaxation of this model compared to standard IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' It states that the instruments are excluded, yet this exclusion may be conditional on an unobserved (vector-valued) random variable U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This is a significant relaxation of the standard exclusion restriction, which is possible only with assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b, we introduce a control function τ and a control variable T = τ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Conditional on the control variable T, the instruments Z are independent from the common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This assumption describing the existence the control function τ is often called index sufficiency, where T is a (multiple) index of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c, we require that conditional on the control variable T, the instruments Z are complete for treatment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This is a standard completeness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' It simply means that keeping the variation of Z described by T fixed, the instruments must remain sufficiently relevant for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In slightly different words, after conditioning on T, enough variation must be left in the instruments Z to infer the effect of treatment A on outcome Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As in standard 3 IV with observed confounders, this relevance requirement is typically testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3a states that the proxies W are independent from instruments Z conditional on the common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The proxies W must also be complete for the unobserved common confounders U, as stated in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Again, completeness means the proxies W are sufficiently relevant for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A different way to understand these assumptions is that the unobserved variable U, which explains all association (correlation) between the proxies W and instruments Z, renders the instruments exogenous when observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this sense, W can be (possibly quite poor) proxies for what we consider the unobserved common confounder U, as long as they are sufficiently relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Conditional on W, the instruments Z are still endogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Common confounders U are never observed, and W could be quite poor proxies for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Yet, we prove that conditioning on a control function T, which makes the instruments Z and proxies W independent, restores the exclusion of instruments Z which holds conditional on the unobserved U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 3 Learning the Confounding Structure In this section, we describe the main idea of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Using only observable information, we find a control variable T, conditional on which the instruments Z are independent from the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We then explain what may be considered the optimal control variable T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 Learning a Control Function The control function τ ∈ L2(Z), described in lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1, generates the control variable T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This control variable renders the instruments Z independent from the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Logically, if the instruments Z and the proxies W are independent condi- tional on U, it follows that any such control variable T also renders Z and W independent conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assume W ⊥⊥ Z | U (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Take any τ ∈ L2(Z), where T := τ(Z), such that U ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, also W ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' One possible such control variable is T = Z, yet this would leave no remaining variation in Z to instrument for A conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Also, lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 does not provide a way to identify any control function τ apart from a function which captures the same information as Z itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For this purpose, we need lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this lemma, we establish that any T = τ(Z), conditional on which the instruments Z and proxies W are independent, also renders Z conditionally independent from the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 4 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assume W ⊥⊥ Z | U (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3a), and for any g(U) ∈ L2(U), E �g(U)|W � = 0 only when g(U) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Take any τ ∈ L2(Z), where T := τ(Z), such that W ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, also U ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Unlike lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1, the conclusion of lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 is not obvious and requires the com- pleteness of proxies W for the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Again, completeness means that the proxies W must be sufficiently relevant for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If this were not the case, it would be impossible to keep all variation in Z that is associated with U fixed, using a control variable T derived only using information about the association of Z and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We can interpret U as all unobserved confounders that associate (correlate) Z and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 is an important result, because it allows the identification of a control function that does not capture all variation in instruments Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For any T that we identify conditional on which instruments Z and proxies W are independent, the instrument exclusion assumption can be relaxed to exclusion conditional on all unobservables U that associate (correlate) Z and W (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The parallels to standard IV are quite clear: The conditional exclusion assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a is untestable, yet relaxed compared to standard IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The relevance requirement of Z for A conditional on T (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c) is testable, yet stricter compared to standard IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The requirement for relevance of Z for A conditional on T implies that only a subset of control functions τ ∈ L2(Z), which leave enough relevant variation in Z conditional on T, enable model identification under assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1: T valid := � τ ∈ L2(Z) : �W ⊥⊥ Z | τ(Z) � and � E �g(A, τ(Z)|Z � = 0 only when g(A, τ(Z)) = 0 �� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1) As both defining relevance conditions of this set T valid are testable, its non-emptiness is testable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 Optimal Control Function Under assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1, the optimal control function τ ∗ ∈ T valid out of the set of valid control functions captures the minimum feasible information in Z in a sense of minimising the variance of the asymptotically unbiased estimator ˆJ of some causal estimand J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Figure 1 illustrates schematically how the bias and variance of the IV estimator conditional on the control variable T depend on the complexity of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In figure 1, the complexity of T = τ(Z) on the x-axis increases from T = 0 on the extreme left to T = Z on the extreme right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Moving further to the right on the x-axis means that the control function captures more information in Z, starting with variation 5 Figure 1: Implied typical estimator properties with control functions τ of varying complexity Notes: This figure illustrates some typical properties of an estimator ˆJ of the causal effect of a treatment A on an outcome Y , using instruments Z while conditioning on a control function T = τ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Moving to the right on the x-axis, the control function captures more information in Z, starting with variation in Z which correlates with the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the left rectangle, the control function is too simple to render Z exogenous conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, the estimator is inconsistent, yet the degree of inconsistency decreases as the complexity of the control function T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the central rectangle, the control function captures enough information for Z to be excluded conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' So, the estimator is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the right rectangle, the control function is too complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Conditional on T, the instruments Z are no longer sufficiently relevant for A and the estimator ˆJ asymptotically does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the first two rectangles, the asymptotic variance of the estimator ˆJ increases with the complexity of the control function, because conditional on T, less information in Z is used to infer the effect of A on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' in Z which correlates with the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the left rectangle, the complexity of τ is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' T does not capture all information in Z that correlates with U, so even conditional on T the instruments Z remain endogenous, and the estimator ˆJ is inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, as all information in Z is used for inference, the asymptotic variance of the estimator ˆJ will be relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As the complexity of τ increases towards the right in the left quadrant, more information about the elements of Z which correlate with U is captured in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Increasing the complexity of τ increases the asymptotic variance of ˆJ as less information in Z is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Importantly, as this information corresponds to variation in Z that correlates with U, inconsistency is being reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the central rectangle of figure 1, the complexity of τ is sufficient for Z to be exogenous conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, the estimator ˆJ is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, as τ increases in complexity, we use less information in Z to infer the causal estimand J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, inevitably the asymptotic variance of the estimator ˆJ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Consequently, the optimal τ would be that of minimal 6 UKZIT UIZIT UIZIT Z relevant Z relevant Z not relevant plim(IJ JI) avar(J) T=0 T=Z T = T(Z) captures more information in Zcomplexity, such that Z is excluded conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In practice, we do not know but can only estimate T valid, so the exact minimum complexity valid τ is unknown and estimated with sampling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, even if a τ is chosen with slightly too little complexity, the resulting inconsistency may still be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The margin of sufficient complexity is at the border of the left and central rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' When a τ with slightly too little complexity is chosen, a small degree of inconsistency is incurred, but depending on the sample size possibly outweighed in terms of mean squared error contribution by the associated standard deviation reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This is an example of a small in-sample bias-variance tradeoff, while we otherwise focus on identification to enable the construction of consistent estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As the complexity of τ increases, at some point the instruments Z are no longer relevant for treatment A conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Asymptotically, the estimator ˆJ no longer exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This is the case in the right rectangle of figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the extreme, τ is simply an identify function and T = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' No variation in Z remains to infer the effect of A on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, even in less extreme cases where there is some variation left in Z conditional on T, it may simply be insufficient variation to be relevant for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 Specification Test A straightforward way to ensure the sufficient complexity of some τ is to test W ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' An alternative to this is a specification test, similar in spirit to specification testing in overidentified IV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Consider the two control functions τ1 and τ2, such that T1 = τ1(Z) and T2 = τ2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Without loss of generality, let τ1 be less complex than τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The null hypothesis is the conditional exogeneity of Z given T1, H0 : Z ⊥⊥ Y (a) | T1, with alternative Ha : Z ̸⊥⊥ Y (a) | T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Let ˆJ1 and ˆJ2 be the two causal estimators of estimand J based on τ1 and τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Suppose that under both control functions, the instruments Z remain conditionally relevant for treatment A, so that both estimators ˆJ1 and ˆJ2 have some probability limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We also still assume that conditional on U, the instruments Z are exogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The conditional exogeneity of Z given U is assumed, because here we only test for the sufficient complexity of τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z ⊥⊥ U | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 Under the null hypothesis H0, both estimators ˆJ1 and ˆJ2 converge to the true causal effect J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, the asymptotic variance of ˆJ2 with the more complex control function τ2 will be larger than that of ˆJ1, as ˆJ2 uses less variation in Z than ˆJ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Under the alternative Ha, the estimators do not have the same probability limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If τ2 still captures enough variation T2 1To test whether some instruments Z are exogenous conditional on U, we can use a standard specification test for different Z, if J is overidentified conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7 in Z for the instruments Z to be conditionally exogenous, ˆJ2 still converges to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' ˆJ1 on the other hand will no longer converge to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Generally, there is no guarantee that T2 still renders Z conditionally exogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this case, ˆJ2 converges to some value other than J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, unless the additional variation that we condition on in T2 compared to T1 is exogenous due to some particularly poor construction of τ2, ˆJ1 and ˆJ2 still have different probability limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A specification test using this logic is generally possible for the sufficient complexity of a control function τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 4 Point Identification Without further parametric restrictions on the outcome or first stage model, at most set identification is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' When the outcome model is linearly separable in the observables and unobservables, we show how to point-identify the model part relating to the observ- ables [Newey and Powell, 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If instead the first stage is monotonous, a control function approach can be used to point-identify average structural functions and thus causal effects with a common support assumption (instead of completeness) [Imbens and Newey, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We construct a control function for the endogenous variation in A while already keeping the endogenous variation in Z fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 Linearly separable outcome model An outcome model with linear separability in the treatment and a disturbance is one special case where point identification is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With a linearly separated disturbance ε, it is straightforward to represent the exclusion of instrument Z as mean-independence conditional on common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 fully describes this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 (Linearly separable outcome model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' There exists some function k0 ∈ L2(A) such that Y = Y (A) = k0(A) + ε, E �ε|Z, U � = E �ε|U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1) The conditional moment describes the mean-independence of instruments Z conditional on the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 (Identification in linearly separable model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Let assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (1/2b/2c/3) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' There is a unique h ∈ L2(A, T) for which E �Y |Z � = E �h(A, T)|Z �, and it satisfies h(A, T) = k0(A) + δT(T), where δT(T) = E �ε|T �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 8 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 establishes point identification of the function k0 of the effect of the ob- servable treatment A on outcome Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' While we do not make this explicit, k0 may also be a function of the proxies W or other observed covariates X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The linear separability in combination with the completeness assumption leads to a straightforward identification in the linearly separable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Unlike in Tien [2022], identification of an average structural function when there are interactions of the observables and unobservable U is much more difficult in this model where the proxies W may also have a direct effect on treatment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We could have considered other model specifications or versions of completeness to es- tablish identification [D’Haultfoeuille, 2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For now, we leave this exercise for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 First stage monotonicity If the outcome model is not linearly separable in treatment and disturbance, monotonicity in the first stage reduced form is an alternative assumption to identify average causal (struc- tural) effects [Imbens and Newey, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If the common confounders U were observed, there would be a simple control function for the endogenous variation in A due to monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 describes the necessary first stage reduced form monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 (Monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A = h(Z, η) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' h(Z, η) is strictly monotonic in η with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' η is a continuously distributed scalar with a strictly increasing conditional CDF Fη|U on the conditional support of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z ⊥⊥ η | U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 describes the strict montonicity of A in the disturbance η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This dis- turbance η is scalar and continuously distributed conditional on the unobservable U, with a strictly increasing conditional CDF according to assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Jointly, these two as- sumptions ensure that for any given Z, any A is associated with a unique η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The unobserved confounders U may affect A, but only through their effect on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This restriction keeps the model monotonous in the unobservables to ensure point identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Finally, assump- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 requires full independence of instruments Z and η conditional on the common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The above setup does not immediately help with identification, because the common confounders U are always unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1, we establish a few useful facts about 9 the conditional distribution of the scalar disturbance η given T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Notably, this conditional distribution Fη|T is also strictly increasing on the conditional support of η, and unsurprisingly the instruments Z are independent from η conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Fη|T := � U FA|Z,U(A, Z, u)fU|T(u, T) dµU(u) is a strictly increasing CDF on the conditional support of η, and Z ⊥⊥ η | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The above lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 implies that Fη|T(η) is a one-to-one function of η conditional on T, just like Fη|U(η) conditional on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This fact is useful, because it is no longer necessary to condition on the unobservable U to identify the endogenous variation in A, which is η, exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instead, if we can identify Fη|T(η), η is held fixed as long as (Fη|T(η), T) is held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The remaining difficulty is to identify Fη|T(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this regard, theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 states that Fη|T(η) is equal to the conditional CDF of A given Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This conditional CDF is defined as VT in equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Let VT := FA|Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='T(A, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3) Under assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2, VT = Fη|T(η), and A ⊥⊥ Y (a) | (VT, T), for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 states that despite the unobservable common confounder U, there exist the observable control functions VT and T conditional on which we retrieve unconfoundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Specifically, we retrieve unconfoundedness because all variation in treatment A stems from instruments Z once we condition on VT and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Fortunately, conditional on T, the instruments Z are fully exogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' So far, our arguments have only been with respect to exogeneity, not yet relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' To describe relevance, we use a common support assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3, with focus on a causal effect of interest J := � A Y (a)π(a) dµA(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This common support assumption requires the sufficient relevance of instruments Z for treatment A, and sufficient variation in Z, both conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In slightly different 10 words, after holding all variation in Z associated with U fixed through T, the variation in Z must still be sufficiently rich and relevant for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 (Common Support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For all a ∈ A, where the contrast function is non- zero (π(a) ̸= 0), the support of (VT, T) equals the support of (VT, T) conditional on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With the common support assusmption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3, average causal quantities J in our model are identified under monotonicity (assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3, we explicitly replace the completeness assumption in assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c by the common support assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3, which is the correct relevance requirement with a control function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 (Average causal quantity identification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Suppose assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (1/2a/2b/3) [relaxed IV model], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 [monotonicity], and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 [common support] hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, any J := � A Y (a)π(a) dµA(a) is identified as J = � VT ,T � A E �Y |A = a, (VT, T) = (vT, t) � π(a) dµA(a) dFVT ,T(vT, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We simply integrate out the control functions (VT, T) without dependence on treatment A to obtain the causal quantity of interest J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Typically, J will be some form of average treatment effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Other functions of interest than the above (weighted) averages of potential outcomes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' quantile structural functions, are also identified as a consequence of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2, but require corresponding common support assumption which will differ from assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 5 Linear Model In this section, we explain identification in the common confounding model in linear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Apart from the illustrative purpose of linear models, their tractability and ease of use make them attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For the common confounding IV approach, the linear model provides useful intuition regarding the relevance and exclusion assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' First, we describe the model assumptions in linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The outcome variable Y ∈ R is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For ease of notation, we let A ∈ R be one-dimensional too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' All other variables X are of some general dimensions dX, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z ∈ RdZ, W ∈ RdW , and U ∈ RdU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As previously, instruments are called Z, proxies W, and the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Y = Aβ + UγY + WυY + εY , E �εY |Z � = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1) A = Zζ + UγA + WυA + εA, E �εA|Z � = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2) 11 Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 simply states that Y is a linear function of A, U, W and a disturbance εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The disturbances εY are mean-independent from the instruments Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With the conditional moment equation, the model parameters would be identified under a conditonal relevance requirement of Z for A if U could be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The parameter of interest in this model is β, the effect of treatment A on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2, A is a linear function of Z, U, W, and a disturbance εA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The dZ-dimensional vector of parameters ζ describes the marginal effect of Z on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 for A describes the model’s first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The conditional relevance requirement of Z for A would simply be ζ ̸= 0, if U were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With observable U, the model would be sufficiently described at this point to point identify β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As the common confounders U are never observed, the model requires further assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z = UγZ + εZ, E �εZ|U, W � = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3) W = UγW + εW, E �εW|U, Z � = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4) Equations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4 imply that all correlation between Z and W stems from the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' There is no direct effect from either on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If there were, we could model this by increasing the dimension of U by the corresponding element of Z or W until Z and W are uncorrelated conditional on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 describes the rank condition for the richness of W with respect to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 (Rank conditions for γW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' rank(γW) ≥ dU (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5) This first rank condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5 implies dW ≥ dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For simplicity, suppose that rank(γW) = dW = dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, we can simply invert γW to write E �U|Z � = E �W|Z � γ−1 W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The expected value of U given Z is proportional to the expected value of W given Z, as long as the rank condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5 for γW holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With this result, we can write both E �Y |Z � and E �A|Z � as functions of the random variables Z, E �W|Z �, and model parameters: E �Y |Z � = Zζβ + E �W|Z � � γ−1 W γAβ + γ−1 W γY + υAβ + υY � , E �A|Z � = Zζ + E �W|Z � � γ−1 W γA + υA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From the above derivations, it follows that the parameter of interest β can be written in the 12 following ratio form: β = E � (Zζ) Y ��� E �W|Z �� E � (Zζ) A ��� E �W|Z ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='6) As we found previously, the instruments Z are endogenous only due to the common con- founders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the linear model, all endogeneity in Z stems from changes in E �U|Z �, as Z varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As discussed previously, E �U|Z � is held fixed with E �W|Z � as long as W is sufficiently relevant for U as defined via the rank condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5 in assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With the conditional exclusion of Z established, the focus shifts to the conditional relevance of Z for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' E �U|Z � is dU-dimensional, and thus is E �W|Z �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Given that the variation in E �W|Z � has dimension dU, there is spare variation in Z to infer the causal effect J of A on Y , as long as dZ > dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' After keeping the dU-dimensional variation in E �W|Z � fixed, the expected predicted values of treatment A given instruments Z, E � Zζ| E �W|Z �� , must be non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The rank condition for any dA is described in assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 (Rank conditions for Zζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' rank � E � (Zζ)A| E �W|Z ��� = dA (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='7) Usually, this slightly involved rank condition can be understood more simply as rank(ζ) − dU ≥ dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' dU dimensions of variation in Z are lost by conditioning on E �W|Z �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The remaining variation in Z after this conditioning step must be sufficiently relevant for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With its transparency, the linear model sheds light on the assumptions in IV with common confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The same intuition for relevance and exclusion assumptions in the linear model carries on to nonparametric models - specifically the idea of a pre-IV control function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the linear model, instrument variation is used conditional on the dU-dimensional control function E �W|Z �, while in nonlinear settings the control function is a general τ ∈ L2(Z), which serves the same purpose: To render the instruments Z conditionally independent from the proxies W and hence from the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Exclusion of the instruments Z conditional on this control function is the desired consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 13 6 Practical Guide Straightforward testing and discussion of model assumptions is key in any application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this section, we provide a practical guide to identification with this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Each of the four steps we describe has its own subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As in standard IV, the relevance assumptions generally remain testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The conditional exclusion assumption is not testable up to a specification test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 Find T and test relevance of Z, W for U In this step, we test the relevance of W for U (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3), as well as the relevance of Z for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The latter is a necessary condition for the relevance of Z for A conditional on T (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c), which is tested explicitly in subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' First, we find some T = τ(Z) such that Z ⊥⊥ W | T is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As long as some T = τ(Z) leads to the conditional independence of Z and W, this control function τ ∈ L2(Z) also renders Z and U conditionally independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' To test for the sufficient relevance of Z and W with respect to U, we can use T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A sufficient condition for relevance of Z and W for U is that both Z and W contain spare information conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This motivates the choice of a valid τ ∈ T valid, which captures the least information about Z while ensuring the conditional exclusion of Z conditional on T, as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' To simplify the argument, suppose that our model is linear and the dimensions for (Z, T, W, U) are (dZ, dT, dW, dU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Often, relevance of Z and W for U imply min{dZ, dW} ≥ dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The minimum dimension of T to ensure conditional independence of Z and W is dU, so we know that any T = τ(Z) such that Z ⊥⊥ W | T must satisfy dT ≥ dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If we can reject a test with the null hypothesis H0 : min{dZ, dW} ≤ dT, and alternative Ha : min{dZ, dW} > dT, this implies min{dZ, dW} > dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, there is a test whether Z and W are relevant for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, min{dZ, dW} > dT is a sufficient, not a necessary condition for the relevance of Z and W for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z and W are still relevant for U when min{dZ, dW} = dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Unfortunately, this hypothesis is not testable with unobserved U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' So, how should an applied researcher proceed when min{dZ, dW} = dT (which could mean min{dZ, dW} = dU)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Here, we need to distinguish dZ = dT from dW = dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' dZ = dT When T contains as much information as Z, there is no point in moving to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The instruments Z contain no variation conditional on T, so Z cannot be relevant for treatment A conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 14 dW = dT We know for sure that dW ≤ dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If dW = dU, the proxies W are exactly relevant for U without spare information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If we are willing to assume dW = dU, we could move forward to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The variation in U associated with Z would still be held fixed with T in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Yet, dW = dU is not testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' It may well be that dW < dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, the variation in U associated with Z is not held fixed with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We can never test the completeness of W for U when dW = dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Accordingly, we do not generally suggest to move to step 2 by relying on the assumption dW = dU, when dW = dT is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 Test relevance of Z for A given T In step 1, the relevance of Z, W for U was confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Now, we test the relevance of the instruments Z for treatment A conditional on T (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Depending on the additional model assumptions, this is either a test of completeness (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c), or common support (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As any T = τ(Z) is simply the control function τ ∈ L2(Z) applied to instruments Z, the test of relevance of Z for A given T is straightforward for any given τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' It is as simple as a test of instrument relevance with observed confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Let us return to a linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If all components of the instrument vector Z are correlated with both U and A, the conditional relevance requirement simplifies to (dZ − dT) ≥ dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' After conditioning on all variation in Z which correlates with U by holding T fixed, dZ − dT dimensions of instruments Z remain to infer the causal effect of treatment A on outcome Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The remaining instrument variation of dimension dZ − dT is relevant for treatment A only if the treatment’s dimension dA is smaller than or equal to dZ − dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If Z is found to be relevant for A given T, it implies that Z is relevant for both U and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Interpreting U as any source of unobserved variation which associates instruments Z and proxies W, Z would have no variation conditional on T if they were not sufficiently relevant for U (dZ < dU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' So, if Z is relevant for A given T, it implies that Z already had to be relevant for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 Exclusion of Z conditional on U In step 1 and 2 we tested all relevance assumptions in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The conditional exclusion assumption for Z conditional on U remains untestable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' To be precise, Y ⊥⊥ Z | (A, U) (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a) can only be justified on theoretic grounds, not observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In order to justify conditional exclusion theoretically, T can be used to understand the unobserved common confounders U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As T captures all variation in Z associated with U, T immediately explains U in terms of its association with Z and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From the association of T and W, we can interpret U even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For example: If subject-specific pre-college GPA measures are 15 used as instruments Z, and T turns out to capture average GPA, then U could be interpreted as general ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Suppose W contains dummies capturing whether someone has engaged in risky behaviour, including drugs and illegal activity, while in high school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From theory and empirical evidence we would expect high ability to lead to less risky behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, if T is an average GPA, it would be expected to negatively correlated with W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Once we used T to understand the variation reflected by the unobserved confounders U, we can construct a theoretic argument with respect to the conditional exclusion of in- struments Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In our example, the common confounder U reflected general ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The conditional exclusion assumption reduces to whether conditional on general ability U, the subject-specific pre-college GPA measures Z are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This argument clearly depends on the respective choice of treatment A and outcome Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As in standard IV, specification tests, which can be revealing about the exclusion of Z, are possible if the model is overidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If a specification test suggests that different subset of instruments Z conditional on T result in estimators with different probability limits, we reject that all instruments Z are excluded conditional on T (unless the estimand is the local average treatment effect which can vary across subpopulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Necessary for any such test is model overidentification for the causal effect J conditional on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In a simple linear model, overidentification would e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' mean (dZ − dT) > dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' After keeping dT dimensions of Z fixed, the instruments must still contain overidentifying information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Ultimately, just as in standard IV, the conditional exclusion assumption for Z remains largely untestable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Therefore, it is crucial to better understand U from the control variable T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4 Estimation In the final fourth step, use Z to instrument for treatment A conditional on control variable T, to identify (and estimate) the structural function or average causal effect J of A on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Having established that all necessary relevance and exclusion requirements hold, an estimator ˆJ can be formulated for the causal effect of interest J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The form of this estimator depends on the type of parametric model assumptions made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7 Example: Linear Returns to Education Interested in the returns to education, we use data from the National Longitudinal Survey of Youth 1997 [Bureau of Labor Statistics, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The variables of interest are introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 16 Y Household net worth at 35: continuous variable, in USD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A BA degree: 1 if individual i obtained a BA degree, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z Pre-college test results: subject-specific and overall GPA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' ASVAB percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' W Risky behaviour dummies: whether i drank, smoked, or engaged in other behaviours considered risky by the age of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' U Ability: Unmeasured intellectual capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' ˜U Other biases: Selection on unobservables into obtaining a BA degree (at least in part result of optimising individuals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' X Covariates: sex, college GPA, parental education/net worth, siblings, region, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A review of the vast literature on returns to education is far beyond the scope of this paper [Psacharopoulos and Patrinos, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instead, we focus on estimation of a very specific return to education: The causal effect of obtaining a bachelor’s degree A on household net worth at 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Even in a simple linear model like Y = αY + Aβ + UγY + WυY + XηY + εY , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1) two distinct potential sources of confounding are easily identified via the unobservable com- ponents of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1: UγY Ability U likely has a positive effect on household net worth Y , by means of salary and non-salary based net worth accumulation [Griliches, 1977].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The vector-valued linear parameter γY captures this positive linear effect of ability on net worth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' εY The disturbance εY captures all variation in Y , which is jointly unexplained by (A, U, W, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This can be understood as individual-specific, heterogeneous characteristics, and chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Any correlation of A with either of these terms leads to biased estimates of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' How does obtaining a BA degree A correlate with ability U and a general dis- turbance εY ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this identification problem, selection bias in inherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' At least to some degree, individuals choose whether to obtain a BA degree as a result of an optimisation problem of expected 17 utility subject to an information set I: A = arg max a∈{0,1} � E �u(Y (a)) − c(a)|A = a, I �� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2) where u : Y → R is a utility function for net worth with diminishing returns, and c : {0, 1} → R a cost function for obtaining a BA degree A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Both utility and cost function can be individual-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For ease of illustration, suppose individuals are perfectly informed with I = (A, U, W, X, εY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, each individual chooses a ∈ {0, 1} to maximise the utility associated with potential outcome Y (a) minus cost c(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this case, there is an easy decision rule to determine optimal A: A = arg max a∈{0,1} u(Y (a)) − c(a), = 1 �u(Y (1)) − u(Y (0)) > c(1) − c(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' UγY Ceteris paribus, an increase in ability U equally increases Y (0) and Y (1) according to model 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Due to diminishing returns in the utility function u, u(Y (1)) − u(Y (0)) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, also c(1) decreases as higher-ability individuals experience a lower utility cost of obtaining a BA degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The overall effect on the choice of A is ambiguous and depends on the utility functions of the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' εY The effect of εY on the choice of A, on the other hand, is unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' An increase in εY reduces u(Y (1)) − u(Y (0)) due to the diminishing returns of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Cost c, however, is unaffected by εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, A inevitably negatively correlates with εY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This logic regarding negative selection bias when treatment is chosen by utility-maximising individuals is by no means novel [Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2006], or unique to the returns to education identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Negative selection bias is inherent to the treatment variable when it is at least in part the result of optimising behaviour by utility-maximising heterogeneous individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Novel in our approach is the ability to explicitly account for certain biases, in this case ability bias, when proxies for them exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Finding excluded instruments can be much more straightforward when pertinent biases, like ability bias, have already been taken care of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In our identification approach, instruments Z are pre-college test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' These results are strongly correlated with ability U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Yet, conditional on ability, and some other covariates, pre-college test results contain random variation, which is excluded with respect to household net worth Y (at age 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Concurrently, even random variation in pre-college test results is 18 a strong predictor of obtaining a BA degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, instrument relevance likely holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The proxies W are dummies for whether an individual engaged in risky behaviours at high school age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Among others, the risky behaviour dummies include drinking, smoking (marijuana), selling drugs and stealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Theory and empirical evidence suggest the correlation of low intelligence and risky behaviour [Loeber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Therefore, ability U both causes instruments Z and proxies W in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Ability U is the common confounder in this causal question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Clearly, additional covariates are necessary to justify instrument exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' These include sex, college GPA, parental education and net worth, the number of siblings, region of residence, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 Assumptions The linear equivalent to the general common confounding IV model in assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 is described as assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Again, for ease of notation assume dA = 1, just as in this returns to education identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 (Linear IV Model with Common Confounding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Linear outcome model projection: Y = αY + Aβ + UγY + WυY + XηY + εY (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instruments (a) Exclusion: E �εY (Z, U, W, X) � = 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (b) Relevance: For the linear projection of A on (Z, U, W, X), A = αA + Zζ + UγA + WυA + XηA + εA, E �εA(Z, U, W, Z) � = 000 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4) rank � E � (Zζ) A �� T, X �� = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Proxies (a) Exclusion: For the linear projection of W on (Z, U, X) and (Z, X), W = αW + UγW + XηW + εW, E �εW(Z, U, X) � = 000, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='6) W = ˜αW + Z˜γW + X˜ηW + ˜εW, E �˜εW(Z, X) � = 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='7) with T := Z˜γW + X˜ηW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 19 (b) Relevance: rank(γW) ≥ dU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='8) To simplify notation, let Z|X be the true residual of a projection of Z onto X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The linearity of the outcome model implies that the covariance Cov � (Z|Xζ), Y � is Cov � (Z|Xζ), Y � = Cov � (Z|Xζ), A � β + Cov � (Z|Xζ), U � γY + Cov � (Z|Xζ), W � υY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The above expression uses the uncorrelatedness of Z and εY in assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If it were not for the linear confounding from the unobserved common confounders U and proxies W, Z would be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Next, we demonstrate how to use the proxies W to keep Cov �(Zζ)U � fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Cov � (Z|Xζ), U � = Cov � (Z|Xζ), W � γ⊺ W � γWγ⊺ W �−1 The inverse � γWγ⊺ W �−1 exists under assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 that the rank of γW is at least dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, slightly rewriting Cov �(Zζ)Y � as Cov � (Z|Xζ), Y � = Var � Z|Xζ � β + Cov � (Z|Xζ), W � ˜υW, ˜υY := υY + υAβ + γ⊺ W � γWγ⊺ W �−1 (γY + γAβ) , implies that any endogeneity of residualised instruments Z|X is controlled for by conditioning on Z˜γW from the linear projection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' To be precise, Cov � (Z|Xζ), Y |Z˜γW � = Var � Z|Xζ|Z˜γW � β + Cov � (Z|Xζ)W|Z˜γW � � �� � =0 ˜υY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The covariance of the first stage can be rewritten as Cov � (Z|Xζ), A|Z˜γW � = Var � Z|Xζ|Z˜γW � + Cov � (Z|Xζ)W|Z˜γW � � �� � =0 ˜υA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Using both of these results, and one-dimensional treatment A to simplify notation, a simple 20 ratio form for the linear effect of A on outcome Y is β = Cov �� Z|Xζ � Y | Z˜γW � Cov �� Z|Xζ � A | Z˜γW � = Cov �(Zζ) Y | T, X � Cov �(Zζ) A | T, X �, with T = Z˜γW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='9) Hence, the estimator differs from standard IV based estimation only by also holding a linear prediction T of W fixed as the partial predicted values Zζ for A change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, the relevance requirement 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b for the instruments Z is conditional on T and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' T can be represented by a dU-dimensional linear function of Z, E �U|Z, X �, multiplied by γW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, a simpler way to understand the relevance requirement 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b is as rank(ζ) ≥ (dU + dA) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='10) A total of dU dimensions of variation in Z are typically needed to account for the dU- dimensional confounding effect of U via E �U|Z, X �, while the remaining variation in Z still needs to be relevant for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Other than in trivial cases2, equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='10 describes this relevance requirement satisfactorily as a rank condition on ζ, the partial linear projection effect of Z on A conditional on (U, W, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 Find T and test relevance of Z, W for U A valid control function is the linear prediction T = Z˜γW under assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3, meaning that conditional on (T, X), instruments Z are still relevant for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, its OLS estimate T = Zˆ˜γW generally is not a valid control function, because T and Z are perfectly correlated due to sampling variation, unless dZ > dW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, even when dZ > dW, the true ˜γW will have rank dU ≤ dW, while its OLS estimate ˆ˜γW always has possibly larger than necessary rank dW due to sampling variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Ultimately, the estimate ˆ˜γW should at best have exactly rank dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A test is needed for the rank r0 of matrix ˜γW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' If E �U|Z, X � = ZγZ + XγX, then ˜γW = γZγW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Sufficient for the rank condition in assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 is r0 < min{dZ, dW}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This condition means that an unobservable variable of smaller dimension than both W and Z can explain all correlation between W and Z conditional on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This unobserved variable is the common confounder U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' By the definition of U as the (minimum information) unobserved variable which renders W and Z mean-independent, γZ has dU ≤ dZ linearly independent rows (rank(γZ) = dU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As γW has dimensions dU × dW and dU ≤ dW, rank(γW) ≤ dU and thus r0 = rank(γZγW) = rank(γW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' While r0 < dW suffices to confirm the relevance of W for U in assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3, r0 < dZ 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' when Z contains perfectly collinear variation conditional on (U, W, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 21 is necessary for Z to be relevant for treatment A in assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A suitable test for some r < min{dZ, dW} has null hypothesis H0 : r0 ≤ r, and alternative Ha : r0 > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='11) With the OLS estimator ˆ˜γW, we apply a bootstrap based test for its rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' First, write the singular value decomposition as ˜γW = P0 dZ×dZ Π0 dZ×dW Q⊺ 0 dW ×dW (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='12) Then, let φr(A) := �mA j=r+1 π2 j(A) be the sum of squared singular values of A from the (r +1) largest to the smallest singular value, which is the mA-th singular value, where mA is the minimum across A’s number of rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, an equivalent test to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='11 is a test with null hypothesis H0 : φr (˜γW) = 0, and alternative Ha : φr (˜γW) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='13) The bootstrap procedure is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For each binary proxy Wj ∈ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' calculate the probability pj := Pr (Wj = 1) under H0 as pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 = Logit � � � � �Z P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='r dZ×r Π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='r r×r Q⊺ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='j r×1 + X ˜ηW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='j dX×1 � � β0 + α0 � � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' where (a) P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='r corresponds to the first r columns of P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (b) Q0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='j corresponds to the first r entries of the j-th row of Q0 (c) Π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='r corresponds to the r × r matrix of of the first r rows and columns of Π0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (d) ˜ηW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='j corresponds to j-th column of ˜ηW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (e) β0 and α0 are univariate coefficients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' which need to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Draw 1000 new bootstrap samples b ∈ B of binary proxies as W b 0 using the n × dW probability matrix (p0,0, p1,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' , pdW ,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For each bootstrap sample b ∈ B: Calculate the sample projection coefficient ˆ˜γb W,0 by projecting W b 0 onto (Z, X) (all demeaned), and the sum of its smallest squared singular values starting at the (r + 1) largest as φr,0,b := φr �ˆ˜γb W,0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Obtain the p-value as 1 − 1 |B| � b∈B 1 � φr,0,b < φr(˜γW) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 22 Figure 2: Bootstrap based test for H0 : rank(˜γW) = r0 ≤ r Notes: This figure illustrates the bootstrap distribution of the test statistic nφr � ˆ˜γW � in the test with null hypothesis H0 : rank(˜γW ) = r0 ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The left figure depicts the test statistic distribution under r = 0, and the left under r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For r = 0, the p-value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' H0 : r0 = 0 is strongly rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' For r = 1, the p-value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' H0 : r0 ≤ 1 cannot be rejected at any meaningful level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In figure 2, the bootstrapped distributions of the test statistic nφr �ˆ˜γW � are depicted under two different null hypotheses: r0 = 0 and r0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Non-rejection of the test is evidence in favour of the low rank r0 of ˜γW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the left diagram of figure 2, where the test concerns r0 = 0, the p-value is at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The test provides strong evidence against r0 = 0, which indicates some correlation between W and Z conditional on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The right diagram of figure 2 depicts the test statistic bootstrap distribution for H0 : r0 ≤ 1, and provides strong evidence against rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The associated p-value is 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, we can conclude that the rank of γW is at most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In the NLS97 data, pre-college test results Z and risky behaviour dummies have dimensions dZ = 7 and dW = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, r0 ≤ 1 allows the conclusion that the common confounder dimension is small: dU ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Conditional on covariates X, all covariance between Z and W is explained by a one-dimensional unobserved U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Successfully, the proximal assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 was tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In addition, the necessary dZ > dU condition for conditional instrument relevance (assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b) was confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 Test relevance of Z for A given T Despite satisfying the necessary dZ > dU condition for IV relevance (assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b), a proper test for the conditional relevance of Z for A given the control function T is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this step, we first explain how to construct the here one-dimensional control variable T after having conducted the tests in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, we test for the conditional relevance of instrument Z for treatment A given this control function T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 23 Ho : ro = 0 Ho : ro < 1 95% CV 95% CV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='005Given the statistical evidence in favour of dU ≤ 1, we construct the variable T N×1 := Z N×dZ ˆP0,1 dZ×1 ˆΠ0,1 1×1 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='14) with the singular value decomposition of the OLS estimator ˆ˜γW = ˆP0 ˆΠ0 ˆQ⊺ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' ˆP0,1 is the first column of ˆP0, and ˆΠ0,1 is the top-left entry of ˆΠ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Aside from sampling error, proxies W are mean-independent from instruments Z conditional on (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With the control T now defined, we can use a bootstrap based test to confirm the relevance of instruments Z for A conditional on (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The null hypothesis can be formulated as H0 : rank � E �(Zζ)A|T, X �� < dA, with alternative Ha : rank � E �(Zζ)A|T, X �� = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='15) Importantly, under H0 the effect of Z on A (given X) would be fully described by a one- dimensional T, as the dimension of U was found to be r0 ≤ 1 in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' When treatment A is one-dimensional, a simple test for this null hypothesis compares the R2 of an unrestricted (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='16) and restricted regression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A = ˜αA,ur + Z ˜ζ + X˜ηA,ur + ˜εA,ur, E � ˜εA,ur|Z, X � = 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='16) A = ˜αA,r + T ˜γA + X˜ηA,r + ˜εA,r, E � ˜εA,r|T, X � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='17) Under H0, both regressions would predict A equally well, despite the dimension reduction on Z in the second regression, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With the uncertainty in estimated T, we use a sim- ple bootstrap-based test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' With 1000 bootstrap samples bt ∈ Bt, we obtain a bootstrap distribution of R2 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Under H0, R2 ur is (asymptotically) distributed as R2 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Figure 3: Test for Relevance of Z for A given T (H0 : rank � E �Zζ|T, X �� < dA) Notes: This figure illustrates the bootstrap distribution of the restricted R2 r in regression 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The dimension of T,dT = 1, is based on the test in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 24 Bootstrap distribution of R2 R2 95% CV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='30Figure 3 depicts the bootstrap distribution of R2 r based on the restricted regression 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The control variable T is constructed for each bootstrap sample as described in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The unrestricted R2 ur based on the unrestricted regression 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='16 fits the data significantly better, which indicates rejection of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The p-value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' There is predictive information in Z for A, beyond that controlled for in (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In other words, Z satisfies the conditional instrument relevance requirement 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4 Exclusion of Z conditional on U While both relevance assumptions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b could be tested successfully, the exclusion of instrument Z conditional on U in assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a remains generally untestable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' To argue whether Z is exogenous conditional on U, it is worth asking: Which information is being held fixed in T, and what does this imply about U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The linear construction of T from Z is illustrated in table 1, where the instruments have been normalised to standard deviation one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z is mean-independent from W conditional on (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' T mostly consists of an average of subject-specific pre-college GPA measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In this sense, T closely measures academic ability, as captured by pre-college GPA measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Without the transcript GPA and ASVAB percentile, the subject-specific GPA measures describe 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5% of variation in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Despite the negative dependence of T on ASVAB percentile in its construction, T positively correlates with ASVAB percentile unconditional on the GPA measures with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='31 correla- tion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The interpretation of T and consequently U is pretty straightforward: It positively reflects (academic) ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Table 1: Construction of T = Z ˆP0,1 ˆΠ0,1 GPA ASVAB English Math SocSci LifeSci percentile T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='262 As U reflects (academic) ability, an increase in T is expected to result in a reduction of risky behaviour [Loeber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Indeed, a one standard deviation increase in T reduces the probability of having engaged in risky behaviour by the age of 17 between 3% and 9%, as illustrated in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' All effects have strong statistical and economic significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Compared to the average probability of engaging in risky behaviour, the estimated effect of a one standard deviation change in T is largest for some of the riskiest behaviour we considered: selling drugs (-54%), running away (-46%), and attacking someone (-41%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' T captures the information we expected based on our suspicion about the unobserved confounder ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' T closely reflects (academic) ability as measured by high-school GPA 25 Table 2: Effect of T on W try run attack sell destroy steal steal drink smoke marijuana away someone drugs property < 50$ > 50$ Pr 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='8% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='9% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0% T 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='9% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='6% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='8% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='9% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='7% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2% Notes: The table contains sample probabilities for engaging in risky behaviour by the age of 17 in the Pr row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The estimated decrease in the probability of engaging in risky behaviour from a linear probability model for a one standard deviation increase in T is noted in the row corresponding to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' measures, which reduces the probability of engaging in risky behaviours during high-school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, we can conclude that the common confounder U contains the unobserved variable ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Now, an argument is required for the conditional exogeneity of instruments Z given unobserved ability U and observed covariates X: Y = αY + Aβ + UγY + WυY + XηY + εY , E �εY | Z, X � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' While ability is the obvious confounder of the effect of pre-college GPA measures on net worth later in life, there are other possible confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Among everyone who goes to college, those with higher pre-college GPA are likely to also have a higher college GPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Even conditional on whether someone obtained a BA degree, a higher college GPA likely leads to higher earnings later in life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Thus, college GPA is an important observed confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Family and individual net worth at young age can affect pre-college GPA measures as more learning resources are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Their effect on net worth later in life is undeniable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Apart from net worth, other family background characteristics likely affect both pre-college test scores and net worth later in life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We include parental education, maternal age at first birth and the individual’s birth, as well as the number of siblings to capture family background characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Individual- specific characteristics are other important confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We include sex and citizenship status based on birth as further covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Conditional on this rich set of covariates X, and the unobserved variable ability U, there is no reason to believe that pre-college test scores Z would affect or be correlated with post-college earnings through any other channel than obtaining a BA degree A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Despite our best efforts in explaining U, and the provided arguments in favour of assumption 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2a, a test or conditional instrument exclusion is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A specification test is not feasible, because in this example the model is not overidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5 Estimation Estimation of the fully linear model is now straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As in Tien [2022], we call the estimator an instrumented common confounding (ICC) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' ˆβICC = � A⊺PZMT,XA �−1 � A⊺PZMT,XY � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='18) Here, PZ = Z (Z⊺Z)−1 Z⊺ is the projection matrix of Z, and MT,X = In − PT,X is the annihilator matrix of (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In table 3, the estimates of four major methods are compared: ordinary least squares (OLS), instrumental variables (IV), proximal learning (PL), and the here suggested ICC estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The row corresponding to T describes the estimated partial effect of T (normalised to standard deviation one) on net worth Y (at 35) in the respective regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' T is only used in proximal learning and ICC, but derived from the covariation of (Z, A) and W in negative control [Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=', 2020], as opposed to Z and W in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The row corresponding to A contains estimates for β, the causal effect of obtaining a BA degree A on net worth Y (at 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Their unit is US Dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Table 3: Estimates with different estimators (Y in thousands (k)) OLS PL IV ICC A 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='18*** 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='90*** 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='97*** 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='15** (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='12) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='40) (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='74) (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='93) T 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='76*** 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='05** (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='82) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='37) Notes: The table contains estimates and their standard errors (in paran- theses) for β in the A row, and the linear parameter on T if used in the method from four estimators: Ordinary Least Squares (OLS), Proximal Learning (NC), Instrumental Variables (IV), and Instrumented Common Confounding (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Asterisks indicate significance at the 1% (***), 5% (**) and 10% (*) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' OLS estimates that obtaining a BA degree increases net worth at 35 by 59k$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The proximal learning estimator conditions on its own T, so implicitly anything fixed that covaries (Z, A) and proxies W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The proximal learning estimate at 31k$, is indeed economically significantly smaller than the OLS estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As hypothesised, this might indicate that unobserved ability, which correlates (Z, A) and W, is a confounder which biases the estimated effect of education on net worth upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' In contrast, the IV estimate is much larger at 223k$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The inherent negative selection bias may thus be quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' However, the IV 27 estimator ignores the strong correlation of the pre-college test score instruments Z with ability U, which may lead to an accentuated ability bias compared to that in OLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As the estimator should be robust to ability bias, we condition on T and obtain the ICC estimator at 125k$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Indeed, conditioning on T attenuates the estimate by the expected ability bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' As relevance is not strongly satisfied for the instruments Z conditional on T in the ICC estimator, the standard error is expectedly large for this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Still, both general selection bias and ability bias appear to be strong confounders in this difficult identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Quantitatively separating ability and general selection bias helps add the necessary cred- ibility to IV, which misses under the original IV exclusion assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 8 Conclusion In this work, we relax instrument exclusion in the presence of mismeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Other observed variables, the proxies, must be relevant for the unobserved confounders, which cause endogeneity in the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The mild parametric index sufficiency assump- tion is also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Importantly, the proxies can be economically meaningful variables, with their own effects on treatment and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This method can be useful in various causal identification problems with observational data, where the unobserved confounders are otherwise unrestricted observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The linear returns to education identification problem illustrates how this method can identify causal effects when instrument exclusion, as often in practice, is a strong and hardly testable assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This paper established two point identification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' When point identification is impossible, this approach can still identify informative bounds on causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This set identification exercise is left to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Further, we have not demonstrated how to construct estimators other than in the linear example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Uncertainty in the control function estimation will be reflected in the performance of any estimator using this identification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The integration of this approach, which at best identifies averages of causal effects across unobservables, with marginal treatment effects, is another remaining task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 28 References Richard W Blundell and James L Powell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Endogeneity in semiparametric binary response models.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Estimation of treatment effects without an exclusion restriction: With an application to the analysis of the school breakfast program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Journal of Applied Econometrics, 28(6):982–1017, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Kenichi Nagasawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Treatment effect estimation with noisy conditioning variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00667, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Whitney K Newey and James L Powell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instrumental variable estimation of nonparametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Econometrica, 71(5):1565–1578, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Whitney K Newey, James L Powell, and Francis Vella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Nonparametric estimation of trian- gular simultaneous equations models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Econometrica, 67(3):565–603, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' George Psacharopoulos and Harry Anthony Patrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Returns to investment in education: a decennial review of the global literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Education Economics, 26(5):445–458, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Christian Tien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Instrumented common confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Emmanuel Selorm Tsyawo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Feasible iv regression without excluded instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='09621, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 30 9 Proofs Section 3 proofs Proof of lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Let t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' fW,Z|T(W, Z|T) = � U fW,Z|U,T(W, Z|u, T)fU|T(u, T) dµU(u) = � U fW|Z,U,T(W|Z, u, T)fZ|U,T(Z|u, T)fU|T(u, T) dµU(u) = � U fW|U(W|u)fZ|T(Z|T)fU|T(u, T) dµU(u) = fZ|T(Z|T) � U fW|U(W|u)fU|T(u, T) dµU(u) = fZ|T(Z|T)fW|T(W|T) From line two to three we use W ⊥⊥ Z | U (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3a) and U ⊥⊥ Z | T (by construction of T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From line four to five we again use W ⊥⊥ τ(Z) | U (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Proof of lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We write fW|Z(W, Z) in two separate ways using T and relate those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' fW|Z(W, Z) = � U fW|U(W, u)fU|Z(u, Z) dµU(u) = � U fW|U(W|u)fU|T,Z(u, t, z) dµU(u) The first line follows from W ⊥⊥ Z | U (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' fW|Z(W, Z) = fW|T(W, T) = � U fW|U(W, u)fU|T(u, t) dµU(u) The first line follows from the construction of T = τ(Z), τ ∈ L2(Z), such that W ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' By the equality of the expressions in steps 1 and 2 above, for all z ∈ Z, � U fW|U(W, u)fU|T,Z(u, t, z) dµU(u) = � U fW|U(W, u)fU|T(u, t) dµU(u), � U fU|T,Z(u, t, z)fW(W) fU(u) fU|W(u, W) dµU(u) = � U fU|T(u, t)fW(W) fU(u) fU|W(u, W) dµU(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 31 Let gt,z(U) := fU|T,Z(U,t,z) fU(U) and gt(U) := fU|T (U,t) fU(U) for any z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, E � gt,z(U)fW(W)|W � = E �gt(U)fW(W)|W � E �� gt,z(U) − gt(U) ����� W � fW(W) = 0 By completeness of W for U (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3b), the above only holds if gt,z(U) = gt(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' This implies fU|T,Z(U, t, z) = fU|T(U, t), and thus U ⊥⊥ Z | T for any T = τ(Z), τ ∈ L2(Z) such that W ⊥⊥ Z | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 proofs Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' E �Y |Z � = E � k(A) + ε �� Z � = E �k(A)|Z � + E � E �ε|U, Z � |Z � = E �k(A)|Z � + E � E �ε|U � |Z � = E �k(A)|Z � + E � E �ε|U � |T � = E �k(A)|Z � + E �ε|T � From line two to three we use the moment E �ε|Z, U � = E �ε|U � formulated in assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From line three to four we use U ⊥⊥ Z | T and T = τ(Z) (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2b/2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Required for this step is the identifiability of T, given by lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 subject to assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Completeness of Z for A given T (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c) ensures that there is a unique h ∈ L2(A, T) for which E �Y |Z � = E �h(A, T)|Z �, where h(A, T) = k(A) + E � E �ε|U � |T � = k(A) + E �ε|T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 32 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 proofs Proof of lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Fη|T(η) = � U Fη|u(η)fU|T(u, T) dµU(u), ∂Fη|T(e) ∂e ����� e=η = � U ∂Fη|u(e) ∂e ����� e=η � �� � >0 ∀u,η fU|T(u, T) dµU(u) > 0, so Fη|T(η) is strictly increasing on the conditional support of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Now prove Z ⊥⊥ η | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' fZ,η|T = � U fZ,η|U,TfU|T dµU(u) = � U fη|Z,U,TfZ|U,TfU|T dµU(u) = � U fη|U,TfZ|TfU|T dµU(u) = fZ|T � U fη|U,TfU|T dµU(u) = fZ|Tfη|T From second to third line we use Z ⊥⊥ η | U (assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3) and Z ⊥⊥ U | T (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We have shown that Z ⊥⊥ η | T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Proof of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' First, we derive a preliminary result as if U were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' FA|Z,U(a, z, u) = Pr �A ≤ a|Z = z, U = u � = Pr �h(z, η) ≤ a|Z = z, U = u � = Pr � η ≤ h−1(a, z)|Z = z, U = u � = Pr � η ≤ h−1(a, z)|U = u � = Fη|u � h−1(a, z) � = Fη|u (η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From line one to two we use the invertibility of h(z, η) (assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='(1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From line 33 two to three we use Z ⊥⊥ η | U (assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, VT := FA|Z(A, Z) = � U FA|Z,U(A, Z, u)fU|T(u, T) dµU(u) = � U Fη|U (η) fU|T(u, T) dµU(u) = Fη|T(η) On line one we use Z ⊥⊥ U | T (assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' From line one to two, we use the result derived at the beginning of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' The final line again follows from τ(Z) ⊥⊥ η | U (assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Hence, VT = FA|Z(A, Z) = Fη|T(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' By lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 (strictly increasing Fη|T on the support of η), (η, T) and (VT, T) = (Fη|T(η), T) are associated with the same sigma algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We show A ⊥⊥ Y (a) | (VT, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' fY (a),A|VT ,T(Y (a), A|VT, T) = � U fY (a)|A,VT ,T,U(Y (a)|h(Z, η), VT, T, u) � �� � =fY (a)|VT ,T,u(Y (a)|VT ,T,u) fA|T,VT ,U(h(Z, η)|VT, T, u) � �� � =fA|VT ,T (h(Z,η)|VT ,T) fU|T(u|T) dµU(u) = fA|VT ,T(h(Z, η)|VT, T) � U fY (a)|VT ,T,u(Y (a)|VT, T, u)fU|T(u|T) dµU(u) � �� � =fY (a)|VT ,T (Y (a)|VT ,T) = fA|VT ,T(A|VT, T)fY (a)|VT ,T(Y (a)|VT, T) =⇒ A ⊥⊥ Y (a) | (VT, T) 34 Proof of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' J := E �� A Y (a)π(a) dµA(a) � is identified as � VT ,T � A E �Y |A = a, (VT, T) = (vT, t) � π(a) dµA(a) dFVT ,T(vT, t) E �� A E �Y (a)|A = a, VT, T � π(a) dµA(a) � E �� A E �g(A, ε)|A = a, VT, T � π(a) dµA(a) � E �� A � E g(A, ε) dFε|A,VT ,T(ε, a, VT, T)π(a) dµA(a) � E �� A � E g(A, ε) dFε|VT ,T(ε, VT, T)π(a) dµA(a) � E �� E � A g(A, ε)π(a) dµA(a) dFε|VT ,T(ε, VT, T) � E �� A g(A, ε)π(a) dµA(a) � E �� A Y (a)π(a) dµA(a) � = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' We let Y (a) = g(a, ε) for some ε ∈ E and g ∈ L2(A, ε) from line two to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Then, A ⊥⊥ Y (a) | (VT, T) from theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2 implies A ⊥⊥ ε | (VT, T), which we use from line four to five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' All other steps are algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 10 Data Description The sample consists of 1,983 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Y : Household net worth at 35 (Z9141400) Household net worth was top-coded at 600,000$ and bottom-coded at -300,000$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0% of individuals were top-coded, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3% bottom-coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' A: Bachelor’s degree obtained (Z9084400) If there is a date of obtaining a bachelor’s degree (Z9084400 ≥ 0 or invalid skip −3), A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0% of individuals in the sample have obtained a BA degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Z: Pre-college ability measures Instruments are credit-weighted high-school GPAs in English (R9872000), Math (R9872200), Social Sciences (R9872300) and Life Sciences (R9872400), as well as the ASVAB percentile in each individual’s respective age group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 35 Figure 4: Histogram for Y Notes: Distribution of household net worth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' W: Pre-college risky behaviour The proxies consist of risky behaviour dummies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' They equal one when an individual has engaged in behaviour considered ”risky” by age 17 or earlier if missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Table 4: Probability of engaging in risky behaviour W by 17 (in %) mari- run attack sell destroy steal steal drink smoke juana away someone drugs property < 50$ > 50$ Pr 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 X: Individual, family, and regional covariates The proxies consist of risky behaviour dummies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' They equal one when an individual has engaged in behaviour considered ”risky” by age 17 or earlier if missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 36 400 300 - 200 - 00T 0 300000 200000 100000 0 100000 200000 300000 400000 500000 600000Figure 5: Histogram for high-school GPA Notes: Distribution of household high-school GPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Figure 6: Histogram for ASVAB percentile in 3-month age cohort Notes: Distribution of ASVAB percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='English ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='Math ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='Social Sciences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='Life Sciences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='400140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='40 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='20 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='100Table 5: Correlation matrix of pre-college risky behaviour W (in %) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='mari- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='sell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='destroy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='steal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='steal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='drink ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='smoke ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='juana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='away ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='someone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='drugs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='property ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='< 50$ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='> 50$ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='8% region south at 17 R1200300 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3% region west at 17 R1200300 (i.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3% non english home R0551900 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='1% 0 poor english interview R2394903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='3% 0 38 Table 7: Additional covariates in outcome model variable NLS97 basis info replace invalid by net worth age 20 Z9048900 35% quantile (2737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='7) gpa college B0004600 400+ as 400 region north central at 35 U0001900 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='5% region south at 35 U0001900 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='7% region west at 35 U0001900 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='2% urban at 35 U0015000 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='9% other non rural at 35 U0015000 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='4% 39 Table 8: OLS coefficients key variables OLS NC IV ICC const 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='36 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='21 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='85 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='52 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='41) (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='21) (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='38) (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='10) T 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='76*** 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='05** (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='82) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='37) A 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='18*** 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='90*** 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='97*** 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='15** (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='12) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='40) (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='74) (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='93) Notes: The table contains estimates and their standard errors (in paran- theses) for β in the A row, and the linear parameter on T if used in the method from four estimators: Ordinary Least Squares (OLS), Proximal Learning (NC), Instrumental Variables (IV), and Instrumented Common Confounding (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Asterisks indicate significance at the 1% (***), 5% (**) and 10% (*) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 40 Table 9: OLS coefficients individual- and family-covariates OLS NC IV ICC hh net worth 1997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='15*** 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='24*** 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='66** 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='18** (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='24) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='88) (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='09) (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='27) bio mom age first birth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='28** 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='60** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='80 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='15) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='13) bio mom age subject birth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='55) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='57) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='62) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='58) bio mom educ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='21* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='53 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='95) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='77) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='04) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03) bio dad educ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='22* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='38 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='84) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='71) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='06) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='32) relation parent figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='81*** 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='29*** 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='41** 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='78*** (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='22) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='44) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='75) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='82) parent religious 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03) non english home 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='64 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='37 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='47 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='23 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='01) (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='20) (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='40) (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='43) poor english interview 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='41 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='96 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='91 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='72 (112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='29) (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='33) (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='58) (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='79) Notes: The table contains estimates and their standard errors (in parantheses) for slope coeffi- cients on individual and family covariates using four estimators: Ordinary Least Squares (OLS), Proximal Learning (NC), Instrumental Variables (IV), and Instrumented Common Confounding (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Asterisks indicate significance at the 1% (***), 5% (**) and 10% (*) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 41 Table 10: Slope coefficients regional covariates OLS NC IV ICC urban 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='59 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='04) (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='68) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='64) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='01) urban 35 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='42*** 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='21*** 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='53*** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='79*** (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='48) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='89) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='00) (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='47) other non rural 17 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='02 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='62 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='34 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='31) (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='07) (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='55) (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='25) other non rural 35 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='49 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='51 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='20 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='35 (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='08) (66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='52) (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='14) (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='57) region north central 17 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='59* 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='51** 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='27* 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='92** (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='91) (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='27) (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='13) (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='68) region north central 35 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='57 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='57 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='45 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='64) (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='07) (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='28) (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='11) region south 17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='45 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='74 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='67 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='96 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='82) (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='22) (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} 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+page_content='03) (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='97) (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='79) (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='34) region west 35 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='37 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='70 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='20 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='98) (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='06) (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='19) (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='22) Notes: The table contains estimates and their standard errors (in parantheses) for slope coefficients on regional covariates using four estimators: Ordinary Least Squares (OLS), Proximal Learning (NC), Instrumental Variables (IV), and Instrumented Common Confounding (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Asterisks indicate significance at the 1% (***), 5% (**) and 10% (*) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 42 Table 11: Slope coefficients risky behaviour proxies OLS IV ever drank 17 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='44 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='56 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='10) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='63) ever smoked 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='40 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='39 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='01) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='90) ever marijuana 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='38 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='90) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='10) ever ran away 17 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='52 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='39 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='91) (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='74) ever attack 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='71 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='39 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='46) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='54) ever sell drugs 17 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='44 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='73 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='87) (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='27) ever destroy property 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='57 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='89) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='59) ever steal bit 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='15 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='50) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='30) ever steal lot 17 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='98 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='98 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='45) (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content='12) Notes: The table contains estimates and their standard errors (in parantheses) for slope coefficients on risky behaviour proxies using four estimators: Ordinary Least Squares (OLS), Proximal Learning (NC), Instrumental Variables (IV), and Instrumented Common Confounding (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' Asterisks indicate significance at the 1% (***), 5% (**) and 10% (*) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} +page_content=' 43' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctA0T4oBgHgl3EQfGv-w/content/2301.02052v1.pdf'} diff --git a/dNAyT4oBgHgl3EQfwvkf/content/tmp_files/2301.00652v1.pdf.txt b/dNAyT4oBgHgl3EQfwvkf/content/tmp_files/2301.00652v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..291319f982ff13961c278fa64fb58604245a5393 --- /dev/null +++ b/dNAyT4oBgHgl3EQfwvkf/content/tmp_files/2301.00652v1.pdf.txt @@ -0,0 +1,975 @@ +EFFICIENT SPEECH REPRESENTATION LEARNING WITH LOW-BIT QUANTIZATION +Ching-Feng Yeh, Wei-Ning Hsu, Paden Tomasello, Abdelrahman Mohamed +Meta AI +{cfyeh,wnhsu,padentomasello,abdo}@meta.com +ABSTRACT +With the development of hardware for machine learning, +newer models often come at the cost of both increased sizes +and computational complexity. In effort to improve the ef- +ficiency for these models, we apply and investigate recent +quantization techniques [1, 2] on speech representation learn- +ing models [3]. The quantization techniques were evaluated +on the SUPERB [4] benchmark. +On the ASR task, with +aggressive quantization to 1 bit, we achieved 86.32% stor- +age reduction (184.42 → 25.23), 88% estimated runtime +reduction (1.00 → 0.12) with increased word error rate +(7.06 → 15.96). +In comparison with DistillHuBERT [5] +which also aims for model compression, the 2-bit configura- +tion yielded slightly smaller storage (35.84 ↔ 46.98), better +word error rate (12.68 ↔ 13.37) and more efficient estimated +runtime (0.15 ↔ 0.73). +Index Terms— Quantization, Representation Learning +1. INTRODUCTION +Modern machine learning technology has pushed the limits +of related applications above and beyond in daily lives. As +the performances improves, the number of parameters and +the computational complexity of the models are also growing +significantly [6, 7, 8, 9]. The growth in resource consump- +tion not only means higher energy usage but also makes these +applications less accessible. On the other hand, with the de- +velopment of mobile and wearable devices, machine learning +applications have been transitioning closer to the device side +over the past few years. Given the growth in complexity of +models and the need from edge devices, improving model ef- +ficiency has gained heavy interests and has bee widely studied +[1, 2, 10, 5, 11, 12]. +Among the numerous directions for improving model effi- +ciency, quantization is particularly appealing, as quantization +aims to keep the original model architecture but replaces the +parameters with lower-precision alternatives for both storage +saving and computation reduction. In addition, quantization +typically casts parameters to lower-precision data types such +as integers, which are more favorable on edge devices since +integer operations are typically much cheaper and faster for +the processors on these devices [11, 13]. However, quanti- +zation in nature converts numbers from continuous domains +to discrete domains, therefore introduces quantization errors +in computation and typically cause the model performance +to degrade. +Therefore, in the field of quantization-related +research, minimal performance loss and maximal efficiency +gain, or a better trade-off, has always been the pursuit. +Recently, speech representation learning has been gaining +popularity due to the high potential in unifying and gen- +eralizing the common components across different speech +tasks such as automatic speech recognition (ASR) and key- +word spotting (KS). Traditionally, the models for individual +tasks are designed and trained independently from each other. +While this practice works well for individual tasks, there ex- +ists a major redundancy between models for these tasks since +many components serve similar purposes. For example, both +ASR and KS models have modules converting speech signals +to higher-level embeddings. Having a shared module instead +of two separate ones for both tasks will minimize the redun- +dancy. In that spirit, speech representation learning aims to +train a unified model to generate embeddings from speech +signals to be adopted by downstream tasks and therefore +reduces the overhead introduced by individual tasks. +In this work, we investigated two recently proposed quan- +tization techniques: 1) robustly binarized Transformer [1] 2) +squashed weight quantization [2]. We analyzed these quan- +tization techniques on top of the HuBERT [3] model for +speech representation learning and evaluated on the SUPERB +[4] benchmark. +From the experimental results, significant +storage reduction (184.42 → 25.23) and estimated runtime +improvement (1.00 → 0.12) were observed from applying +an extreme 1-bit quantization (binarization) with a word error +rate degradation (7.06 → 15.96) on the ASR task. Although +the degradation is non-trivial, compared with recent compres- +sion approaches such as DistillHuBERT [5], quantization still +offers a better trade-off between resource consumption and +model performance. +2. EFFICIENT LOW-BIT QUANTIZATION +Quantization converts tensors from high-precision domains +(typically floating-point numbers) to low-precision (typically +integers or binaries) domains. While quantization provides +benefits such as reductions both on storage and computation, +arXiv:2301.00652v1 [eess.AS] 14 Dec 2022 + +it also presents challenges to be applied with minimal perfor- +mance degradation compared with the original high-precision +models. In this section, we summarize the techniques adopted +in this work to produce a good trade-off between efficiency +and performance for quantized models. +2.1. Quantization Aware Training (QAT) and Straight- +Through Estimator (STE) +Among the wide variety of quantization strategies, quanti- +zation aware training (QAT) [14, 15] has been popular for +closing the performance gap between original and quantized +models. Different from post-training quantization, QAT in- +corporates quantization operations during both the inference +and gradient computation during training. This enables the +model parameters to simulate quantization effects along with +the training data so that they are more robust to quantization +errors in later stages [12]. +A major challenge for QAT is how to propagate the gra- +dients and update the model parameters with quantization in +place. As quantization is in nature a clipping operation, the- +oretically the gradients are either zeros or indifferentiables at +most operating points, meaning the model parameters won’t +be effectively updated. To resolve this, straight-through es- +timator (STE) [16] was proposed as an estimation for gradi- +ent updates. In STE, the forward pass still utilizes the model +parameters in discrete domain, but in the backward pass the +gating from quantization is bypassed in the chain rule and the +gradients are directly applied onto the model parameters, as +in equations (1) and (2). From equation (2), the gradients are +simply passed to the unquantized parameters as an approxi- +mation. In this work, both QAT and STE were adopted in +model training. +forward : y = Q(W) ∗ x + b. +(1) +backward : ∂y +∂W = +∂y +∂Q(W) ∗ ∂Q(W) +∂W +STE +≈ +∂y +∂Q(W). +(2) +2.2. Robustly Binarized Transformer (BiT) +Conventionally, for n-bit quantization, real-valued numbers +are converted into discrete counterparts, such as {0, 1, ..., 2n− +1} for asymmetric cases and {−2n−1, 1, ..., 2n−1 − 1} for +symmetric cases with optionally a scaling factor α. Recently, +new quantization techniques has emerged beyond this simple +formulation [1, 2, 12]. Among the techniques, the robustly +binarized transformer (BiT) [1] demonstrates smaller quan- +tization errors and more aggressive yet more efficient model +inference by applying quantization not only on parameters but +also on activations. +The core idea of BiT is the two-set elastic quantization, +where different formulations are applied to different numeri- +cal ranges. For example, the outputs from softmax operations +will be positive only, while the weight in linear operations can +be either positive or negative. This is referred to as the ”two- +set” quantization scenario, in which one is for asymmetric +(positive only) and the other is for symmetric (both positive +and negative). This enables better utilization for the precious +bits in quantization. Given a scaling factor α ∈ R+ and a +threshold β ∈ R, a tensor X can be quantized as in equation +(3). Both α and β can be stored as additional model parame- +ters and updated during QAT with gradients through STE. +XQ = +� +α ∗ Clip( X−β +α , 0, 1), +if X ∈ R+ +α ∗ Clip(X − β, −1, 1), +if X ∈ R +. +(3) +Two-set elastic quantization provides a generic way to +quantize any tensor, as shown in equation (3). In addition +to quantization on model parameters to reduce the storage +size, application on activations can also reduce floating op- +erations further. For example, for low-parameter but high- +computation operations such as multi-head attention, major +computations happen between intermediate activations such +as the query, key and values. Quantizing such activations can +move significant amount of floating operations to quantized +domains and improve computational efficiency, as will be dis- +cussed further in experiments. +2.3. Squashed Weight Quantization (SqWQ) +Recently, squashed weight quantization (SqWQ) [2] was also +proposed to reduce the quantization error. Squashed weight +quantization aims to re-distribute the parameters into uniform +distributions, as in equation (4), where g is a gain factor in +form of a vector. +y = Q(tanh(W)) ∗ x ∗ eg + b. +(4) +To achieve the re-distribution, the additional regulariza- +tion loss LQ is added to the loss function as defined in equa- +tion (5), where λq is the weight of regularization loss and σt +is the target standard deviation, both as hyper-parameters to +be tuned. +LQ = λq ∗ ((stddev(W) − σt)2 + mean(W)2). +(5) +By enforcing the parameters to be uniformly distributed, +squashed weight quantization also preserves the utilization +of the precious bits and demonstrated great performance for +lower-bit models. + +Base Model +Quant +Precision +SUPERB Tasks +Storage +(MBs)↓ +FLOPs +(Gs)↓ +QuantOPs +(GBits)↓ +Runtime +(Est. x)↓ +ASR↓ +KS↑ +SF↑ +PR↓ +QbE↑ +IC↑ +ASV↓ +SD↓ +ER↑ +HuBERT (Base)[3] +– +fp16 +6.42 +96.59 +0.88 +5.41 +7.36 +97.15 +5.11 +6.20 +64.92 +189.14 +153.14 +0.00 +1.38 +HuBERT +(+FastConv[17]) +– +fp16 +7.06 +96.62 +0.89 +6.05 +6.91 +97.28 +5.30 +6.32 +65.00 +184.42 +110.79 +0.00 +1.00 +SqWQ[2] +w8 +9.69 +96.88 +0.88 +7.30 +6.19 +96.65 +5.88 +6.52 +62.83 +99.65 +82.24 +1898.44 +1.00 +w4 +9.98 +96.59 +0.88 +8.03 +5.86 +96.26 +6.06 +6.73 +62.79 +57.19 +82.24 +1054.69 +0.89 +w2 +12.56 +94.22 +0.86 +11.79 +5.27 +94.02 +6.31 +7.12 +62.38 +35.95 +82.24 +632.81 +0.83 +w1 +25.37 +85.07 +0.73 +41.77 +4.74 +64.88 +18.23 +11.26 +54.40 +25.34 +82.24 +421.88 +0.80 +BiT-L[1] +(Linear +Only) +w8a8 +7.03 +96.85 +0.88 +6.22 +6.36 +98.23 +5.54 +6.36 +65.94 +99.49 +82.29 +1898.44 +1.00 +w4a4 +8.58 +96.56 +0.88 +7.15 +6.40 +96.10 +5.55 +6.26 +64.12 +57.02 +82.29 +527.34 +0.81 +w2a2 +10.80 +95.88 +0.86 +8.79 +5.62 +97.47 +5.68 +6.55 +63.49 +35.79 +82.29 +158.20 +0.76 +w1a1 +12.23 +94.94 +0.86 +10.49 +5.99 +96.49 +6.55 +6.87 +63.06 +25.17 +82.29 +52.73 +0.75 +BiT-LA[1] +(Linear ++Attention) +w8a8 +7.07 +97.21 +0.89 +6.30 +6.40 +98.10 +5.56 +6.24 +65.77 +99.54 +11.82 +3868.56 +0.63 +w4a4 +9.35 +96.62 +0.88 +7.76 +6.37 +96.92 +5.75 +6.09 +66.58 +57.08 +11.82 +1074.60 +0.25 +w2a2 +12.68 +95.07 +0.85 +12.56 +5.23 +95.02 +7.40 +6.94 +63.00 +35.84 +11.82 +322.38 +0.15 +w1a1 +15.96 +93.83 +0.78 +22.96 +5.63 +93.01 +6.83 +7.62 +61.68 +25.23 +11.82 +107.46 +0.12 +DistillHuBERT[5] +– +fp16 +13.37 +95.98 +0.83 +16.27 +5.11 +94.99 +8.55 +6.19 +63.02 +46.98 +80.34 +0.00 +0.73 +Table 1. Evaluation of Quantization Techniques on SUPERB Tasks and Profiling Results. +3. KNOWLEDGE DISTILLATION +During quantization-aware training, along with the quanti- +zation errors accumulated through operations, the gradient +can also degrade through back-propagation [1, 2]. To miti- +gate the gradient degradation through operators, knowledge +distillation [15, 18] has proven to be effective where the +”student” model aims to imitate the outputs of the ”teacher” +model, regardless of the original loss function of the teacher +model. +There are different strategies to apply knowledge +distillation for different scenarios and domains. Since quan- +tization keeps the model architecture, meaning the student +(quantized) model shares the same tensor shapes with the +teacher (unquantized) model, we aim to distill not only the +final output of the models but also the intermediate outputs +and attention weights from each inner Transformer layers, as +described in equation (6) and (7), where MSE() is the mean +square error operator, yT and yS are model outputs, oT,i and +oS,i are intermediate outputs for layer i, aT,i and aS,i are +attention weights for layer i in teacher and student models. +Lfinal = MSE(yT , yS) + Llayers. (6) +Llayers = � +i(MSE(oT,i, oS,i) + MSE(aT,i, aS,i)). (7) +4. EXPERIMENTS +4.1. Experimental Setup +We adopt the HuBERT[3] (base) model as the baseline. For +QAT, the same 960 hours of training set from LibriSpeech[19] +for building the baseline model is used. The evaluation was +performed on the 9 downstream tasks in the SUPERB[4] +challenge including automatic speech recognition (ASR), +keyword spotting (KS), slot filling (SF), phoneme recogni- +tion (PR), query by example (QbE), intent classification (IC), +automatic speaker verification (ASV), speaker diarization +(SD) and emotion recognition (ER). The tasks are labeled +with up/down arrows showing the goals of the metrics as in +Tables 1 and 2. For example, ASR is measure in word error +rates (WERs) therefore lower is better. In all tasks, the model +serves as a speech representation extractor with parameters +fixed after training. The implementation was built on top of +fairseq[20] and torchaudio[21]. +For evaluating the resource consumption of the models, +we extended the DeepSpeed [22] tool to profile the models for +1) on-disk storage 2) floating point operations 3) quantization +operations. The definition of these metrics are: +• Storage: The required space to store all model parame- +ters, measured in megabytes (MBs). Parameters in fp16 +are estimated to take 16 bits each, while quantized pa- +rameters take the same bits for annotated weight bits +(e.g. 8 bits for w8). +• FLOPs: The sum of floating operations (FLOPs) dur- +ing the forward pass, measured in gigas (Gs). +• QuantOPs: The sum of quantization (integer or bi- +nary) operations during the forward pass, measured in +gigabits (GBits). QuantOPs are subjective to the num- +ber of bits in quantization. For example, for operations +between a 8-bit integer and a 2-bit integer, a multiplica- +tion would take 2∗8 = 16 QuantOPs, while an addition +would take max(2, 8) = 8 QuantOPs. +• Runtime: The estimated runtime, measured in relative +proportion (x) to the baseline HuBERT(+FastConv) +model in fp16. Since FLOPs and QuantOPs are fun- +damentally different, their execution speeds are highly +dependent on the hardware [11]. In attempt to integrate +the two types of operations in a hardware-agnostic +fashion, we created an anchor point for conversion rate +between FLOPs and QuantOPs. +In Table 1, we as- +sume the SqWQ-w8 model runs as fast as its fp16 +counterpart and both have runtime 1.00, meaning +(110.79 − 82.24) = 28.55 Gs in FLOPs would run +as fast as 1898.44 GBits in QuantOPs. This conversion + +Base Model +Quant +Loss +Precision +SUPERB Tasks +ASR↓ +KS↑ +SF↑ +PR↓ +QbE↑ +IC↑ +ASV↓ +SD↓ +ER↑ +HuBERT +(+FastConv[17]) +– +– +fp16 +7.06 +96.62 +0.89 +6.05 +6.91 +97.28 +5.30 +6.32 +65.00 +BiT-LA[1] +(Linear ++Attention) +HuBERT[3] +w8a8 +8.32 +93.34 +0.89 +7.30 +6.31 +90.26 +6.51 +7.25 +63.16 +w4a4 +10.59 +92.25 +0.85 +8.68 +5.89 +90.82 +6.57 +7.14 +64.22 +w2a2 +14.29 +91.72 +0.81 +14.46 +4.80 +88.68 +8.71 +7.85 +60.29 +w1a1 +18.93 +91.07 +0.77 +25.92 +5.22 +83.80 +8.05 +8.73 +58.50 +Knowledge +Distillation +w8a8 +7.07 +97.21 +0.89 +6.30 +6.40 +98.10 +5.56 +6.24 +65.77 +w4a4 +9.35 +96.62 +0.88 +7.76 +6.37 +96.92 +5.75 +6.09 +66.58 +w2a2 +12.68 +95.07 +0.85 +12.56 +5.23 +95.02 +7.40 +6.94 +63.00 +w1a1 +15.96 +93.83 +0.78 +22.96 +5.63 +93.01 +6.83 +7.62 +61.68 +Table 2. Comparison between Loss Functions for Quantized Model Training: HuBERT and Knowledge Distillation. +rate is applied to all models for runtime estimation in +Table 1. +4.2. Experimental Results +4.2.1. SUPERB Tasks and Profiling +Table 1 presents the results for SUPERB tasks and model pro- +filing. The HuBERT (Base) model [3] as the first baseline +has a relative high FLOPs given similar storage size com- +pared with the alternative version (+FastConv) in which the +convolutional feature extractor is replaced with a more effi- +cient configuration from [17] on top of [23]. Therefore, we +applied quantization on top of the more efficient version Hu- +BERT(+FastConv) and refer to it as the new baseline, which +is 38% more efficient in FLOPs. The precision for quantiza- +tion in table 1 are labeled with the number of bits for both +model weights and activations. For example, w4a2 refers to 4 +bits for model weights and 2 bits for activations for BiT. Since +SqWQ focuses on quantizing the model weights, 8 bits were +applied for activations across all setups {w8, w4, w2, w1}. +Based on the HuBERT(+FastConv) model, 3 different +quantization strategies were applied: 1) SqWQ 2) BiT (Lin- +ear Only) 3) BiT (Linear+Attention), for each different num- +ber of bits {8, 4, 2, 1} were all experimented. Since most +SUPERB tasks demonstrate similar trends, we focus on the +ASR task as the main metric. +To evaluate squashed weight quantization (SqWQ), re- +sults showed that the SqWQ-w8 model effectively reduced +the storage size (184.42 → 99.65) with some degradation +(7.06 → 9.69) on the ASR task. The SqWQ-w4 further re- +duced the storage (99.65 → 57.19) and also had lower Quan- +tOPs (1898.44 → 1054.69) due to lower bits applied. The +further degradation from SqWQ-w8 to SqWQ-w4 is milder +(9.69 → 9.98) compared with fp16 to SqWQ-w8 (7.06 → +9.69), potentially benefiting from better utilization of bit by +fitting the parameters into uniform distributions. The same +trend applies to SqWQ-w2 and SqWQ-w1 models. +For models with BiT quantization applied to linear only +(BiT-L), results showed similar trends as the SqWQ mod- +els. +Both SqWQ and BiT-L applies to linear operations +only and share similar trends in which the degradation in- +creases and the number of bits becomes lower, with BiT-L +showing smaller gaps from the fp16 baseline. +For exam- +ple, BiT-L-w1a1 is similar to SqWQ-w2 (12.23 ↔ 12.56) +and lower than SqWQ-w1 (12.23 ↔ 25.37). +In addi- +tion, since BiT applies quantization to activations as well, +the models with lower bits also show lower complexity on +QuantOPs, as observed between BiT-L-w1a1 and SqWQ-w1 +(52.73 ↔ 421.88). From the FLOPs and estimated runtimes, +it is worth noting that although linear operations were moved +to quantizated domains, a significant portion of FLOPs (82.24 +out of 110.79) remained, which dominated the estimated run- +time (1.00 → 0.75) even though the QuantOPs were reduced +significantly (1898.44 → 52.73) comparing the BiT-L-w8a8 +and BiT-L-w1a1 models. +For models with BiT quantization applied to both lin- +ear and attention operations (BiT-LA), the storage did not +reduce but slightly increased due to the scaling factors α +and thresholds β introduced in elastic quantization. +For +example, BiT-LA-w1a1 took more space than BiT-L-w1a1 +(25.23 ↔ 25.17) given similar bits. +As quantization ad- +ditionally applied to attention operations, the quantization +error increased across all precisions, but tended to be worse +for model with lower bits. +For example, results showed +that slight degradation from BiT-L-w8a8 to BiT-LA-w8a8 +(7.03 → 7.07) but the gap was larger from BiT-L-w1a1 to +Bit-LA-w1a1 (12.23 → 15.96). +The major benefit from +quantizing attention additionally is in computation. Compar- +ing BiT-L-w1a1 and BiT-LA-w1a1, the FLOPs was signifi- +cantly reduced (82.29 → 11.82) since the attention operators +were moved to quantization domains. +Although this also +increased QuantOPs (52.73 → 107.46), it was insignificant +for the low bits and the overall reduced estimated runtime +(0.75 → 0.12). +In addition to quantization, other approaches such as neu- +ral architecture search are also of wide interest for efficient +modeling. To compare the effectiveness of quantization, we +included the results from DistillHuBERT [5], which also aims +to improve the efficiency for HuBERT models. From the Ta- +ble, comparable models included all three quantization strate- + +Fig. 1. Comparison between One-step and Scheduled Quan- +tization on SUPERB-ASR Task. +gies with 2 bits (SqWQ-w2, BiT-L-w2a2 and BiT-LA-w2a2), +which were all around 35 MBs in storage. Results showed +that all three quantization strategies provide lower word er- +ror rates (12.56, 10.80, 12.68 ↔ 13.37) and lighter storage +(35.95, 35.79, 35.84 ↔ 46.98), with BiT-LA-w2a2 signif- +icantly outperformed on the FLOPs (11.82 ↔ 80.34) and +estimated runtime (0.15 ↔ 0.73), which demonstrated that +quantization is effective among efficient modeling techniques. +4.2.2. Knowledge Distillation for Quantized Model Training +As described in section 3, model training with quantization +is challenging since quantization error accumulates through +back-propagation. Therefore, we adopted knowledge distilla- +tion for minimizing not only the mean square error between +model outputs but also intermediate layer outputs and atten- +tion weights. To investigate the impact from knowledge distil- +lation, we trained another set of models with exactly the same +quantization configurations, the same initialization from the +baseline fp16 models, the same dataset but the loss function +was changed to the original HuBERT loss [3] for speech rep- +resentation learning. These models were learning to predict +masked frames instead of imitating the baseline fp16 model +and there was no intermediate outputs involved. +BiT quantization on both linear and attention operations +(BiT-LA) was applied and the results were summarized in Ta- +ble 2. We see that the trends are similar across precisions with +models trained with HuBERT loss degrading more than mod- +els trained with knowledge distillation. For example, word +error rates are (18.93 ↔ 15.96) between HuBERT loss and +knowledge distillation. The profiling metrics such as storage +and runtime were omitted since they are independent from +loss functions and can be found in Table 1. It is evident that +knowledge distillation is effective for training quantized mod- +els by minimizing intermediate layer outputs to mitigate ac- +cumulated quantization errors through back-propagation. +4.2.3. One-Step vs. Scheduled Quantization +In section 4.2.1, the models were quantized directly from the +original model in fp16 to the target precision. For example, +BiT-LA-w1a1 was distilled directly from HuBERT(+FastConv)- +fp16. This is referred to as ”one-step” quantization, as op- +posed to the scheduled quantization suggested in [1], where +findings indicated that multi-step quantization involving in- +termediate precisions may improve the performance degrada- +tion. There are endless combinations to perform multi-step +distillation, depending on the number of bits for weights +and activations and how many bits to reduce at a time. In +comparison, one-step quantization provides simpler setup but +would potentially suffer from larger performance degrada- +tion. To understand the impact of scheduled quantization, +two quantization schedules were experimented against the +one-step results: 1) fp16 →w8a8 →w4a4 →w2a2 →w1a1 +and 2) fp16 →w1a2 →w1a1 as suggested in [1]. The results +were plotted in Figure 1 where the X-axis is for prevision +and the Y-axis is the word error rate in the ASR task. Results +showed that the models with scheduled quantization offered +similar performance as their one-step counterparts, especially +at the target precision w1a1. Given the many combinations of +scheduled quantization, we draw the preliminary conclusion +that no significant improvement was observed from scheduled +quantization. +5. CONCLUSION +In this work, we investigated two novel quantization tech- +niques: 1) robustly binarized Transformer [1] 2) squashed +weight quantization [2]. +The quantization was applied to +HuBERT [3] models for speech representation learning tasks. +The experiments were evaluated on the SUPERB [4] bench- +mark and significant savings were observed both on storage +and estimated runtime through quantization. For the aggres- +sively binarized models, storage was saved by 86.32% in +megabytes (MBs) (184.42 → 25.23), estimated runtime was +reduced by 88% (1.00 → 0.12). For comparable configura- +tions to DistillHuBERT [5], 2-bit models offered lower word +rates (12.68 ↔ 13.37) and estimate runtime (0.15 ↔ 0.73) +while still smaller in storage (35.84 ↔ 46.98). With the +growing sizes and computational complexity of modern mod- +els, we believe it is crucial to make models both compact +and efficient so they are more accessible and deployable to +resource-constrained environments such as edge devices. +6. ACKNOWLEDGEMENT +We would like to express our sincere gratitude for the follow- +ing contributors: 1) Zechun Liu and Barlas Oguz from Meta +AI for providing technical details and discussions on the ”Ro- +bustly Binarized Transformer” work. 2) Xiaohui Zhang and +Zhaoheng Ni from Meta AI for techincal discussions and im- +plementation on top of fairseq and torchaudio. 3) Anuj Diwan +from University Texas at Austin for technical discussions and +optimization. + +1-step +fp16->w8a8->w4a4->w2a2->wla1 +fp16->wla2->wla1 +18 +(%) +16 +WER +14 +SUPERB-ASR +12 +10 +8 +0 +fp16 +8e8m +w4a4 +w2a2 +wla2 +wlal7. 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file mode 100644 index 0000000000000000000000000000000000000000..e20fcbb39ac2c102af251ca794a2a6ae1abbeba5 --- /dev/null +++ b/dNAyT4oBgHgl3EQfwvkf/content/tmp_files/load_file.txt @@ -0,0 +1,625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf,len=624 +page_content='EFFICIENT SPEECH REPRESENTATION LEARNING WITH LOW-BIT QUANTIZATION Ching-Feng Yeh, Wei-Ning Hsu, Paden Tomasello, Abdelrahman Mohamed Meta AI {cfyeh,wnhsu,padentomasello,abdo}@meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='com ABSTRACT With the development of hardware for machine learning, newer models often come at the cost of both increased sizes and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In effort to improve the ef- ficiency for these models, we apply and investigate recent quantization techniques [1, 2] on speech representation learn- ing models [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The quantization techniques were evaluated on the SUPERB [4] benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' On the ASR task, with aggressive quantization to 1 bit, we achieved 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='32% stor- age reduction (184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='42 → 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23), 88% estimated runtime reduction (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='12) with increased word error rate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='06 → 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In comparison with DistillHuBERT [5] which also aims for model compression, the 2-bit configura- tion yielded slightly smaller storage (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='84 ↔ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='98), better word error rate (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 ↔ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='37) and more efficient estimated runtime (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='15 ↔ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Index Terms— Quantization, Representation Learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' INTRODUCTION Modern machine learning technology has pushed the limits of related applications above and beyond in daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' As the performances improves, the number of parameters and the computational complexity of the models are also growing significantly [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The growth in resource consump- tion not only means higher energy usage but also makes these applications less accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' On the other hand, with the de- velopment of mobile and wearable devices, machine learning applications have been transitioning closer to the device side over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Given the growth in complexity of models and the need from edge devices, improving model ef- ficiency has gained heavy interests and has bee widely studied [1, 2, 10, 5, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Among the numerous directions for improving model effi- ciency, quantization is particularly appealing, as quantization aims to keep the original model architecture but replaces the parameters with lower-precision alternatives for both storage saving and computation reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In addition, quantization typically casts parameters to lower-precision data types such as integers, which are more favorable on edge devices since integer operations are typically much cheaper and faster for the processors on these devices [11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' However, quanti- zation in nature converts numbers from continuous domains to discrete domains, therefore introduces quantization errors in computation and typically cause the model performance to degrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Therefore, in the field of quantization-related research, minimal performance loss and maximal efficiency gain, or a better trade-off, has always been the pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Recently, speech representation learning has been gaining popularity due to the high potential in unifying and gen- eralizing the common components across different speech tasks such as automatic speech recognition (ASR) and key- word spotting (KS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Traditionally, the models for individual tasks are designed and trained independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' While this practice works well for individual tasks, there ex- ists a major redundancy between models for these tasks since many components serve similar purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, both ASR and KS models have modules converting speech signals to higher-level embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Having a shared module instead of two separate ones for both tasks will minimize the redun- dancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In that spirit, speech representation learning aims to train a unified model to generate embeddings from speech signals to be adopted by downstream tasks and therefore reduces the overhead introduced by individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In this work, we investigated two recently proposed quan- tization techniques: 1) robustly binarized Transformer [1] 2) squashed weight quantization [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' We analyzed these quan- tization techniques on top of the HuBERT [3] model for speech representation learning and evaluated on the SUPERB [4] benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' From the experimental results, significant storage reduction (184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='42 → 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23) and estimated runtime improvement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='12) were observed from applying an extreme 1-bit quantization (binarization) with a word error rate degradation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='06 → 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='96) on the ASR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Although the degradation is non-trivial, compared with recent compres- sion approaches such as DistillHuBERT [5], quantization still offers a better trade-off between resource consumption and model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' EFFICIENT LOW-BIT QUANTIZATION Quantization converts tensors from high-precision domains (typically floating-point numbers) to low-precision (typically integers or binaries) domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' While quantization provides benefits such as reductions both on storage and computation, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00652v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='AS] 14 Dec 2022 it also presents challenges to be applied with minimal perfor- mance degradation compared with the original high-precision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In this section, we summarize the techniques adopted in this work to produce a good trade-off between efficiency and performance for quantized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Quantization Aware Training (QAT) and Straight- Through Estimator (STE) Among the wide variety of quantization strategies, quanti- zation aware training (QAT) [14, 15] has been popular for closing the performance gap between original and quantized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Different from post-training quantization, QAT in- corporates quantization operations during both the inference and gradient computation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' This enables the model parameters to simulate quantization effects along with the training data so that they are more robust to quantization errors in later stages [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' A major challenge for QAT is how to propagate the gra- dients and update the model parameters with quantization in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' As quantization is in nature a clipping operation, the- oretically the gradients are either zeros or indifferentiables at most operating points, meaning the model parameters won’t be effectively updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' To resolve this, straight-through es- timator (STE) [16] was proposed as an estimation for gradi- ent updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In STE, the forward pass still utilizes the model parameters in discrete domain, but in the backward pass the gating from quantization is bypassed in the chain rule and the gradients are directly applied onto the model parameters, as in equations (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' From equation (2), the gradients are simply passed to the unquantized parameters as an approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In this work, both QAT and STE were adopted in model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' forward : y = Q(W) ∗ x + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (1) backward : ∂y ∂W = ∂y ∂Q(W) ∗ ∂Q(W) ∂W STE ≈ ∂y ∂Q(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Robustly Binarized Transformer (BiT) Conventionally, for n-bit quantization, real-valued numbers are converted into discrete counterparts, such as {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=', 2n− 1} for asymmetric cases and {−2n−1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=', 2n−1 − 1} for symmetric cases with optionally a scaling factor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Recently, new quantization techniques has emerged beyond this simple formulation [1, 2, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Among the techniques, the robustly binarized transformer (BiT) [1] demonstrates smaller quan- tization errors and more aggressive yet more efficient model inference by applying quantization not only on parameters but also on activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The core idea of BiT is the two-set elastic quantization, where different formulations are applied to different numeri- cal ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, the outputs from softmax operations will be positive only, while the weight in linear operations can be either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' This is referred to as the ”two- set” quantization scenario, in which one is for asymmetric (positive only) and the other is for symmetric (both positive and negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' This enables better utilization for the precious bits in quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Given a scaling factor α ∈ R+ and a threshold β ∈ R, a tensor X can be quantized as in equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Both α and β can be stored as additional model parame- ters and updated during QAT with gradients through STE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' XQ = � α ∗ Clip( X−β α , 0, 1), if X ∈ R+ α ∗ Clip(X − β, −1, 1), if X ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (3) Two-set elastic quantization provides a generic way to quantize any tensor, as shown in equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In addition to quantization on model parameters to reduce the storage size, application on activations can also reduce floating op- erations further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, for low-parameter but high- computation operations such as multi-head attention, major computations happen between intermediate activations such as the query, key and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Quantizing such activations can move significant amount of floating operations to quantized domains and improve computational efficiency, as will be dis- cussed further in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Squashed Weight Quantization (SqWQ) Recently, squashed weight quantization (SqWQ) [2] was also proposed to reduce the quantization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Squashed weight quantization aims to re-distribute the parameters into uniform distributions, as in equation (4), where g is a gain factor in form of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' y = Q(tanh(W)) ∗ x ∗ eg + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (4) To achieve the re-distribution, the additional regulariza- tion loss LQ is added to the loss function as defined in equa- tion (5), where λq is the weight of regularization loss and σt is the target standard deviation, both as hyper-parameters to be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' LQ = λq ∗ ((stddev(W) − σt)2 + mean(W)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (5) By enforcing the parameters to be uniformly distributed, squashed weight quantization also preserves the utilization of the precious bits and demonstrated great performance for lower-bit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Base Model Quant Precision SUPERB Tasks Storage (MBs)↓ FLOPs (Gs)↓ QuantOPs (GBits)↓ Runtime (Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' x)↓ ASR↓ KS↑ SF↑ PR↓ QbE↑ IC↑ ASV↓ SD↓ ER↑ HuBERT (Base)[3] – fp16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='42 96.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='62 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='82 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='12 DistillHuBERT[5] – fp16 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='37 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='83 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='11 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='99 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='19 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='02 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='98 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Evaluation of Quantization Techniques on SUPERB Tasks and Profiling Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' KNOWLEDGE DISTILLATION During quantization-aware training, along with the quanti- zation errors accumulated through operations, the gradient can also degrade through back-propagation [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' To miti- gate the gradient degradation through operators, knowledge distillation [15, 18] has proven to be effective where the ”student” model aims to imitate the outputs of the ”teacher” model, regardless of the original loss function of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' There are different strategies to apply knowledge distillation for different scenarios and domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Since quan- tization keeps the model architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' meaning the student (quantized) model shares the same tensor shapes with the teacher (unquantized) model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' we aim to distill not only the final output of the models but also the intermediate outputs and attention weights from each inner Transformer layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' as described in equation (6) and (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' where MSE() is the mean square error operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' yT and yS are model outputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' oT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='i and oS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='i are intermediate outputs for layer i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' aT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='i and aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='i are attention weights for layer i in teacher and student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Lfinal = MSE(yT , yS) + Llayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (6) Llayers = � i(MSE(oT,i, oS,i) + MSE(aT,i, aS,i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' (7) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Experimental Setup We adopt the HuBERT[3] (base) model as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For QAT, the same 960 hours of training set from LibriSpeech[19] for building the baseline model is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The evaluation was performed on the 9 downstream tasks in the SUPERB[4] challenge including automatic speech recognition (ASR), keyword spotting (KS), slot filling (SF), phoneme recogni- tion (PR), query by example (QbE), intent classification (IC), automatic speaker verification (ASV), speaker diarization (SD) and emotion recognition (ER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The tasks are labeled with up/down arrows showing the goals of the metrics as in Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, ASR is measure in word error rates (WERs) therefore lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In all tasks, the model serves as a speech representation extractor with parameters fixed after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The implementation was built on top of fairseq[20] and torchaudio[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For evaluating the resource consumption of the models, we extended the DeepSpeed [22] tool to profile the models for 1) on-disk storage 2) floating point operations 3) quantization operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The definition of these metrics are: Storage: The required space to store all model parame- ters, measured in megabytes (MBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Parameters in fp16 are estimated to take 16 bits each, while quantized pa- rameters take the same bits for annotated weight bits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 8 bits for w8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' FLOPs: The sum of floating operations (FLOPs) dur- ing the forward pass, measured in gigas (Gs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' QuantOPs: The sum of quantization (integer or bi- nary) operations during the forward pass, measured in gigabits (GBits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' QuantOPs are subjective to the num- ber of bits in quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, for operations between a 8-bit integer and a 2-bit integer, a multiplica- tion would take 2∗8 = 16 QuantOPs, while an addition would take max(2, 8) = 8 QuantOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Runtime: The estimated runtime, measured in relative proportion (x) to the baseline HuBERT(+FastConv) model in fp16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Since FLOPs and QuantOPs are fun- damentally different, their execution speeds are highly dependent on the hardware [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In attempt to integrate the two types of operations in a hardware-agnostic fashion, we created an anchor point for conversion rate between FLOPs and QuantOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In Table 1, we as- sume the SqWQ-w8 model runs as fast as its fp16 counterpart and both have runtime 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00, meaning (110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='79 − 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='24) = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='55 Gs in FLOPs would run as fast as 1898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='44 GBits in QuantOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' This conversion Base Model Quant Loss Precision SUPERB Tasks ASR↓ KS↑ SF↑ PR↓ QbE↑ IC↑ ASV↓ SD↓ ER↑ HuBERT (+FastConv[17]) – – fp16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='06 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='91 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='32 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 BiT-LA[1] (Linear +Attention) HuBERT[3] w8a8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='32 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='89 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='31 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='25 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='16 w4a4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='59 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='85 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='89 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='82 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='14 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='22 w2a2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='29 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='80 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='71 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='85 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='29 w1a1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='93 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='77 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='92 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='80 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='50 Knowledge Distillation w8a8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='07 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='40 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='24 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='77 w4a4 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='09 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='58 w2a2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='85 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='56 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='94 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 w1a1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='96 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='78 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='96 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='63 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='83 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='62 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Comparison between Loss Functions for Quantized Model Training: HuBERT and Knowledge Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' rate is applied to all models for runtime estimation in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Experimental Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' SUPERB Tasks and Profiling Table 1 presents the results for SUPERB tasks and model pro- filing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The HuBERT (Base) model [3] as the first baseline has a relative high FLOPs given similar storage size com- pared with the alternative version (+FastConv) in which the convolutional feature extractor is replaced with a more effi- cient configuration from [17] on top of [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Therefore, we applied quantization on top of the more efficient version Hu- BERT(+FastConv) and refer to it as the new baseline, which is 38% more efficient in FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The precision for quantiza- tion in table 1 are labeled with the number of bits for both model weights and activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, w4a2 refers to 4 bits for model weights and 2 bits for activations for BiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Since SqWQ focuses on quantizing the model weights, 8 bits were applied for activations across all setups {w8, w4, w2, w1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Based on the HuBERT(+FastConv) model, 3 different quantization strategies were applied: 1) SqWQ 2) BiT (Lin- ear Only) 3) BiT (Linear+Attention), for each different num- ber of bits {8, 4, 2, 1} were all experimented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Since most SUPERB tasks demonstrate similar trends, we focus on the ASR task as the main metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' To evaluate squashed weight quantization (SqWQ), re- sults showed that the SqWQ-w8 model effectively reduced the storage size (184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='42 → 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='65) with some degradation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='06 → 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='69) on the ASR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The SqWQ-w4 further re- duced the storage (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='65 → 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='19) and also had lower Quan- tOPs (1898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='44 → 1054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='69) due to lower bits applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The further degradation from SqWQ-w8 to SqWQ-w4 is milder (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='69 → 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='98) compared with fp16 to SqWQ-w8 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='06 → 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='69), potentially benefiting from better utilization of bit by fitting the parameters into uniform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The same trend applies to SqWQ-w2 and SqWQ-w1 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For models with BiT quantization applied to linear only (BiT-L), results showed similar trends as the SqWQ mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Both SqWQ and BiT-L applies to linear operations only and share similar trends in which the degradation in- creases and the number of bits becomes lower, with BiT-L showing smaller gaps from the fp16 baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For exam- ple, BiT-L-w1a1 is similar to SqWQ-w2 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23 ↔ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='56) and lower than SqWQ-w1 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23 ↔ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In addi- tion, since BiT applies quantization to activations as well, the models with lower bits also show lower complexity on QuantOPs, as observed between BiT-L-w1a1 and SqWQ-w1 (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73 ↔ 421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' From the FLOPs and estimated runtimes, it is worth noting that although linear operations were moved to quantizated domains, a significant portion of FLOPs (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='24 out of 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='79) remained, which dominated the estimated run- time (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='75) even though the QuantOPs were reduced significantly (1898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='44 → 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73) comparing the BiT-L-w8a8 and BiT-L-w1a1 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For models with BiT quantization applied to both lin- ear and attention operations (BiT-LA), the storage did not reduce but slightly increased due to the scaling factors α and thresholds β introduced in elastic quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, BiT-LA-w1a1 took more space than BiT-L-w1a1 (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23 ↔ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='17) given similar bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' As quantization ad- ditionally applied to attention operations, the quantization error increased across all precisions, but tended to be worse for model with lower bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, results showed that slight degradation from BiT-L-w8a8 to BiT-LA-w8a8 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='03 → 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='07) but the gap was larger from BiT-L-w1a1 to Bit-LA-w1a1 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23 → 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The major benefit from quantizing attention additionally is in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Compar- ing BiT-L-w1a1 and BiT-LA-w1a1, the FLOPs was signifi- cantly reduced (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='29 → 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='82) since the attention operators were moved to quantization domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Although this also increased QuantOPs (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73 → 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='46), it was insignificant for the low bits and the overall reduced estimated runtime (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='75 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In addition to quantization, other approaches such as neu- ral architecture search are also of wide interest for efficient modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' To compare the effectiveness of quantization, we included the results from DistillHuBERT [5], which also aims to improve the efficiency for HuBERT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' From the Ta- ble, comparable models included all three quantization strate- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Comparison between One-step and Scheduled Quan- tization on SUPERB-ASR Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' gies with 2 bits (SqWQ-w2, BiT-L-w2a2 and BiT-LA-w2a2), which were all around 35 MBs in storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Results showed that all three quantization strategies provide lower word er- ror rates (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='56, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='80, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 ↔ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='37) and lighter storage (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='95, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='79, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='84 ↔ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='98), with BiT-LA-w2a2 signif- icantly outperformed on the FLOPs (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='82 ↔ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='34) and estimated runtime (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='15 ↔ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73), which demonstrated that quantization is effective among efficient modeling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Knowledge Distillation for Quantized Model Training As described in section 3, model training with quantization is challenging since quantization error accumulates through back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Therefore, we adopted knowledge distilla- tion for minimizing not only the mean square error between model outputs but also intermediate layer outputs and atten- tion weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' To investigate the impact from knowledge distil- lation, we trained another set of models with exactly the same quantization configurations, the same initialization from the baseline fp16 models, the same dataset but the loss function was changed to the original HuBERT loss [3] for speech rep- resentation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' These models were learning to predict masked frames instead of imitating the baseline fp16 model and there was no intermediate outputs involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' BiT quantization on both linear and attention operations (BiT-LA) was applied and the results were summarized in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' We see that the trends are similar across precisions with models trained with HuBERT loss degrading more than mod- els trained with knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, word error rates are (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='93 ↔ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='96) between HuBERT loss and knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The profiling metrics such as storage and runtime were omitted since they are independent from loss functions and can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' It is evident that knowledge distillation is effective for training quantized mod- els by minimizing intermediate layer outputs to mitigate ac- cumulated quantization errors through back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' One-Step vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Scheduled Quantization In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='1, the models were quantized directly from the original model in fp16 to the target precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For example, BiT-LA-w1a1 was distilled directly from HuBERT(+FastConv)- fp16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' This is referred to as ”one-step” quantization, as op- posed to the scheduled quantization suggested in [1], where findings indicated that multi-step quantization involving in- termediate precisions may improve the performance degrada- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' There are endless combinations to perform multi-step distillation, depending on the number of bits for weights and activations and how many bits to reduce at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' In comparison, one-step quantization provides simpler setup but would potentially suffer from larger performance degrada- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' To understand the impact of scheduled quantization, two quantization schedules were experimented against the one-step results: 1) fp16 →w8a8 →w4a4 →w2a2 →w1a1 and 2) fp16 →w1a2 →w1a1 as suggested in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The results were plotted in Figure 1 where the X-axis is for prevision and the Y-axis is the word error rate in the ASR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Results showed that the models with scheduled quantization offered similar performance as their one-step counterparts, especially at the target precision w1a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' Given the many combinations of scheduled quantization, we draw the preliminary conclusion that no significant improvement was observed from scheduled quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' CONCLUSION In this work, we investigated two novel quantization tech- niques: 1) robustly binarized Transformer [1] 2) squashed weight quantization [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The quantization was applied to HuBERT [3] models for speech representation learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' The experiments were evaluated on the SUPERB [4] bench- mark and significant savings were observed both on storage and estimated runtime through quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For the aggres- sively binarized models, storage was saved by 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='32% in megabytes (MBs) (184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='42 → 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='23), estimated runtime was reduced by 88% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='00 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' For comparable configura- tions to DistillHuBERT [5], 2-bit models offered lower word rates (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='68 ↔ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='37) and estimate runtime (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='15 ↔ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='73) while still smaller in storage (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='84 ↔ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content='98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' With the growing sizes and computational complexity of modern mod- els, we believe it is crucial to make models both compact and efficient so they are more accessible and deployable to resource-constrained environments such as edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' ACKNOWLEDGEMENT We would like to express our sincere gratitude for the follow- ing contributors: 1) Zechun Liu and Barlas Oguz from Meta AI for providing technical details and discussions on the ”Ro- bustly Binarized Transformer” work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 2) Xiaohui Zhang and Zhaoheng Ni from Meta AI for techincal discussions and im- plementation on top of fairseq and torchaudio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 3) Anuj Diwan from University Texas at Austin for technical discussions and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' 1-step fp16->w8a8->w4a4->w2a2->wla1 fp16->wla2->wla1 18 (%) 16 WER 14 SUPERB-ASR 12 10 8 0 fp16 8e8m w4a4 w2a2 wla2 wlal7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' REFERENCES [1] Zechun Liu, Barlas Oguz, Aasish Pappu, Lin Xiao, Scott Yih, Meng Li, Raghuraman Krishnamoorthi, and Yashar Mehdad, “Bit: Robustly binarized multi-distilled trans- former,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNAyT4oBgHgl3EQfwvkf/content/2301.00652v1.pdf'} +page_content=' [2] Nikko Strom, Haidar Khan, and 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Doser1, 2, Marc Kéry3, Andrew O. Finley2,4, Sarah P. Saunders5, Aaron S. +Weed6, Elise F. Zipkin1, 2 +1Department of Integrative Biology, Michigan State University, East Lansing, MI, USA +2Ecology, Evolution, and Behavior Program, Michigan State University, East Lansing, MI, USA +3Swiss Ornithological Institute, Sempach, Switzerland +4Department of Forestry, Michigan State University, East Lansing, MI, USA +5Science Division, National Audubon Society, New York, NY, USA +6Northeast Temperate Inventory and Monitoring Network, National Park Service, Woodstock, +VT, USA +Corresponding Author: Jeffrey W. Doser, email: doserjef@msu.edu; ORCID ID: 0000-0002- +8950-9895 +Data Availability Statement +All data and code associated with this manuscript are available on GitHub (https://github. +com/doserjef/Doser_et__2023_SVC) and will be posted on Zenodo upon acceptance. +Acknowledgements +This work was supported by National Science Foundation (NSF) grants DBI-1954406, DMS- +1916395, and DEB-2213565. +1 +arXiv:2301.05645v1 [stat.AP] 13 Jan 2023 + +Abstract +Aim +Species distribution models (SDMs) are increasingly applied across macroscales using widely +available detection-nondetection data sources. However, assumptions of stationarity in species- +environment relationships or population trends inherent to most SDM techniques are frequently +violated at broad spatial scales. Bayesian spatially-varying coefficient (SVC) models can readily +account for nonstationarity, yet their use is relatively scarce, due, in part, to a gap in under- +standing both the data requirements needed to fit SVC SDMs, as well as the inferential benefits +of applying a more complex modeling framework. +Innovation +Using simulations, we present guidelines and recommendations for fitting single-season and +multi-season SVC SDMs. We display the inferential benefits of SVC SDMs using an empirical +case study assessing spatially-varying trends of 51 forest birds in the eastern US from 2000- +2019. We provide user-friendly and computationally efficient software to fit SVC SDMs in the +spOccupancy R package. +Main conclusions +While all datasets are unique, we recommend a minimum sample size of ∼500 spatial locations +when fitting single-season SVC SDMs, while for multi-season SVC SDMs, ∼100 sites is adequate +for even moderate amounts of temporal replication (e.g., 5 years). Within our case study, we +found 88% (45 of 51) of species had strong support for spatially-varying occurrence trends. +Further, SVC SDMs revealed spatial patterns in occurrence trends that were not evident in +simpler models that assumed a constant trend or separate trends across distinct ecoregions. We +suggest five guidelines: (1) only fit single-season SVC SDMs with more than ∼500 sites; (2) +consider using informative priors on spatial parameters to improve spatial process estimates; (3) +use data from multiple seasons if available; (4) use model selection to compare SVC SDMs with +simpler alternatives; and (5) develop simulations to assess the reliability of inferences. These +guidelines provide a comprehensive foundation for using SVC SDMs to evaluate the presence and +2 + +impact of nonstationary environmental factors that drive species distributions at macroscales. +3 + +Introduction +Elucidating the factors that drive the occurrence patterns and ranges of species is a funda- +mental objective of ecology. Species distribution models (SDMs) are the primary tool used +to study where species occur across both space and time to inform a wide-range of questions +in ecology and conservation (Guisan and Zimmermann, 2000). While SDMs can leverage a +variety of data types (e.g., presence-only, abundance), they are commonly used with detection- +nondetection data in a generalized-linear model (GLM)-based framework, allowing for quantifi- +cation of species-environment relationships and probabilities of local-level occurrence. Numer- +ous methodological approaches have been developed to accommodate complexities that arise in +modeling species distributions, such as imperfect detection (i.e., the failure to observe a species +at a site when it is truly present; MacKenzie et al. 2002), range dynamics (MacKenzie et al., +2003), spatial autocorrelation (e.g., Latimer et al. 2006), and false-positive errors (Royle and +Link, 2006; Miller et al., 2011). +SDMs are increasingly applied across large spatial extents (i.e., macroscales) as a result of +a growing interest in macroecological patterns, the emergence of large-scale citizen science pro- +grams (e.g., eBird, iNaturalist), and increasing availability of data from regional- to continental- +scale monitoring programs. As the spatial extent of analysis increases, the common assumption +of stationarity in species-environment relationships becomes less realistic (Pease et al., 2022b). +Stationarity implies that the effects of environmental covariates are constant throughout the +modeled spatial and/or temporal domain. Accounting for such nonstationarity, or differential +effects of environmental factors across space and/or time (Rollinson et al., 2021), is important +to accurately reflect species-environment relationships and to identify the relative effects of dif- +ferent environmental drivers across a species range (Martínez-Minaya et al., 2018; Sultaire et al., +2022). For example, Sultaire et al. (2022) found spatial variation in the effects of increasing tem- +perature and snow cover duration on snowshoe hare (Lepus americanus) occurrence, suggesting +that climate limits their distribution in multiple different ways across the species range. Such +insight can lead to multi-scale management and conservation actions that adequately address +both large-scale and local-level drivers of species distributions (Pease et al., 2022a). +Nonstationarity in species-environment relationships can arise from a variety of abiotic and +biotic processes (Miller, 2012). Abiotic factors such as historical disturbance regimes, landscape +composition/configuration, fine-scale habitat characteristics (e.g., vegetation quality), and en- +4 + +vironmental conditions (e.g., soil content, temperature) can result in varying effects of environ- +mental factors on species across their ranges (Rollinson et al., 2021). Given spatial heterogeneity +in resource availability, effects of environmental factors on species occurrence may be stronger +in areas with limited resources compared to areas with a large amount of resources (Pease +et al., 2022a). Similarly, the effect of environmental factors may be non-linear, resulting in non- +stationarity in the effect across different spatial regions that span an environmental gradient +(Rousseau and Betts, 2022). Alternatively, nonstationarity may arise from biotic processes such +as local genetic adaptations or spatial variation in species interactions (e.g., predation, compe- +tition). For example, Pease et al. (2022a) found that nonstationarity in the effect of forest cover +on white-tailed deer (Odocoileus virginianus) occurrence across North Carolina was partially +driven by variation in predation pressure across the state. This interplay of abiotic and biotic +factors with species-environment relationships makes nonstationarity a common phenomenon +across ecology (Foody, 2004; Finley, 2011; Jarzyna et al., 2014). +In addition to understanding nonstationarity in species-environment relationships, monitor- +ing programs are often interested in determining whether population trends vary across space +(Bled et al., 2013; Babcock et al., 2016; Meehan et al., 2019). Quantifying spatial variabil- +ity in population trends can help generate hypotheses as to the drivers of population changes +(e.g., Crossley et al. 2021), identify priority areas for conservation or restoration (e.g., Ethier +et al. 2017), and provide insights into where additional monitoring effort is needed to reduce +uncertainty in population trends. +Statistical frameworks exist to accommodate nonstationarity, such as geographically weighted +regression (GWR; Fotheringham et al. 2003), generalized additive models (GAMs; Wood 2006), +and spatio-temporal exploratory models (e.g., Fink et al. 2010). While these approaches are +useful, they often require a priori specification of stratification grids and fixing certain param- +eter values prior to model fitting, which can unduly impact model results and interpretation +(Thorson, 2019). Machine learning approaches such as random forests (Liaw et al., 2002) and +MaxEnt (Phillips et al., 2006) are commonly used to model species distributions and estimate +complex, nonlinear species-environment relationships, but these approaches do not explicitly +estimate spatial nonstationarity in species-environments through spatial covariance functions +(although see Georganos et al. (2021) and Saha et al. (2021) for recent spatial extensions of +random forests). Bayesian spatially varying coefficient (SVC) models (Gelfand et al., 2003) are +5 + +an attractive alternative as their hierarchical construction provides great flexibility for com- +plex, hierarchically-structured ecological data (Finley, 2011; Finley and Banerjee, 2020). In +particular, SVC models are a straightforward extension of spatial GLMs that allow regression +coefficients to vary continuously across space, most commonly using some form of Gaussian +process specification (Banerjee et al., 2014), in addition to allowing the intercept to vary spa- +tially. While Bayesian SVC models are more complex than many of the previously mentioned +alternatives, they allow for full uncertainty propagation, do not require a priori decisions on +grid size or parameter values, and have been shown to outperform geographically weighted re- +gression, the most commonly used alternative, in a variety of simulation and empirical examples +(Wheeler and Calder, 2007; Finley, 2011). +Despite increased recognition of the prevalence of nonstationary in macrosystems (Rollinson +et al., 2021), the use of SVCs within SDMs is still scarce in many fields including wildlife ecology +and conservation. Recent applications suggest increasing interest in this flexible framework for +a variety of ecological applications (e.g., Meehan et al. 2019; Pease et al. 2022b; Sultaire et al. +2022), yet a comprehensive understanding of the data requirements needed to estimate SVCs in +SDMs is lacking. Typical SDM applications present many unique data complexities (e.g., binary +response variable, imperfect detection) and there is little guidance on how such complexities +impact the sample size requirements and reliability of predictions and inference from SDMs with +SVCs (hereafter SVC SDMs). Additionally, the limited implementation of SVC SDMs precludes +demonstration of the inferential gains from such a modeling framework relative to SDMs that +assume stationarity. The R packages sdmTMB (Anderson et al., 2022), VAST (Thorson, 2019), +and INLA (Lindgren and Rue, 2015) provide functionality to fit a variety of SVC SDMs, but +these approaches do not directly account for imperfect detection, which likely contributes to +the limited adoption of SVC SDMs in wildlife ecology (Chapter 9; Kéry and Royle 2021). +Using simulations and an empirical case study, we present guidelines and recommendations +for fitting SVC SDMs via the spOccupancy R package (Doser et al., 2022b) while explicitly +accounting for imperfect detection. We focus our guidelines on practical issues that are likely to +be encountered by practitioners when using SVCs in a species distribution modeling framework. +We use simulations to understand the data requirements necessary to yield reasonably accurate +and precise estimates from SVC SDMs. As an empirical example, we quantify spatially-varying +occurrence trends in 51 forest birds across the eastern U.S. from 2000-2019. By providing explicit +6 + +guidelines and recommendations for the use of SVC SDMs, accompanied with user-friendly +software options that readily accommodate imperfect detection, we hope to increase the careful +application of SVC SDMs in wildlife ecology to produce more accurate species distribution maps +and provide improved inference into the processes affecting populations across spatial scales. +Models +Here we present SVC single-season and multi-season occupancy models, with details on two +SVC SDM variants that do not account for imperfect detection in Appendix S2. We fit all +models in a Bayesian framework with priors and numerical algorithms implemented to maximize +computational efficiency (see Appendix S1 for full MCMC details). +SVC single-season occupancy model +Let sj denote the spatial coordinates of site j, where j = 1, . . . , J. We define z(sj) as the true +presence (1) or absence (0) of the target species at site j with spatial coordinates sj. We model +z(sj) as +z(sj) ∼ Bernoulli(ψ(sj)), +(1) +where ψ(sj) is the occupancy probability of the species at site j. We model ψ(sj) according +to +logit(ψ(sj)) = x(sj)β + ˜x(sj)w(sj), +(2) +where β is a vector of regression coefficients (including an intercept) that describe the non- +spatial effects of covariates x(sj), and w(sj) is a vector of spatially-varying effects of covariates +˜x(sj). Thus ˜x(sj) may be identical to x(sj) if all covariate effects are assumed to vary spatially, +or a subset of x(sj) if some effects are assumed to be constant across space. Note that the model +reduces to a traditional single-species occupancy model when all covariate effects are assumed +constant across space and a spatial occupancy model (Johnson et al., 2013; Doser et al., 2022b) +when only the intercept is assumed to vary across space. +The spatially-varying effects w(sj) serve as local adjustments of the covariate effects at each +7 + +site j from the overall non-spatial effects β, resulting in the covariate having a unique effect on +species occupancy probability at each site j. Following Gelfand et al. (2003), we model each +r = 1, . . . , ˜q spatially-varying effect wr(sj) using a zero-mean spatial Gaussian process. More +specifically, we have +wr(s) ∼ N(0, Cr(s, s′, θr)), +(3) +where Cr(s, s′, θr) is a J × J covariance matrix that is a function of the distances between +any pair of site coordinates s and s′ and a set of parameters (θr) that govern the spatial +process according to a spatial correlation function. Our associated software implementation in +spOccupancy supports four correlation functions: exponential, spherical, Gaussian, and Matérn +(Banerjee et al. 2014; see Appendix S1 for guidance on determining which correlation function +to use). +For the exponential, spherical, and Gaussian correlation functions, θr = {σ2 +r, φr}, +where σ2 +r is a spatial variance parameter and φr is a spatial decay parameter. Large values +of σ2 +r indicate large variation in the magnitude of a covariate effect across space, while values +of σ2 +r close to 0 suggest little spatial variability in the magnitude of the effect. φr controls +the distance-dependent decay of the spatial dependence in the covariate effect and is inversely +related to the spatial range, such that when φr is small, the covariate effect has a larger range +of spatial dependence and varies more smoothly across space compared to larger values of φr +(Appendix S2: Figure S1). +The Matérn correlation function has an additional smoothness +parameter νr, which provides further flexibility in the smoothness and decay of the spatial +process. As computational complexity of the full Gaussian process specification in Equation 3 +increases in cubic order with the number of spatial locations, we use Nearest Neighbor Gaussian +Processes (NNGPs; Datta et al. 2016) as a computationally efficient alternative that provides +nearly identical inference to the full Gaussian process. See Appendix S1 for further details. +To account for imperfect detection in an occupancy modeling framework, k = 1, . . . , K(sj) +sampling replicates are obtained at each site j to estimate whether a nondetection of the target +species is truly an absence (MacKenzie et al., 2002; Tyre et al., 2003). We model the observed +detection (1) or nondetection (0) of a study species at site j, denoted yk(sj), conditional on the +true latent occupancy process z(sj), following +yk(sj) | z(sj) ∼ Bernoulli(pk(sj)z(sj)), +(4) +8 + +where pk(s) is the probability of detecting the species at site j during replicate k given the +species is truly present at the site. We model detection probability as a function of site and/or +observation-level covariates according to +logit(pk(sj)) = vk(sj)α, +(5) +where α is a vector of regression coefficients (including an intercept) that describe the effect +of site and/or observation covariates vk(sj) on detection. Note that effects of covariates on +detection probability may also be nonstationary, but we assume they are stationary throughout +our simulations, case study, and software implementation. However, spatially-varying covariate +effects could in principle be added to the detection model using the same process described +above. +SVC multi-season occupancy model +Consider the case where detection-nondetection data are collected across multiple seasons (e.g., +years, breeding periods) during which the true occupancy status can change. +Such spatio- +temporal data can be used to understand occupancy trends over time (e.g., Isaac et al. 2014), +as well as the environmental variables that drive spatio-temporal shifts in species distributions +(e.g., Rushing et al. 2020). As an extension to the SVC occupancy model, we present a SVC +multi-season occupancy model that estimates spatially-varying effects (as described previously) +of covariates and accounts for temporal autocorrelation using temporal random effects. +Let yk,t(sj) denote the detection or nondetection of the target species at site j during +replicate survey k during time period t, with t = 1, . . . , T. We model yk,t(sj) conditional on the +true occupancy status, zt(sj) of the species at site j during time t according to +zt(sj) ∼ Bernoulli(ψt(sj)), +(6) +where ψt(sj) is the occupancy probability at site j during primary time period t. We model +ψt(sj) following +logit(ψt(sj)) = xt(sj)β + ˜xt(sj)w(sj) + ηt, +(7) +9 + +where ηt is a random temporal effect and the other parameters are defined as before. Note +that the covariates can now vary across space and/or time period. For example, by including +year as a covariate in ˜X we can estimate a spatially-varying trend, and ultimately make predic- +tions to generate trend maps across a region of interest. Because we assume the non-spatial (β) +and spatially-varying (w(sj)) coefficients are constant over the T primary time periods, they +represent the average covariate effects across the temporal extent of the data. We model ηt as +either an unstructured random effect (i.e., ηt ∼ Normal(0, σ2 +T )) or using a first-order autore- +gressive (i.e., AR(1)) covariance structure in which we estimate a temporal variance parameter, +σ2 +T , and a temporal correlation parameter, ρ. +The data yk,t(sj) are modeled conditional on the true occupancy status zt(sj) analogous +to the SVC occupancy model in Equations 4 and 5, with detection probability now allowed to +vary across sites, replicates, and/or primary time periods. +Software implementation and prediction +We implement single-season and multi-season Bayesian SVC SDMs with new functionality in +v0.5.2 of the spOccupancy R package (Doser et al. 2022b; see Appendix S2: Table S1 for +function names). We assign Gaussian priors to all non-spatial regression coefficients, inverse- +Gamma priors for the temporal variance parameter, and uniform priors for all correlation pa- +rameters. +For the spatial variance parameters, we allow for either an inverse-Gamma prior +or uniform prior. We obtain computationally efficient implementations via NNGPs and Pólya- +Gamma data augmentation (Polson et al., 2013). See Appendix S1 for full Markov Chain Monte +Carlo (MCMC) details and Appendix S3 for a detailed spOccupancy vignette. Additionally, +the Bayesian framework readily enables prediction from the posterior predictive distribution, +which facilitates mapping species-specific occupancy probabilities and any SVCs along with +fully propagated uncertainty. We include functionality for prediction in our associated software +implementation (Appendix S3). +Simulation studies +We performed four simulation studies using spOccupancy to evaluate the consequences of failing +to address nonstationarity in SDMs and of how differing data characteristics influence inference +10 + +and predictive performance of SVC SDMs. Below we briefly present four simulation studies, +with additional details on a fifth simulation study in Appendix S2 that assesses the influence of +detection probability on estimates from SVC SDMs. For all simulations, we ran three chains, +each with 15,000 MCMC iterations with a burn-in period of 10,000 iterations and a thinning +rate of 5, resulting in a total of 3000 posterior draws. We assessed convergence using visual +assessment of trace plots with the coda package (Plummer et al., 2006) and the Gelman-Rubin +R-hat diagnostic (Brooks and Gelman, 1998). We used weakly informative priors for all model +parameters (Appendix S2). +Simulation study 1: proof of concept +As a proof of concept, we used simulations to assess how failing to account for nonstationarity +affected inference for various levels of spatial dependence in the covariate effect (e.g., small vs. +high variability, short vs. long range spatial dependence). We simulated data from J = 400 +sites across a unit square and K = 5 replicate surveys at each site for use in an occupancy +modeling framework, where detection probability was set to moderate (average = 0.45) and +varying positively (0.4 on the logit scale) with a standardized observation-level covariate (see +Appendix S2 for an additional simulation study assessing bias and precision across varying +levels of detection probability). We generated species occupancy as a function of an intercept +(logit-scale β0 = 0) and a single covariate (logit-scale β1 = 0), whose effects both varied across +space following Equations 2 and 3 using an exponential correlation function. For the intercept, +we set the spatial variance to 1 and the spatial decay parameter to φ = 3/0.4. When using +an exponential correlation function, the effective range, or the distance at which the spatial +correlation between points drops to 0.05 (Banerjee et al., 2014), corresponds to approximately +3 / φ (i.e., since 3 ≈ −log(0.05)), and so a spatial decay parameter of φ = 3/0.4 yields an effective +range of 0.4 (i.e., 40% of the simulated study area across the unit square). For the SVC of the +covariate effect, we varied the spatial variance parameter across four values (σ2 = {0.1, 0.5, 1, 2}) +and the spatial decay parameter across three values (φ = {3/0.1, 3/0.5, 3/0.8}) to assess how +failing to account for nonstationarity is related to the spatial characteristics of the effect. See +Appendix S2: Figure S1 for the resulting spatial pattern in the SVC under all combinations of +these parameter values. We simulated 50 data sets for each combination of parameter values, +resulting in 600 simulated data sets. We compared the performance of three models: (1) a basic +11 + +occupancy model with no spatially-varying intercept or SVC; (2) a spatial occupancy model +with a spatially-varying intercept but no SVC; and (3) the full SVC occupancy model. We used +the Widely Applicable Information Criterion (WAIC) as a measure of model fit (Watanabe, +2010) and four-fold cross-validation using model deviance as a metric of predictive performance +(Hooten and Hobbs, 2015). +Additionally, we generated data using the same criteria as above but now with perfect +detection. We then repeated the simulation study using a SVC generalized linear model (SVC +GLM) to assess differences in bias between occupancy models and GLMs that do not explicitly +account for imperfect detection. +We compared the absolute bias and 95% credible interval +widths for occupancy probability estimates from the SVC occupancy model and SVC GLM to +determine if there were any differences in model performance for the two model types. +Simulation study 2: sample size requirements for single-season +SVC models +We used simulations to assess how the total number of sampled sites and the number of spatially +varying coefficients influenced bias and precision from SVC occupancy models. We generated +data with eight different amounts of spatial locations (J = {200, 400, 800, 1200, 1600, 2000, 4000, 6000}) +and under different numbers of SVCs (1, 2, and 3) with moderate spatial variance, resulting in +24 simulation scenarios. For all scenarios, the model intercept was space-varying. We simulated +50 data sets for each combination of parameter values, yielding 1200 total simulated data sets. +We fit a SVC occupancy model to each data set with the number of SVCs used to generate the +data. We assessed bias and precision of occupancy probability estimates (ψ(sj)) using mean +absolute bias and 95% credible interval widths. We further computed the Pearson’s correlation +coefficient between the estimated SVCs and the true values to assess the reliability of inferences +drawn from SVC occupancy models. +Simulation study 3: sample size requirements for multi-season +SVC models +For our third simulation study, we repeated simulation study 2 but now we assessed the per- +formance of a multi-season SVC occupancy model under varying amounts of spatial locations +12 + +(J = {100, 200, 400, 800, 1600}, number of SVCs (1, 2, 3), and number of time periods (5, 10, +15). We generated data using an AR(1) temporal autocorrelation structure. We simulated 50 +data sets for each combination of parameter values, resulting in 2250 total simulated data sets. +We assessed bias and precision using the same criteria as in simulation study 2. +Simulation study 4: spatially varying population trends +As quantification of spatially-varying trends in occupancy probability is a common objective of +monitoring programs, we used simulations to understand the data requirements necessary for +reliable estimation of a spatially-varying temporal trend using an SVC multi-season occupancy +model. We assessed how the total number of sampled sites and the number of primary time +periods influenced the accuracy of temporal trend estimates from SVC multi-season occupancy +models. We varied the total number of sites (J = {100, 200, 400, 800, 1600}) and the number +of time periods (T = {5, 10, 15}). We simulated 50 data sets under each of the 15 scenarios, +resulting in 750 total simulated data sets. We used Pearson correlation coefficients with the +true values to assess accuracy of the spatially-varying trend estimates from an SVC multi-season +occupancy model. +Case study +Quantifying spatially-explicit trends can help identify areas of conservation interest (e.g., climate- +change refugia) as well as provide insights on species range dynamics (e.g., trailing vs. leading +edge trends). We applied the SVC multi-season occupancy model to assess trends in 51 bird +species across a ∼4.04 million km2 region of the eastern US (i.e., the continental US east of +the 100th meridian) from 2000-2019 (T = 20 years) using detection-nondetection data from +the North American Breeding Bird Survey (Pardieck et al., 2020). We restricted our analysis +to a community of 66 eastern forest bird species following the classification of Bateman et al. +(2020), which consisted of species with large variations in abundance and breeding range size. +We only assessed trends for 51 of the 66 species whose breeding ranges (derived from BirdLife +International 2021) had at least 50% overlap with the study area (see Appendix S2 Table S3 +for species names). Our objectives for this case study were to (1) develop spatially-explicit +maps of occupancy trends for each of the 51 species across the eastern US, and (2) assess the +13 + +performance of the SVC multi-season occupancy model to (i) a model with a constant trend +over space and (ii) a model that allows the trend to vary across coarse ecological strata (i.e., +Bird Conservation Regions (BCRs)). +We used data from J = 1846 BBS routes (i.e., sites) sampled at least once between 2000- +2019 (mean number of sampled years per route = 15). BBS observers performed a three-minute +point count survey at each of 50 stops along each route, counting all birds seen or heard within +a 0.4 km radius. We summarized the data for each species at each site into K = 5 spatial +replicates (each comprising data from 10 of the 50 stops), where each replicate took value 1 if +the species was detected at any of the 10 stops in that replicate, and value 0 if the species was +not detected. While such an approach has been used in previous studies with BBS data (e.g., +Rushing et al. 2020), this use of spatial replicates in an occupancy modeling framework may +lead to violation of the closure assumption (Kendall and White, 2009), and so we refer to our +response as species-specific occurrence rather than occupancy. +For each of the 51 species, we fit a SVC multi-season occupancy model in which we modeled +occurrence probability as a function of a spatially-varying intercept, a spatially-varying trend, +and an AR(1) temporal random effect. +Because our interest was solely in assessing spatial +variation in occurrence trends, we did not estimate any SVCs for spatially-varying covariates +(e.g., forest cover, temperature) and rather used a spatially-varying intercept to account for +variability in occurrence probability across space, as is commonly done when inference focuses +on trends (e.g., Bled et al. 2013; Meehan et al. 2019). We modeled detection probability using +a separate intercept for each year, linear and quadratic effects of day of survey to account for +seasonal variation in detection probability, linear and quadratic effects of survey replicate to +account for variability in detection probability over a day of sampling, and a random observer +effect. The occurrence trend covariate and all detection covariates were standardized to have +mean 0 and standard deviation 1. For each species, we used pre-existing breeding ranges from +BirdLife International (BirdLife International, 2021) to only use routes that fell inside a 50 km +buffer of the species range when fitting the occupancy model. We compared the full SVC multi- +season occupancy model to a model that assumed the trend was constant across space (i.e., +constant model) and a model that estimated a separate trend parameter for 21 BCRs in the +study region (i.e., BCR model). The BCR model represents a simpler alternative to the SVC +model that allows for spatial variability in the occurrence trend at a coarse, pre-determined +14 + +resolution, i.e., by regional stratification of the model coefficients. If occurrence trends vary +spatially at a coarse resolution, the BCR model may be a simpler, more computationally-efficient +alternative to the full SVC model (Pease et al., 2022a). We used WAIC as a measure of model +fit to compare the three candidate models for each species. We fit all models in spOccupancy, +running three chains of each model for 50,000 MCMC iterations with a burn-in period of 30,000 +iterations and a thinning rate of 20, yielding 3000 posterior samples. We used vague priors for all +non-spatial parameters. We used a uniform prior for the spatial variance parameters to restrict +the spatial variance from taking large values that are unreasonable on the logit scale (Wright +et al., 2021). We additionally used an informative uniform prior on the spatial decay parameter +to prevent unreasonably small estimates of the spatial range. Specifically, we restricted the +upper bound of the uniform distribution to 3/667.93, which resulted in a minimum effective +spatial range of 667.93km. We found this was sufficiently flexible to account for nonstationarity +in species trends while simultaneously preventing the model from overfitting. See Appendix S2 +for additional discussion on prior distributions. After fitting the models, we predicted across +the range of each species in the study area to generate maps of spatially-varying occurrence +trends across the twenty year period. +Results +Simulation studies +The benefits of modeling nonstationarity in SVC SDMs were dependent on the characteristics +of the spatial dependence in the covariate effect (Table 1). Improvements in model performance +of the SVC occupancy model were highest for covariates effects with a large spatial variance and +large spatial range according to both WAIC and four-fold cross-validation deviance (Table 1, +Appendix S2: Table S2). As expected, when the spatial variance in the covariate effect was small +(0.1, 0.5), an SVC occupancy model only yielded small improvements in WAIC compared to a +spatial occupancy model that assumed a stationary covariate effect, and either no improvements +or very small improvements in cross-validation deviance. Interestingly, predictive performance +was generally worse or only marginally improved when the effective range of the covariate effect +was small (10% of study area), regardless of the spatial variance. +Accuracy and precision were only marginally higher for an SVC GLM compared to an SVC +15 + +occupancy model, with bias on average being 2.22% lower for the SVC GLM simulations and +95% credible intervals on average being 2.85% smaller for the SVC GLM simulations. These +small differences indicate minimal differences in the ability of SVC GLMs and SVC occupancy +models to estimate occurrence probability when data were generated according to the respective +model. +Bias and uncertainty in occupancy probability estimates from an SVC occupancy model de- +creased as the number of spatial locations increased, and increased as the number of estimated +SVCs increased (Figure 1). Widths of 95% credible intervals on occupancy probabilities spanned +more than 70% of the possible range (i.e., 0-1) when fitting an SVC occupancy model with 200 +or 400 sites, indicating large uncertainty in SVC occupancy model estimates from data sets with +fewer than ∼ 500 sites. Accuracy of the estimated SVCs showed a similar pattern, with accuracy +strongly increasing as the number of spatial locations increases. In particular, the correlation +coefficient of the estimated SVCs with the true simulated values was extremely low when fitting +models with 200 or 400 sites (Figure 1C), indicating inference on nonstationary covariate effects +in SVC occupancy models might be unreliable with data sets comprised of fewer than approx- +imately 500 spatial locations. Accuracy and precision of occupancy probability estimates in +SVC occupancy models clearly increased with increasing amounts of temporal replication and +increasing detection probability, while accuracy and precision of SVC estimates showed no clear +patterns with varying amounts of replication and detection probability (Appendix S2: Tables +S7, S8). +Bias and uncertainty in occupancy probability and SVC estimates from a multi-season SVC +occupancy model showed similar patterns to the single-season SVC occupancy model, with +bias and uncertainty decreasing as the number of spatial locations and number of time periods +increases, and increasing as more SVCs were estimated in the model (Figure 2). Importantly, +bias and uncertainty for a given number of spatial locations was much smaller in a multi- +season SVC occupancy model compared to a single-season SVC occupancy model. For example, +absolute bias in occupancy probability for a multi-season SVC occupancy model with 200 sites, +5 time periods, and a single SVC was 0.12, while for a single-season model with the same +number of sites and SVCs was 0.17, a 29% decrease in bias. Accuracy of the estimated SVCs in +multi-season occupancy models was also much higher than single-season occupancy models with +the same number of spatial locations, in particular when the number of time periods is high +16 + +(Figure 2). This result suggests that the additional information provided by temporal replication +makes it possible to assess nonstationary covariate effects on occupancy probability with a more +modest number of spatial locations (∼ 100) compared to single-season SVC occupancy models. +Additionally, accuracy of spatially-varying trend estimates from a SVC multi-season occupancy +model showed an identical pattern, with correlation coefficients with the true values all high (i.e., +>0.6), even for a modest number of spatial locations (i.e., 100) and time periods (5; Appendix +S2: Supplemental Figure S2). +Case study +There was strong support for nonstationarity in twenty-year occurrence trends across the east- +ern US for the majority of the forest bird species that we included in our analysis (Figure 3), +with many species showing both positive and negative trends across their breeding range (Fig- +ure 4, Appendix S2: Supplemental Figures S3-S53). The SVC multi-season occupancy model +substantially outperformed (i.e., ∆WAIC > 2) the BCR model and the constant trend model +for 88% of all species (45 of 51) according to the WAIC. Of the six species with less support for +the SVC model, five of them (Golden-winged Warbler, American Woodcock, Mississippi Kite, +Red-cockaded Woodpecker, Eastern Screech Owl) had very low raw occurrence probabilities +(i.e., < 0.05), while the sixth species (Broad-winged Hawk) showed a fairly constant positive +trend across its range (Appendix S2: Supplemental Table S4). Generally, the SVC model re- +vealed spatial heterogeneity in occurrence trends not evident in either the constant or BCR +models (Figure 5). More specifically, the constant model averaged over any spatial variability +in occurrence trends (Figure 5A, D, G), while the BCR model was able to adequately capture +some spatial variability in trends, but was not able to capture heterogeneity in trends occurring +within a given BCR (Figure 5B, E, H). +Discussion +Accounting for nonstationarity in species-environment relationships is increasingly important as +the scope of ecological research expands in spatial and temporal extent (Rollinson et al., 2021). +Bayesian SVC models are a flexible approach to account for nonstationarity (Gelfand et al., +2003), yet their use in species distribution modeling and wildlife ecology has been limited, +17 + +perhaps due to their increased complexity compared to alternative approaches and a lack of +user-friendly software to fit SVC SDMs that simultaneously accommodate other common data +complexities in wildlife ecology (i.e., imperfect detection; Chapter 9, Kéry and Royle 2021). +In a case study on a forest bird community in the eastern US, we found strong support for +nonstationarity in occurrence trends for 45 out of 51 eastern forest bird species, illustrating the +potentially high prevalence of nonstationarity in model coefficients within empirical data sets. +Using simulations, we found that reasonable accuracy and precision of SVC SDMs requires a +large number of spatial locations (i.e., >500 spatial locations in single-season SDMs), while the +additional temporal data generated from multi-season sampling leads to accurate and precise +estimates from multi-season SVC SDMs with a more modest number of spatial locations (i.e., +100 spatial locations). +When simulating data with a single SVC, improvements in model performance of an SVC +occupancy model were highest when the effect had a large spatial variance and long effective +range (Table 1). As the spatial variance decreased toward zero, the magnitude of the spatial +component of the covariate effect became increasingly small, ultimately resulting in negligible +changes in the estimated occupancy probability between a model that includes an SVC and a +model that assumes a constant covariate effect. This is analogous to fitting a model with a +traditional random effect, where the importance of the random effect decreases as the random +effect variance approaches zero and model comparison approaches will often favor the simpler +model. As the effective spatial range decreases toward zero, the spatial correlation in the SVCs +occurs over a smaller distance, and thus the estimate of the SVC at any given location is +informed by a smaller number of data points. For binary data, it is not possible to estimate +an unstructured site-level random effect (Bolker, 2022), which likely contributed to why we +saw negligible differences in predictive performance between the SVC occupancy model and +the spatial occupancy model when the effective spatial range was small (Table 1). Thus, SVC +SDMs that use binary data will be able to better accommodate covariates whose effects are +presumed to show high correlation across relatively large spatial regions compared to those that +vary across spatial scales that are close to the minimum distance between spatial locations in +the data set. +Bias of occupancy probability and SVC estimates from SVC SDMs was highly dependent +on the number of spatial locations in the data set and, to a lesser extent, the number of SVCs +18 + +estimated in the model (Figures 1, 2). For single-season SVC SDMs, correlation coefficients +of the estimated SVCs with the true simulated values were noticeably low (i.e., ≤ 0.5) when +simulating data with fewer than approximately 500 spatial locations, but steadily increased to +greater than 0.8 as the number of spatial locations increased (Figure 1C). This suggests that +SVC SDMs can provide reasonable inference on nonstationary covariate effects, but only for +data sets with a large number of spatial locations (i.e., > 500), which is substantially larger +than the number of spatial locations needed to fit a standard occupancy model with covariates +(Kéry and Royle, 2016; MacKenzie et al., 2017). For multi-season SVC SDMs, the temporal +replication provides additional information to estimate SVCs, which resulted in more accurate +and precise estimates of occupancy probability and SVCs for a given number of spatial locations +than we found in single-season SVC SDMs (Figures 2, Appendix S2: Supplemental Figure S4). +In the eastern forest bird case study, we found substantial support for nonstationarity in +occurrence trends from 2000-2019 for 45 out of 51 modeled species (i.e., 88%) (Figure 3, Ap- +pendix S2: Supplemental Table S4). Occurrence trends varied from species to species, with some +species showing clear declines across much of their range (e.g., Chimney Swift, Appendix S2: +Supplemental Figure S15), others showing widespread increases in occurrence (e.g., Mississippi +Kite, Appendix S2: Supplemental Figure S30), and others showing patterns consistent with +latitudinal range shifts (e.g., Eastern Wood-pewee, Wood Thrush, Figure 4B, C; Rushing et al. +2020). Our findings of substantial spatial variability in occurrence trends largely align with the +variability demonstrated by spatially-explicit relative abundance trends estimated using eBird +data (Fink et al., 2022). Only one species (Eastern Screech Owl) had substantial support (i.e., +∆WAIC > 2) of the BCR model over the SVC model, suggesting that generally across species +in the community there was substantial variation of occurrence trends within BCRs. The BCR +model, which estimates a separate occurrence trend for each BCR within the species range, is +a simpler alternative to the SVC model that allows trends to vary across pre-defined regions. +While the BCR model provides improved inference compared to the constant model, it averages +over any heterogeneity in trends within a BCR, which can mask more fine-scale patterns in +population trends. For example, the BCR model did not capture an increasing trend of Gray +Catbird in Louisiana and southern Mississippi that was found in the SVC model (Figure 5). +Smith and Edwards (2021) recently developed a similar model for BBS count data that esti- +mates separate relative abundance trends for pre-defined spatial units as unstructured random +19 + +effects. Our results here, although for occurrence and not relative abundance, suggest an SVC +model may provide additional flexibility and detailed inference for estimating spatially-varying +relative abundance trends from BBS data. +We envision several extensions to the SVC SDMs presented here and in the associated +software implementation in spOccupancy. First, the SVC models could be extended to a multi- +species framework that uses a spatial factor modeling approach to model SVCs and account for +correlations between species (Doser et al., 2022a). Second, we could incorporate SVCs in the +detection portion of the occupancy model, which would allow effects of covariates that influence +detectability to vary spatially (Thorson, 2019). Lastly, we could extend the multi-season SVC +models to allow the covariates to vary both spatially and temporally using a dynamic linear +modeling framework (Finley et al., 2012). +Guidelines when fitting spatially-varying coefficient species dis- +tribution models +Below we present a set of five practical guidelines for practitioners to consider when fitting SVC +SDMs. We provide this set of guidelines along with user-friendly and computationally efficient +implementations of single-season and multi-season SVC SDMs in the spOccupancy R package +(Doser et al., 2022b) and an associated vignette that provides a detailed walk through for fitting +these models (Appendix S3). +1. Avoid fitting single-season SVC SDMs with fewer than ∼500 sites. The limi- +tations of binary data make it difficult to accurately estimate SVCs without a relatively +large number of spatial locations (Figure 1). When inference on nonstationary covari- +ate effects is a primary objective, we recommend only fitting single-season SVC SDMs +(with or without imperfect detection) using data sources that have at least ∼ 500 spatial +locations. +2. Consider using an informative prior to prevent small effective spatial range +estimates. Estimating an SVC that has a small effective spatial range (or equivalently +a large spatial decay parameter φ) is inherently difficult when using binary data in SVC +SDMs as a result of limitations in estimating site-level random effects with binary data +(Bolker, 2022). In the eastern forest bird case study, we found that using an informative +20 + +prior on the spatial decay parameters restricted small estimates of the effective spatial +range and eliminated overfitting of the estimated SVCs while still providing sufficiently +flexible estimates of the spatially-varying trends. See Appendix S1 for a discussion on +specifying the prior distribution for the spatial decay parameters. +3. Use data from multiple seasons when available. The additional replication provided +by data collected over multiple seasons substantially improves the ability to estimate +SVCs with a smaller number of spatial locations (e.g., 100 sites; Figures 2, Appendix S2: +Supplemental Figure S4). When available, we suggest using data from multiple seasons +to estimate SVC SDMs, in which case the estimated SVCs are interpreted as the average +effect of the covariate across the temporal extent of the data on species occupancy at any +given location. +4. Use model selection to compare SVC SDMs with simpler submodels. As we did +in the eastern forest bird case study, comparing SVC SDMs to simpler models that allow +covariate effects to vary across designated spatial units (e.g., ecoregions, management +units) can provide insight into the spatial scale at which covariate effects vary across +space. Our associated software implementation in spOccupancy provides functionality to +perform model selection with WAIC and k-fold cross-validation. +5. Use simulations to assess reliability of predictions and inference. We performed +simulations to assess the reliability of SVC SDMs under a variety of scenarios. However, +we suggest performing simulations prior to fitting an SVC SDM based on the character- +istics of the data set at hand (e.g., in terms of sample sizes). This can provide insight on +the amount of bias and uncertainty to expect in estimates, and if reliable inference of the +SVCs can be made given a set number of spatial locations. We provide multiple functions +in spOccupancy to simulate data with SVCs for such assessments, which we describe in +detail in the associated vignette (Appendix S3). +Nonstationarity in species-environment relationships is prevalent throughout ecology (Rollinson +et al., 2021). As we demonstrated, the use of spatially-varying coefficients in species distribution +models can help elucidate the environmental factors that drive species distribution dynamics, +especially across broad spatial scales. For accurate conclusions regarding the drivers of species +distributions over space and time, we require reliable, accessible, and efficient computational +21 + +methods to quantify nonstationary relationships. 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Ecology and Evolution, 11(13):8516–8527. +27 + +Tables and Figures +Table 1: Model comparison results for a non-spatial occupancy model (OCC), a spatially- +varying intercept occupancy model (SVI), and a spatially-varying coefficients occupancy +model (SVC) using simulated data with varying degrees of spatial autocorrelation. Values +represent the difference in four-fold cross-validation deviance and WAIC for the SVC +model compared to the two candidate models. Lower values indicate better performance, +with boldface indicating the best performing model for a given scenario. +Values are +averaged across 50 simulated data sets. +Effective spatial +Spatial +Deviance +WAIC +range (%) +variance +OCC +SVI +SVC +OCC +SVI +SVC +10 +0.1 +24.07 +-0.83 +0.00 +31.34 +5.85 +0.00 +10 +0.5 +18.54 +0.34 +0.00 +25.86 +2.89 +0.00 +10 +1.0 +17.66 +-0.49 +0.00 +28.79 +6.84 +0.00 +10 +2.0 +14.48 +2.43 +0.00 +22.40 +5.90 +0.00 +30 +0.1 +23.84 +0.40 +0.00 +29.78 +4.80 +0.00 +30 +0.5 +22.20 +4.39 +0.00 +31.01 +9.76 +0.00 +30 +1.0 +28.24 +9.36 +0.00 +35.54 +15.56 +0.00 +30 +2.0 +35.13 +22.66 +0.00 +38.83 +22.35 +0.00 +80 +0.1 +24.19 +-0.29 +0.00 +31.29 +5.90 +0.00 +80 +0.5 +26.93 +4.29 +0.00 +33.40 +4.63 +0.00 +80 +1.0 +39.28 +16.13 +0.00 +38.97 +15.37 +0.00 +80 +2.0 +44.62 +33.01 +0.00 +46.16 +28.64 +0.00 +28 + +0.10 +0.12 +0.14 +0.16 +0.18 +0 +2000 +4000 +6000 +Number of sites +Absolute bias +(A) Occupancy probability bias +0.5 +0.6 +0.7 +0.8 +0 +2000 +4000 +6000 +Number of sites +95% CI Width +(B) Occupancy probability uncertainty +0.4 +0.5 +0.6 +0.7 +0.8 +0 +2000 +4000 +6000 +Number of sites +Correlation coefficient +(C) SVC Accuracy +Number of SVCs +1 +2 +3 +Figure 1: Absolute bias in occupancy probability estimates (A), 95% credible interval +(CI) widths of occupancy probability estimates (B), and accuracy of spatially-varying +coefficient estimates from a spatially-varying coefficient occupancy model (C). Values in +(C) are Pearson correlation coefficients between the estimated SVCs and the true values +used to simulate the data, such that higher values indicate higher accuracy of the model. +Values are averaged across 50 simulated data sets. +29 + +5 years +10 years +15 years +400 +800 +1200 +1600 +400 +800 +1200 +1600 +400 +800 +1200 +1600 +0.06 +0.09 +0.12 +0.15 +Number of sites +Absolute bias +(A) Occupancy probability bias +5 years +10 years +15 years +400 +800 +1200 +1600 +400 +800 +1200 +1600 +400 +800 +1200 +1600 +0.3 +0.4 +0.5 +0.6 +0.7 +Number of sites +95% CI Width +(B) Occupancy probability uncertainty +5 years +10 years +15 years +400 +800 +1200 +1600 +400 +800 +1200 +1600 +400 +800 +1200 +1600 +0.6 +0.7 +0.8 +0.9 +Number of sites +Correlation coefficient +(C) SVC Accuracy +Number of SVCs +1 +2 +3 +Figure 2: Absolute bias in occupancy probability estimates (A), 95% credible interval +(CI) widths of occupancy probability estimates (B), and accuracy of spatially-varying +coefficient estimates from a spatially-varying coefficient multi-season occupancy model +under differing numbers of primary time periods (i.e., years). Values in (C) are Pearson +correlation coefficients between the estimated SVCs and the true values used to simulate +the data, such that higher values indicate higher accuracy of the model. +Values are +averaged across 50 simulated data sets. +30 + +GWWA +KEWA +AMWO +BBCU +BAOR +BWWA +EASO +WOTH +BACS +BLJA +CERW +FISP +CHSW +PROW +GRCA +SCTA +WEWA +GCFL +TUTI +BRTH +RBGR +EATO +EAWP +CACH +CWWI +EABL +INBU +PRAW +WEVI +RCWO +BHNU +YBCU +RHWO +ACFL +EAPH +SUTA +RTHU +OROR +YTWA +YTVI +PIWA +NOPA +BWHA +HOWA +LOWA +NOCA +CARW +RSHA +SWWA +RBWO +MIKI +0.00 +0.25 +0.50 +0.75 +1.00 +Proportion of sites with positive trend +Species +Figure 3: Proportion of BBS routes within a given species range with a positive trend +estimate from a spatially-varying coefficient occupancy model for 51 eastern forest bird +species. Points represent posterior median and gray lines denote 95% credible intervals. +See Appendix S2: Table S3 for definition of species codes. +31 + +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +−0.5 +0.0 +0.5 +Trend +(logit scale) +(A) Tufted Titmouse +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +−0.8 +−0.4 +0.0 +0.4 +Trend +(logit scale) +(B) Eastern Wood−pewee +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +−1.0 +−0.5 +0.0 +0.5 +1.0 +Trend +(logit scale) +(C) Wood Thrush +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +−0.5 +0.0 +0.5 +Trend +(logit scale) +(D) Blue−winged Warbler +Figure 4: Mean predictions of the spatially-varying occurrence trend from 2000-2019 from +a spatially-varying coefficient occupancy model for four example species that show varying +degrees of heterogeneity in trends across the study region. Trends are only shown within +each species range. +Panel A: Tufted Titmouse (3.15 million km2); Panel B: Eastern +Wood-pewee (3.74 million km2); Panel C: Wood Thrush (3.12 million km2); Panel D: +Blue-winged Warbler (1.73 million km2). +32 + +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(A) Gray Catbird Constant +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(B) Gray Catbird BCR +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(C) Gray Catbird SVC +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(D) Eastern Phoebe Constant +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(E) Eastern Phoebe BCR +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(F) Eastern Phoebe SVC +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(G) Scarlet Tanager Constant +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(H) Scarlet Tanager BCR +25°N +30°N +35°N +40°N +45°N +50°N +100°W + 95°W + 90°W + 85°W + 80°W + 75°W +Longitude +Latitude +(I) Scarlet Tanager SVC +−1 +0 +1 +Trend +(logit scale) +Figure 5: Mean predictions of an occurrence trend from 2000-2019 for three example +species from three different models: a spatial occupancy model with a constant trend +across the species range (Constant), a spatial occupancy model with a separate trend for +each Bird Conservation Region (BCR), and an SVC occupancy model (SVC) estimating +a spatially-varying trend. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' USA 5Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' National Audubon Society,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' USA 6Northeast Temperate Inventory and Monitoring Network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' National Park Service,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Woodstock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' VT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' USA Corresponding Author: Jeffrey W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Doser, email: doserjef@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' ORCID ID: 0000-0002- 8950-9895 Data Availability Statement All data and code associated with this manuscript are available on GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' com/doserjef/Doser_et__2023_SVC) and will be posted on Zenodo upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Acknowledgements This work was supported by National Science Foundation (NSF) grants DBI-1954406, DMS- 1916395, and DEB-2213565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='05645v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='AP] 13 Jan 2023 Abstract Aim Species distribution models (SDMs) are increasingly applied across macroscales using widely available detection-nondetection data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' However, assumptions of stationarity in species- environment relationships or population trends inherent to most SDM techniques are frequently violated at broad spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Bayesian spatially-varying coefficient (SVC) models can readily account for nonstationarity, yet their use is relatively scarce, due, in part, to a gap in under- standing both the data requirements needed to fit SVC SDMs, as well as the inferential benefits of applying a more complex modeling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Innovation Using simulations, we present guidelines and recommendations for fitting single-season and multi-season SVC SDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We display the inferential benefits of SVC SDMs using an empirical case study assessing spatially-varying trends of 51 forest birds in the eastern US from 2000- 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We provide user-friendly and computationally efficient software to fit SVC SDMs in the spOccupancy R package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Main conclusions While all datasets are unique, we recommend a minimum sample size of ∼500 spatial locations when fitting single-season SVC SDMs, while for multi-season SVC SDMs, ∼100 sites is adequate for even moderate amounts of temporal replication (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 5 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Within our case study, we found 88% (45 of 51) of species had strong support for spatially-varying occurrence trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Further, SVC SDMs revealed spatial patterns in occurrence trends that were not evident in simpler models that assumed a constant trend or separate trends across distinct ecoregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We suggest five guidelines: (1) only fit single-season SVC SDMs with more than ∼500 sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2) consider using informative priors on spatial parameters to improve spatial process estimates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (3) use data from multiple seasons if available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (4) use model selection to compare SVC SDMs with simpler alternatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' and (5) develop simulations to assess the reliability of inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' These guidelines provide a comprehensive foundation for using SVC SDMs to evaluate the presence and 2 impact of nonstationary environmental factors that drive species distributions at macroscales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 3 Introduction Elucidating the factors that drive the occurrence patterns and ranges of species is a funda- mental objective of ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Species distribution models (SDMs) are the primary tool used to study where species occur across both space and time to inform a wide-range of questions in ecology and conservation (Guisan and Zimmermann, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' While SDMs can leverage a variety of data types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', presence-only, abundance), they are commonly used with detection- nondetection data in a generalized-linear model (GLM)-based framework, allowing for quantifi- cation of species-environment relationships and probabilities of local-level occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Numer- ous methodological approaches have been developed to accommodate complexities that arise in modeling species distributions, such as imperfect detection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', the failure to observe a species at a site when it is truly present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' MacKenzie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2002), range dynamics (MacKenzie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2003), spatial autocorrelation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Latimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2006), and false-positive errors (Royle and Link, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' SDMs are increasingly applied across large spatial extents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', macroscales) as a result of a growing interest in macroecological patterns, the emergence of large-scale citizen science pro- grams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', eBird, iNaturalist), and increasing availability of data from regional- to continental- scale monitoring programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As the spatial extent of analysis increases, the common assumption of stationarity in species-environment relationships becomes less realistic (Pease et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Stationarity implies that the effects of environmental covariates are constant throughout the modeled spatial and/or temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Accounting for such nonstationarity, or differential effects of environmental factors across space and/or time (Rollinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2021), is important to accurately reflect species-environment relationships and to identify the relative effects of dif- ferent environmental drivers across a species range (Martínez-Minaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Sultaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For example, Sultaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2022) found spatial variation in the effects of increasing tem- perature and snow cover duration on snowshoe hare (Lepus americanus) occurrence, suggesting that climate limits their distribution in multiple different ways across the species range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Such insight can lead to multi-scale management and conservation actions that adequately address both large-scale and local-level drivers of species distributions (Pease et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Nonstationarity in species-environment relationships can arise from a variety of abiotic and biotic processes (Miller, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Abiotic factors such as historical disturbance regimes, landscape composition/configuration, fine-scale habitat characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', vegetation quality), and en- 4 vironmental conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', soil content, temperature) can result in varying effects of environ- mental factors on species across their ranges (Rollinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Given spatial heterogeneity in resource availability, effects of environmental factors on species occurrence may be stronger in areas with limited resources compared to areas with a large amount of resources (Pease et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Similarly, the effect of environmental factors may be non-linear, resulting in non- stationarity in the effect across different spatial regions that span an environmental gradient (Rousseau and Betts, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Alternatively, nonstationarity may arise from biotic processes such as local genetic adaptations or spatial variation in species interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', predation, compe- tition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For example, Pease et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2022a) found that nonstationarity in the effect of forest cover on white-tailed deer (Odocoileus virginianus) occurrence across North Carolina was partially driven by variation in predation pressure across the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' This interplay of abiotic and biotic factors with species-environment relationships makes nonstationarity a common phenomenon across ecology (Foody, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Finley, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Jarzyna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' In addition to understanding nonstationarity in species-environment relationships, monitor- ing programs are often interested in determining whether population trends vary across space (Bled et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Babcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Meehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Quantifying spatial variabil- ity in population trends can help generate hypotheses as to the drivers of population changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Crossley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2021), identify priority areas for conservation or restoration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Ethier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2017), and provide insights into where additional monitoring effort is needed to reduce uncertainty in population trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Statistical frameworks exist to accommodate nonstationarity, such as geographically weighted regression (GWR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Fotheringham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2003), generalized additive models (GAMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Wood 2006), and spatio-temporal exploratory models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Fink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' While these approaches are useful, they often require a priori specification of stratification grids and fixing certain param- eter values prior to model fitting, which can unduly impact model results and interpretation (Thorson, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Machine learning approaches such as random forests (Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2002) and MaxEnt (Phillips et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2006) are commonly used to model species distributions and estimate complex, nonlinear species-environment relationships, but these approaches do not explicitly estimate spatial nonstationarity in species-environments through spatial covariance functions (although see Georganos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2021) and Saha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2021) for recent spatial extensions of random forests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Bayesian spatially varying coefficient (SVC) models (Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2003) are 5 an attractive alternative as their hierarchical construction provides great flexibility for com- plex, hierarchically-structured ecological data (Finley, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Finley and Banerjee, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' In particular, SVC models are a straightforward extension of spatial GLMs that allow regression coefficients to vary continuously across space, most commonly using some form of Gaussian process specification (Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2014), in addition to allowing the intercept to vary spa- tially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' While Bayesian SVC models are more complex than many of the previously mentioned alternatives, they allow for full uncertainty propagation, do not require a priori decisions on grid size or parameter values, and have been shown to outperform geographically weighted re- gression, the most commonly used alternative, in a variety of simulation and empirical examples (Wheeler and Calder, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Finley, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Despite increased recognition of the prevalence of nonstationary in macrosystems (Rollinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2021), the use of SVCs within SDMs is still scarce in many fields including wildlife ecology and conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Recent applications suggest increasing interest in this flexible framework for a variety of ecological applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Meehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Pease et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Sultaire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2022), yet a comprehensive understanding of the data requirements needed to estimate SVCs in SDMs is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Typical SDM applications present many unique data complexities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', binary response variable, imperfect detection) and there is little guidance on how such complexities impact the sample size requirements and reliability of predictions and inference from SDMs with SVCs (hereafter SVC SDMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Additionally, the limited implementation of SVC SDMs precludes demonstration of the inferential gains from such a modeling framework relative to SDMs that assume stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The R packages sdmTMB (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022), VAST (Thorson, 2019), and INLA (Lindgren and Rue, 2015) provide functionality to fit a variety of SVC SDMs, but these approaches do not directly account for imperfect detection, which likely contributes to the limited adoption of SVC SDMs in wildlife ecology (Chapter 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Kéry and Royle 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Using simulations and an empirical case study, we present guidelines and recommendations for fitting SVC SDMs via the spOccupancy R package (Doser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022b) while explicitly accounting for imperfect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We focus our guidelines on practical issues that are likely to be encountered by practitioners when using SVCs in a species distribution modeling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We use simulations to understand the data requirements necessary to yield reasonably accurate and precise estimates from SVC SDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As an empirical example, we quantify spatially-varying occurrence trends in 51 forest birds across the eastern U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' from 2000-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' By providing explicit 6 guidelines and recommendations for the use of SVC SDMs, accompanied with user-friendly software options that readily accommodate imperfect detection, we hope to increase the careful application of SVC SDMs in wildlife ecology to produce more accurate species distribution maps and provide improved inference into the processes affecting populations across spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Models Here we present SVC single-season and multi-season occupancy models, with details on two SVC SDM variants that do not account for imperfect detection in Appendix S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We fit all models in a Bayesian framework with priors and numerical algorithms implemented to maximize computational efficiency (see Appendix S1 for full MCMC details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' SVC single-season occupancy model Let sj denote the spatial coordinates of site j, where j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' , J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We define z(sj) as the true presence (1) or absence (0) of the target species at site j with spatial coordinates sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model z(sj) as z(sj) ∼ Bernoulli(ψ(sj)), (1) where ψ(sj) is the occupancy probability of the species at site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model ψ(sj) according to logit(ψ(sj)) = x(sj)β + ˜x(sj)w(sj), (2) where β is a vector of regression coefficients (including an intercept) that describe the non- spatial effects of covariates x(sj), and w(sj) is a vector of spatially-varying effects of covariates ˜x(sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Thus ˜x(sj) may be identical to x(sj) if all covariate effects are assumed to vary spatially, or a subset of x(sj) if some effects are assumed to be constant across space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Note that the model reduces to a traditional single-species occupancy model when all covariate effects are assumed constant across space and a spatial occupancy model (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Doser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022b) when only the intercept is assumed to vary across space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The spatially-varying effects w(sj) serve as local adjustments of the covariate effects at each 7 site j from the overall non-spatial effects β, resulting in the covariate having a unique effect on species occupancy probability at each site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Following Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2003), we model each r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' , ˜q spatially-varying effect wr(sj) using a zero-mean spatial Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' More specifically, we have wr(s) ∼ N(0, Cr(s, s′, θr)), (3) where Cr(s, s′, θr) is a J × J covariance matrix that is a function of the distances between any pair of site coordinates s and s′ and a set of parameters (θr) that govern the spatial process according to a spatial correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Our associated software implementation in spOccupancy supports four correlation functions: exponential, spherical, Gaussian, and Matérn (Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' see Appendix S1 for guidance on determining which correlation function to use).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For the exponential, spherical, and Gaussian correlation functions, θr = {σ2 r, φr}, where σ2 r is a spatial variance parameter and φr is a spatial decay parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Large values of σ2 r indicate large variation in the magnitude of a covariate effect across space, while values of σ2 r close to 0 suggest little spatial variability in the magnitude of the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' φr controls the distance-dependent decay of the spatial dependence in the covariate effect and is inversely related to the spatial range, such that when φr is small, the covariate effect has a larger range of spatial dependence and varies more smoothly across space compared to larger values of φr (Appendix S2: Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The Matérn correlation function has an additional smoothness parameter νr, which provides further flexibility in the smoothness and decay of the spatial process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As computational complexity of the full Gaussian process specification in Equation 3 increases in cubic order with the number of spatial locations, we use Nearest Neighbor Gaussian Processes (NNGPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2016) as a computationally efficient alternative that provides nearly identical inference to the full Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' See Appendix S1 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' To account for imperfect detection in an occupancy modeling framework, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' , K(sj) sampling replicates are obtained at each site j to estimate whether a nondetection of the target species is truly an absence (MacKenzie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Tyre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model the observed detection (1) or nondetection (0) of a study species at site j, denoted yk(sj), conditional on the true latent occupancy process z(sj), following yk(sj) | z(sj) ∼ Bernoulli(pk(sj)z(sj)), (4) 8 where pk(s) is the probability of detecting the species at site j during replicate k given the species is truly present at the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model detection probability as a function of site and/or observation-level covariates according to logit(pk(sj)) = vk(sj)α, (5) where α is a vector of regression coefficients (including an intercept) that describe the effect of site and/or observation covariates vk(sj) on detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Note that effects of covariates on detection probability may also be nonstationary, but we assume they are stationary throughout our simulations, case study, and software implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' However, spatially-varying covariate effects could in principle be added to the detection model using the same process described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' SVC multi-season occupancy model Consider the case where detection-nondetection data are collected across multiple seasons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', years, breeding periods) during which the true occupancy status can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Such spatio- temporal data can be used to understand occupancy trends over time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Isaac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2014), as well as the environmental variables that drive spatio-temporal shifts in species distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Rushing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As an extension to the SVC occupancy model, we present a SVC multi-season occupancy model that estimates spatially-varying effects (as described previously) of covariates and accounts for temporal autocorrelation using temporal random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Let yk,t(sj) denote the detection or nondetection of the target species at site j during replicate survey k during time period t, with t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model yk,t(sj) conditional on the true occupancy status, zt(sj) of the species at site j during time t according to zt(sj) ∼ Bernoulli(ψt(sj)), (6) where ψt(sj) is the occupancy probability at site j during primary time period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model ψt(sj) following logit(ψt(sj)) = xt(sj)β + ˜xt(sj)w(sj) + ηt, (7) 9 where ηt is a random temporal effect and the other parameters are defined as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Note that the covariates can now vary across space and/or time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For example, by including year as a covariate in ˜X we can estimate a spatially-varying trend, and ultimately make predic- tions to generate trend maps across a region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Because we assume the non-spatial (β) and spatially-varying (w(sj)) coefficients are constant over the T primary time periods, they represent the average covariate effects across the temporal extent of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We model ηt as either an unstructured random effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', ηt ∼ Normal(0, σ2 T )) or using a first-order autore- gressive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', AR(1)) covariance structure in which we estimate a temporal variance parameter, σ2 T , and a temporal correlation parameter, ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The data yk,t(sj) are modeled conditional on the true occupancy status zt(sj) analogous to the SVC occupancy model in Equations 4 and 5, with detection probability now allowed to vary across sites, replicates, and/or primary time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Software implementation and prediction We implement single-season and multi-season Bayesian SVC SDMs with new functionality in v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='2 of the spOccupancy R package (Doser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' see Appendix S2: Table S1 for function names).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We assign Gaussian priors to all non-spatial regression coefficients, inverse- Gamma priors for the temporal variance parameter, and uniform priors for all correlation pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For the spatial variance parameters, we allow for either an inverse-Gamma prior or uniform prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We obtain computationally efficient implementations via NNGPs and Pólya- Gamma data augmentation (Polson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' See Appendix S1 for full Markov Chain Monte Carlo (MCMC) details and Appendix S3 for a detailed spOccupancy vignette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Additionally, the Bayesian framework readily enables prediction from the posterior predictive distribution, which facilitates mapping species-specific occupancy probabilities and any SVCs along with fully propagated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We include functionality for prediction in our associated software implementation (Appendix S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Simulation studies We performed four simulation studies using spOccupancy to evaluate the consequences of failing to address nonstationarity in SDMs and of how differing data characteristics influence inference 10 and predictive performance of SVC SDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Below we briefly present four simulation studies, with additional details on a fifth simulation study in Appendix S2 that assesses the influence of detection probability on estimates from SVC SDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For all simulations, we ran three chains, each with 15,000 MCMC iterations with a burn-in period of 10,000 iterations and a thinning rate of 5, resulting in a total of 3000 posterior draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We assessed convergence using visual assessment of trace plots with the coda package (Plummer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2006) and the Gelman-Rubin R-hat diagnostic (Brooks and Gelman, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used weakly informative priors for all model parameters (Appendix S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Simulation study 1: proof of concept As a proof of concept, we used simulations to assess how failing to account for nonstationarity affected inference for various levels of spatial dependence in the covariate effect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', small vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' high variability, short vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' long range spatial dependence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We simulated data from J = 400 sites across a unit square and K = 5 replicate surveys at each site for use in an occupancy modeling framework, where detection probability was set to moderate (average = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='45) and varying positively (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 on the logit scale) with a standardized observation-level covariate (see Appendix S2 for an additional simulation study assessing bias and precision across varying levels of detection probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We generated species occupancy as a function of an intercept (logit-scale β0 = 0) and a single covariate (logit-scale β1 = 0), whose effects both varied across space following Equations 2 and 3 using an exponential correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For the intercept, we set the spatial variance to 1 and the spatial decay parameter to φ = 3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' When using an exponential correlation function, the effective range, or the distance at which the spatial correlation between points drops to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='05 (Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2014), corresponds to approximately 3 / φ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', since 3 ≈ −log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='05)), and so a spatial decay parameter of φ = 3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 yields an effective range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 40% of the simulated study area across the unit square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For the SVC of the covariate effect, we varied the spatial variance parameter across four values (σ2 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5, 1, 2}) and the spatial decay parameter across three values (φ = {3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='1, 3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5, 3/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='8}) to assess how failing to account for nonstationarity is related to the spatial characteristics of the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' See Appendix S2: Figure S1 for the resulting spatial pattern in the SVC under all combinations of these parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We simulated 50 data sets for each combination of parameter values, resulting in 600 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We compared the performance of three models: (1) a basic 11 occupancy model with no spatially-varying intercept or SVC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2) a spatial occupancy model with a spatially-varying intercept but no SVC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' and (3) the full SVC occupancy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used the Widely Applicable Information Criterion (WAIC) as a measure of model fit (Watanabe, 2010) and four-fold cross-validation using model deviance as a metric of predictive performance (Hooten and Hobbs, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Additionally, we generated data using the same criteria as above but now with perfect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We then repeated the simulation study using a SVC generalized linear model (SVC GLM) to assess differences in bias between occupancy models and GLMs that do not explicitly account for imperfect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We compared the absolute bias and 95% credible interval widths for occupancy probability estimates from the SVC occupancy model and SVC GLM to determine if there were any differences in model performance for the two model types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Simulation study 2: sample size requirements for single-season SVC models We used simulations to assess how the total number of sampled sites and the number of spatially varying coefficients influenced bias and precision from SVC occupancy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We generated data with eight different amounts of spatial locations (J = {200, 400, 800, 1200, 1600, 2000, 4000, 6000}) and under different numbers of SVCs (1, 2, and 3) with moderate spatial variance, resulting in 24 simulation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For all scenarios, the model intercept was space-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We simulated 50 data sets for each combination of parameter values, yielding 1200 total simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We fit a SVC occupancy model to each data set with the number of SVCs used to generate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We assessed bias and precision of occupancy probability estimates (ψ(sj)) using mean absolute bias and 95% credible interval widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We further computed the Pearson’s correlation coefficient between the estimated SVCs and the true values to assess the reliability of inferences drawn from SVC occupancy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Simulation study 3: sample size requirements for multi-season SVC models For our third simulation study, we repeated simulation study 2 but now we assessed the per- formance of a multi-season SVC occupancy model under varying amounts of spatial locations 12 (J = {100, 200, 400, 800, 1600}, number of SVCs (1, 2, 3), and number of time periods (5, 10, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We generated data using an AR(1) temporal autocorrelation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We simulated 50 data sets for each combination of parameter values, resulting in 2250 total simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We assessed bias and precision using the same criteria as in simulation study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Simulation study 4: spatially varying population trends As quantification of spatially-varying trends in occupancy probability is a common objective of monitoring programs, we used simulations to understand the data requirements necessary for reliable estimation of a spatially-varying temporal trend using an SVC multi-season occupancy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We assessed how the total number of sampled sites and the number of primary time periods influenced the accuracy of temporal trend estimates from SVC multi-season occupancy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We varied the total number of sites (J = {100, 200, 400, 800, 1600}) and the number of time periods (T = {5, 10, 15}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We simulated 50 data sets under each of the 15 scenarios, resulting in 750 total simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used Pearson correlation coefficients with the true values to assess accuracy of the spatially-varying trend estimates from an SVC multi-season occupancy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Case study Quantifying spatially-explicit trends can help identify areas of conservation interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', climate- change refugia) as well as provide insights on species range dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', trailing vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' leading edge trends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We applied the SVC multi-season occupancy model to assess trends in 51 bird species across a ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='04 million km2 region of the eastern US (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', the continental US east of the 100th meridian) from 2000-2019 (T = 20 years) using detection-nondetection data from the North American Breeding Bird Survey (Pardieck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We restricted our analysis to a community of 66 eastern forest bird species following the classification of Bateman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2020), which consisted of species with large variations in abundance and breeding range size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We only assessed trends for 51 of the 66 species whose breeding ranges (derived from BirdLife International 2021) had at least 50% overlap with the study area (see Appendix S2 Table S3 for species names).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Our objectives for this case study were to (1) develop spatially-explicit maps of occupancy trends for each of the 51 species across the eastern US, and (2) assess the 13 performance of the SVC multi-season occupancy model to (i) a model with a constant trend over space and (ii) a model that allows the trend to vary across coarse ecological strata (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Bird Conservation Regions (BCRs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used data from J = 1846 BBS routes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', sites) sampled at least once between 2000- 2019 (mean number of sampled years per route = 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' BBS observers performed a three-minute point count survey at each of 50 stops along each route, counting all birds seen or heard within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 km radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We summarized the data for each species at each site into K = 5 spatial replicates (each comprising data from 10 of the 50 stops), where each replicate took value 1 if the species was detected at any of the 10 stops in that replicate, and value 0 if the species was not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' While such an approach has been used in previous studies with BBS data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Rushing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2020), this use of spatial replicates in an occupancy modeling framework may lead to violation of the closure assumption (Kendall and White, 2009), and so we refer to our response as species-specific occurrence rather than occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For each of the 51 species, we fit a SVC multi-season occupancy model in which we modeled occurrence probability as a function of a spatially-varying intercept, a spatially-varying trend, and an AR(1) temporal random effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Because our interest was solely in assessing spatial variation in occurrence trends, we did not estimate any SVCs for spatially-varying covariates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', forest cover, temperature) and rather used a spatially-varying intercept to account for variability in occurrence probability across space, as is commonly done when inference focuses on trends (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Bled et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Meehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We modeled detection probability using a separate intercept for each year, linear and quadratic effects of day of survey to account for seasonal variation in detection probability, linear and quadratic effects of survey replicate to account for variability in detection probability over a day of sampling, and a random observer effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The occurrence trend covariate and all detection covariates were standardized to have mean 0 and standard deviation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For each species, we used pre-existing breeding ranges from BirdLife International (BirdLife International, 2021) to only use routes that fell inside a 50 km buffer of the species range when fitting the occupancy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We compared the full SVC multi- season occupancy model to a model that assumed the trend was constant across space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', constant model) and a model that estimated a separate trend parameter for 21 BCRs in the study region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', BCR model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The BCR model represents a simpler alternative to the SVC model that allows for spatial variability in the occurrence trend at a coarse, pre-determined 14 resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', by regional stratification of the model coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' If occurrence trends vary spatially at a coarse resolution, the BCR model may be a simpler, more computationally-efficient alternative to the full SVC model (Pease et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used WAIC as a measure of model fit to compare the three candidate models for each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We fit all models in spOccupancy, running three chains of each model for 50,000 MCMC iterations with a burn-in period of 30,000 iterations and a thinning rate of 20, yielding 3000 posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used vague priors for all non-spatial parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We used a uniform prior for the spatial variance parameters to restrict the spatial variance from taking large values that are unreasonable on the logit scale (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We additionally used an informative uniform prior on the spatial decay parameter to prevent unreasonably small estimates of the spatial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Specifically, we restricted the upper bound of the uniform distribution to 3/667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='93, which resulted in a minimum effective spatial range of 667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='93km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We found this was sufficiently flexible to account for nonstationarity in species trends while simultaneously preventing the model from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' See Appendix S2 for additional discussion on prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' After fitting the models, we predicted across the range of each species in the study area to generate maps of spatially-varying occurrence trends across the twenty year period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Results Simulation studies The benefits of modeling nonstationarity in SVC SDMs were dependent on the characteristics of the spatial dependence in the covariate effect (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Improvements in model performance of the SVC occupancy model were highest for covariates effects with a large spatial variance and large spatial range according to both WAIC and four-fold cross-validation deviance (Table 1, Appendix S2: Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As expected, when the spatial variance in the covariate effect was small (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5), an SVC occupancy model only yielded small improvements in WAIC compared to a spatial occupancy model that assumed a stationary covariate effect, and either no improvements or very small improvements in cross-validation deviance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Interestingly, predictive performance was generally worse or only marginally improved when the effective range of the covariate effect was small (10% of study area), regardless of the spatial variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Accuracy and precision were only marginally higher for an SVC GLM compared to an SVC 15 occupancy model, with bias on average being 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='22% lower for the SVC GLM simulations and 95% credible intervals on average being 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='85% smaller for the SVC GLM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' These small differences indicate minimal differences in the ability of SVC GLMs and SVC occupancy models to estimate occurrence probability when data were generated according to the respective model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Bias and uncertainty in occupancy probability estimates from an SVC occupancy model de- creased as the number of spatial locations increased, and increased as the number of estimated SVCs increased (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Widths of 95% credible intervals on occupancy probabilities spanned more than 70% of the possible range (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 0-1) when fitting an SVC occupancy model with 200 or 400 sites, indicating large uncertainty in SVC occupancy model estimates from data sets with fewer than ∼ 500 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Accuracy of the estimated SVCs showed a similar pattern, with accuracy strongly increasing as the number of spatial locations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' In particular, the correlation coefficient of the estimated SVCs with the true simulated values was extremely low when fitting models with 200 or 400 sites (Figure 1C), indicating inference on nonstationary covariate effects in SVC occupancy models might be unreliable with data sets comprised of fewer than approx- imately 500 spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Accuracy and precision of occupancy probability estimates in SVC occupancy models clearly increased with increasing amounts of temporal replication and increasing detection probability, while accuracy and precision of SVC estimates showed no clear patterns with varying amounts of replication and detection probability (Appendix S2: Tables S7, S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Bias and uncertainty in occupancy probability and SVC estimates from a multi-season SVC occupancy model showed similar patterns to the single-season SVC occupancy model, with bias and uncertainty decreasing as the number of spatial locations and number of time periods increases, and increasing as more SVCs were estimated in the model (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Importantly, bias and uncertainty for a given number of spatial locations was much smaller in a multi- season SVC occupancy model compared to a single-season SVC occupancy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For example, absolute bias in occupancy probability for a multi-season SVC occupancy model with 200 sites, 5 time periods, and a single SVC was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='12, while for a single-season model with the same number of sites and SVCs was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='17, a 29% decrease in bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Accuracy of the estimated SVCs in multi-season occupancy models was also much higher than single-season occupancy models with the same number of spatial locations, in particular when the number of time periods is high 16 (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' This result suggests that the additional information provided by temporal replication makes it possible to assess nonstationary covariate effects on occupancy probability with a more modest number of spatial locations (∼ 100) compared to single-season SVC occupancy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Additionally, accuracy of spatially-varying trend estimates from a SVC multi-season occupancy model showed an identical pattern, with correlation coefficients with the true values all high (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='6), even for a modest number of spatial locations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 100) and time periods (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Appendix S2: Supplemental Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Case study There was strong support for nonstationarity in twenty-year occurrence trends across the east- ern US for the majority of the forest bird species that we included in our analysis (Figure 3), with many species showing both positive and negative trends across their breeding range (Fig- ure 4, Appendix S2: Supplemental Figures S3-S53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The SVC multi-season occupancy model substantially outperformed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', ∆WAIC > 2) the BCR model and the constant trend model for 88% of all species (45 of 51) according to the WAIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Of the six species with less support for the SVC model, five of them (Golden-winged Warbler, American Woodcock, Mississippi Kite, Red-cockaded Woodpecker, Eastern Screech Owl) had very low raw occurrence probabilities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='05), while the sixth species (Broad-winged Hawk) showed a fairly constant positive trend across its range (Appendix S2: Supplemental Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Generally, the SVC model re- vealed spatial heterogeneity in occurrence trends not evident in either the constant or BCR models (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' More specifically, the constant model averaged over any spatial variability in occurrence trends (Figure 5A, D, G), while the BCR model was able to adequately capture some spatial variability in trends, but was not able to capture heterogeneity in trends occurring within a given BCR (Figure 5B, E, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Discussion Accounting for nonstationarity in species-environment relationships is increasingly important as the scope of ecological research expands in spatial and temporal extent (Rollinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Bayesian SVC models are a flexible approach to account for nonstationarity (Gelfand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2003), yet their use in species distribution modeling and wildlife ecology has been limited, 17 perhaps due to their increased complexity compared to alternative approaches and a lack of user-friendly software to fit SVC SDMs that simultaneously accommodate other common data complexities in wildlife ecology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', imperfect detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Chapter 9, Kéry and Royle 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' In a case study on a forest bird community in the eastern US, we found strong support for nonstationarity in occurrence trends for 45 out of 51 eastern forest bird species, illustrating the potentially high prevalence of nonstationarity in model coefficients within empirical data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Using simulations, we found that reasonable accuracy and precision of SVC SDMs requires a large number of spatial locations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', >500 spatial locations in single-season SDMs), while the additional temporal data generated from multi-season sampling leads to accurate and precise estimates from multi-season SVC SDMs with a more modest number of spatial locations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 100 spatial locations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' When simulating data with a single SVC, improvements in model performance of an SVC occupancy model were highest when the effect had a large spatial variance and long effective range (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As the spatial variance decreased toward zero, the magnitude of the spatial component of the covariate effect became increasingly small, ultimately resulting in negligible changes in the estimated occupancy probability between a model that includes an SVC and a model that assumes a constant covariate effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' This is analogous to fitting a model with a traditional random effect, where the importance of the random effect decreases as the random effect variance approaches zero and model comparison approaches will often favor the simpler model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As the effective spatial range decreases toward zero, the spatial correlation in the SVCs occurs over a smaller distance, and thus the estimate of the SVC at any given location is informed by a smaller number of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For binary data, it is not possible to estimate an unstructured site-level random effect (Bolker, 2022), which likely contributed to why we saw negligible differences in predictive performance between the SVC occupancy model and the spatial occupancy model when the effective spatial range was small (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Thus, SVC SDMs that use binary data will be able to better accommodate covariates whose effects are presumed to show high correlation across relatively large spatial regions compared to those that vary across spatial scales that are close to the minimum distance between spatial locations in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Bias of occupancy probability and SVC estimates from SVC SDMs was highly dependent on the number of spatial locations in the data set and, to a lesser extent, the number of SVCs 18 estimated in the model (Figures 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For single-season SVC SDMs, correlation coefficients of the estimated SVCs with the true simulated values were noticeably low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5) when simulating data with fewer than approximately 500 spatial locations, but steadily increased to greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='8 as the number of spatial locations increased (Figure 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' This suggests that SVC SDMs can provide reasonable inference on nonstationary covariate effects, but only for data sets with a large number of spatial locations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', > 500), which is substantially larger than the number of spatial locations needed to fit a standard occupancy model with covariates (Kéry and Royle, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' MacKenzie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For multi-season SVC SDMs, the temporal replication provides additional information to estimate SVCs, which resulted in more accurate and precise estimates of occupancy probability and SVCs for a given number of spatial locations than we found in single-season SVC SDMs (Figures 2, Appendix S2: Supplemental Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' In the eastern forest bird case study, we found substantial support for nonstationarity in occurrence trends from 2000-2019 for 45 out of 51 modeled species (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 88%) (Figure 3, Ap- pendix S2: Supplemental Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Occurrence trends varied from species to species, with some species showing clear declines across much of their range (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Chimney Swift, Appendix S2: Supplemental Figure S15), others showing widespread increases in occurrence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Mississippi Kite, Appendix S2: Supplemental Figure S30), and others showing patterns consistent with latitudinal range shifts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Eastern Wood-pewee, Wood Thrush, Figure 4B, C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Rushing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Our findings of substantial spatial variability in occurrence trends largely align with the variability demonstrated by spatially-explicit relative abundance trends estimated using eBird data (Fink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Only one species (Eastern Screech Owl) had substantial support (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', ∆WAIC > 2) of the BCR model over the SVC model, suggesting that generally across species in the community there was substantial variation of occurrence trends within BCRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The BCR model, which estimates a separate occurrence trend for each BCR within the species range, is a simpler alternative to the SVC model that allows trends to vary across pre-defined regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' While the BCR model provides improved inference compared to the constant model, it averages over any heterogeneity in trends within a BCR, which can mask more fine-scale patterns in population trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For example, the BCR model did not capture an increasing trend of Gray Catbird in Louisiana and southern Mississippi that was found in the SVC model (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Smith and Edwards (2021) recently developed a similar model for BBS count data that esti- mates separate relative abundance trends for pre-defined spatial units as unstructured random 19 effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Our results here, although for occurrence and not relative abundance, suggest an SVC model may provide additional flexibility and detailed inference for estimating spatially-varying relative abundance trends from BBS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We envision several extensions to the SVC SDMs presented here and in the associated software implementation in spOccupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' First, the SVC models could be extended to a multi- species framework that uses a spatial factor modeling approach to model SVCs and account for correlations between species (Doser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Second, we could incorporate SVCs in the detection portion of the occupancy model, which would allow effects of covariates that influence detectability to vary spatially (Thorson, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Lastly, we could extend the multi-season SVC models to allow the covariates to vary both spatially and temporally using a dynamic linear modeling framework (Finley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Guidelines when fitting spatially-varying coefficient species dis- tribution models Below we present a set of five practical guidelines for practitioners to consider when fitting SVC SDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We provide this set of guidelines along with user-friendly and computationally efficient implementations of single-season and multi-season SVC SDMs in the spOccupancy R package (Doser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2022b) and an associated vignette that provides a detailed walk through for fitting these models (Appendix S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Avoid fitting single-season SVC SDMs with fewer than ∼500 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The limi- tations of binary data make it difficult to accurately estimate SVCs without a relatively large number of spatial locations (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' When inference on nonstationary covari- ate effects is a primary objective, we recommend only fitting single-season SVC SDMs (with or without imperfect detection) using data sources that have at least ∼ 500 spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Consider using an informative prior to prevent small effective spatial range estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Estimating an SVC that has a small effective spatial range (or equivalently a large spatial decay parameter φ) is inherently difficult when using binary data in SVC SDMs as a result of limitations in estimating site-level random effects with binary data (Bolker, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' In the eastern forest bird case study, we found that using an informative 20 prior on the spatial decay parameters restricted small estimates of the effective spatial range and eliminated overfitting of the estimated SVCs while still providing sufficiently flexible estimates of the spatially-varying trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' See Appendix S1 for a discussion on specifying the prior distribution for the spatial decay parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Use data from multiple seasons when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' The additional replication provided by data collected over multiple seasons substantially improves the ability to estimate SVCs with a smaller number of spatial locations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 100 sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Figures 2, Appendix S2: Supplemental Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' When available, we suggest using data from multiple seasons to estimate SVC SDMs, in which case the estimated SVCs are interpreted as the average effect of the covariate across the temporal extent of the data on species occupancy at any given location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Use model selection to compare SVC SDMs with simpler submodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As we did in the eastern forest bird case study, comparing SVC SDMs to simpler models that allow covariate effects to vary across designated spatial units (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', ecoregions, management units) can provide insight into the spatial scale at which covariate effects vary across space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Our associated software implementation in spOccupancy provides functionality to perform model selection with WAIC and k-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Use simulations to assess reliability of predictions and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We performed simulations to assess the reliability of SVC SDMs under a variety of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' However, we suggest performing simulations prior to fitting an SVC SDM based on the character- istics of the data set at hand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', in terms of sample sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' This can provide insight on the amount of bias and uncertainty to expect in estimates, and if reliable inference of the SVCs can be made given a set number of spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' We provide multiple functions in spOccupancy to simulate data with SVCs for such assessments, which we describe in detail in the associated vignette (Appendix S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Nonstationarity in species-environment relationships is prevalent throughout ecology (Rollinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' As we demonstrated, the use of spatially-varying coefficients in species distribution models can help elucidate the environmental factors that drive species distribution dynamics, especially across broad spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' For accurate conclusions regarding the drivers of species distributions over space and time, we require reliable, accessible, and efficient computational 21 methods to quantify nonstationary relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Additional developments that help facilitate the incorporation of nonstationarity into ecological analyses will lead to more useful inferences on the nuanced pressures facing biodiversity in a rapidly changing world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' References Anderson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Generalized additive models: an introduction with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Chapman and Hall/CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Wright, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Irvine, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', Rodhouse, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', and Litt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Spatial gaussian pro- cesses improve multi-species occupancy models when range boundaries are uncertain and nonoverlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Ecology and Evolution, 11(13):8516–8527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 27 Tables and Figures Table 1: Model comparison results for a non-spatial occupancy model (OCC), a spatially- varying intercept occupancy model (SVI), and a spatially-varying coefficients occupancy model (SVC) using simulated data with varying degrees of spatial autocorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Values represent the difference in four-fold cross-validation deviance and WAIC for the SVC model compared to the two candidate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Lower values indicate better performance, with boldface indicating the best performing model for a given scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Values are averaged across 50 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Effective spatial Spatial Deviance WAIC range (%) variance OCC SVI SVC OCC SVI SVC 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='07 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='8 0 2000 4000 6000 Number of sites 95% CI Width (B) Occupancy probability uncertainty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='8 0 2000 4000 6000 Number of sites Correlation coefficient (C) SVC Accuracy Number of SVCs 1 2 3 Figure 1: Absolute bias in occupancy probability estimates (A), 95% credible interval (CI) widths of occupancy probability estimates (B), and accuracy of spatially-varying coefficient estimates from a spatially-varying coefficient occupancy model (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Values in (C) are Pearson correlation coefficients between the estimated SVCs and the true values used to simulate the data, such that higher values indicate higher accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Values are averaged across 50 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 29 5 years 10 years 15 years 400 800 1200 1600 400 800 1200 1600 400 800 1200 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='15 Number of sites Absolute bias (A) Occupancy probability bias 5 years 10 years 15 years 400 800 1200 1600 400 800 1200 1600 400 800 1200 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='7 Number of sites 95% CI Width (B) Occupancy probability uncertainty 5 years 10 years 15 years 400 800 1200 1600 400 800 1200 1600 400 800 1200 1600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='9 Number of sites Correlation coefficient (C) SVC Accuracy Number of SVCs 1 2 3 Figure 2: Absolute bias in occupancy probability estimates (A), 95% credible interval (CI) widths of occupancy probability estimates (B), and accuracy of spatially-varying coefficient estimates from a spatially-varying coefficient multi-season occupancy model under differing numbers of primary time periods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=', years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Values in (C) are Pearson correlation coefficients between the estimated SVCs and the true values used to simulate the data, such that higher values indicate higher accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Values are averaged across 50 simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 30 GWWA KEWA AMWO BBCU BAOR BWWA EASO WOTH BACS BLJA CERW FISP CHSW PROW GRCA SCTA WEWA GCFL TUTI BRTH RBGR EATO EAWP CACH CWWI EABL INBU PRAW WEVI RCWO BHNU YBCU RHWO ACFL EAPH SUTA RTHU OROR YTWA YTVI PIWA NOPA BWHA HOWA LOWA NOCA CARW RSHA SWWA RBWO MIKI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='00 Proportion of sites with positive trend Species Figure 3: Proportion of BBS routes within a given species range with a positive trend estimate from a spatially-varying coefficient occupancy model for 51 eastern forest bird species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Points represent posterior median and gray lines denote 95% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' See Appendix S2: Table S3 for definition of species codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 31 25°N 30°N 35°N 40°N 45°N 50°N 100°W 95°W 90°W 85°W 80°W 75°W Longitude Latitude −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 Trend (logit scale) (A) Tufted Titmouse 25°N 30°N 35°N 40°N 45°N 50°N 100°W 95°W 90°W 85°W 80°W 75°W Longitude Latitude −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='4 Trend (logit scale) (B) Eastern Wood−pewee 25°N 30°N 35°N 40°N 45°N 50°N 100°W 95°W 90°W 85°W 80°W 75°W Longitude Latitude −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='0 Trend (logit scale) (C) Wood Thrush 25°N 30°N 35°N 40°N 45°N 50°N 100°W 95°W 90°W 85°W 80°W 75°W Longitude Latitude −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='5 Trend (logit scale) (D) Blue−winged Warbler Figure 4: Mean predictions of the spatially-varying occurrence trend from 2000-2019 from a spatially-varying coefficient occupancy model for four example species that show varying degrees of heterogeneity in trends across the study region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Trends are only shown within each species range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panel A: Tufted Titmouse (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='15 million km2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panel B: Eastern Wood-pewee (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='74 million km2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panel C: Wood Thrush (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='12 million km2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panel D: Blue-winged Warbler (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='73 million km2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='25°N ' metadata={'source': 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occurrence trend from 2000-2019 for three example ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='species from three different models: a spatial occupancy model with a constant trend ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='across the species range (Constant),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' a spatial occupancy model with a separate trend for each Bird Conservation Region (BCR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' and an SVC occupancy model (SVC) estimating a spatially-varying trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panels A-C: Gray Catbird (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='79 million km2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panels D-F: Eastern Phoebe (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='55 million km2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' Panels G-I: Scarlet Tanager (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content='59 million km2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/g9E5T4oBgHgl3EQfiA9u/content/2301.05645v1.pdf'} diff --git a/gdFLT4oBgHgl3EQfZC_Z/content/tmp_files/2301.12068v1.pdf.txt b/gdFLT4oBgHgl3EQfZC_Z/content/tmp_files/2301.12068v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..38f5a7b73d21bafa129308164bf3bc88a8468723 --- /dev/null +++ b/gdFLT4oBgHgl3EQfZC_Z/content/tmp_files/2301.12068v1.pdf.txt @@ -0,0 +1,2847 @@ +Physics-Inspired Protein Encoder Pre-Training via +Siamese Sequence-Structure Diffusion Trajectory Prediction +Zuobai Zhang * 1 2 Minghao Xu * 1 2 Aur´elie Lozano 3 Vijil Chenthamarakshan 3 Payel Das 3 Jian Tang 1 4 5 +Abstract +Pre-training methods on proteins are recently gain- +ing interest, leveraging either protein sequences +or structures, while modeling their joint energy +landscape is largely unexplored. In this work, in- +spired by the success of denoising diffusion mod- +els, we propose the DiffPreT approach to pre- +train a protein encoder by sequence-structure mul- +timodal diffusion modeling. DiffPreT guides the +encoder to recover the native protein sequences +and structures from the perturbed ones along the +multimodal diffusion trajectory, which acquires +the joint distribution of sequences and structures. +Considering the essential protein conformational +variations, we enhance DiffPreT by a physics- +inspired method called Siamese Diffusion Trajec- +tory Prediction (SiamDiff) to capture the corre- +lation between different conformers of a protein. +SiamDiff attains this goal by maximizing the mu- +tual information between representations of dif- +fusion trajectories of structurally-correlated con- +formers. We study the effectiveness of DiffPreT +and SiamDiff on both atom- and residue-level +structure-based protein understanding tasks. Ex- +perimental results show that the performance of +DiffPreT is consistently competitive on all tasks, +and SiamDiff achieves new state-of-the-art perfor- +mance, considering the mean ranks on all tasks. +The source code will be released upon acceptance. +1. Introduction +Machine learning based methods have achieved impressive +results in predicting protein structures (Jumper et al., 2021; +Baek et al., 2021; Lin et al., 2022) and functionality (Meier +et al., 2021; Gligorijevi´c et al., 2021). Among these meth- +ods, various pre-training approaches (Elnaggar et al., 2021; +*Equal contribution +1Mila - Qu´ebec AI Institute 2Universit´e +de Montr´eal 3IBM Research 4HEC Montr´eal 5CIFAR AI Chair. +Correspondence to: Payel Das , Jian Tang +. +Preprint. +Rives et al., 2021; Zhang et al., 2022) succeed in learning ef- +fective protein representations from amount of available pro- +tein sequences or from their experimental/predicted struc- +tures. Sequence-based pre-training methods (Elnaggar et al., +2021; Rives et al., 2021) can well acquire co-evolutionary in- +formation, and structure-based pre-training methods (Zhang +et al., 2022; Hermosilla & Ropinski, 2022) are able to cap- +ture protein structural characteristics sufficient for tasks like +function prediction and fold classification. These two types +of information are both useful to indicate underlying pro- +tein functions, and they are complementary to each other. +Therefore, modeling the multimodal energy landscape of +protein sequences and structures for pre-training could be a +more promising way than the unimodal pre-training meth- +ods, which is largely unexplored. +To capture the energy landscape, denoising diffusion models +can be a very natural choice, as they are essentially learning +the energy manifold of data through noising and denois- +ing process (Ho et al., 2020). However, to the best of our +knowledge, diffusion models are mostly studied in the con- +text of generative tasks (Luo et al., 2022; Anand & Achim, +2022), rather than learning the effective representation of a +complex data manifold like protein landscape. +These facts motivates us to propose an approach called Diff- +PreT to pre-train structure-informed protein encoders by +sequence-structure multimodal diffusion models. Specifi- +cally, we first design the multimodal diffusion trajectory of +a protein by transforming both its structure and sequence +towards random distribution via gradually adding noises. Pa- +rameterized with the output of the protein encoder, a noise +prediction network is defined to denoise the corrupted pro- +tein structure and sequence step by step. With the diffusion +models for pre-training, we facilitate the encoder to learn +informative representations that capture (1) the inter-atomic +interactions within protein structure, (2) the residue type de- +pendencies along protein sequence, and (3) the joint effect +of sequence and structure variations. +In spite of these advantages, DiffPreT ignores the fact that +any protein structure exists as a population of interconvert- +ing conformers, elucidating this conformational heterogene- +ity is essential for predicting protein function and ligand +binding (Grant et al., 2010). In both DiffPreT and previous +arXiv:2301.12068v1 [cs.LG] 28 Jan 2023 + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +studies, no explicit physical constraints are added to acquire +the structural correlation between different conformers of +a specific protein or between structural homologs, which +prohibits capturing the conformational energy landscape of +a protein (Nienhaus et al., 1997). Therefore, to consider +the physics underlying the conformational change, We pro- +pose Siamese Diffusion Trajectory Prediction (SiamDiff) +to augment the DiffPreT by maximizing the mutual infor- +mation between representations of diffusion trajectories of +structurally-correlated conformers (i.e., siamese diffusion +trajectories). We first adopt a torsional perturbation scheme +on the side chain to generate randomly simulated conform- +ers (Ho & Agard, 2009). Then, for each protein, we generate +diffusion trajectories for a pair of its conformers. We theo- +retically prove that the problem can be transformed to the +mutual prediction of the trajectories using representations +from their counterparts. In this way, the model can keep +the advantages of DiffPreT and inject conformer-related +information into representations as well. +Both DiffPreT and SiamDiff can be flexibly applied to +residue-level and atom-level structures for effective rep- +resentation learning. We employ various self-supervised +algorithms to pre-train residue-level and atom-level struc- +ture encoders, and the pre-trained models are extensively +evaluated on Enzyme Commission number prediction (Glig- +orijevi´c et al., 2021) and ATOM3D (Townshend et al., 2020) +benchmarks. Experimental results demonstrate that Diff- +PreT can consistently achieve competitive performance on +all benchmark tasks and on both atomic and residue-level, +in contrast to existing baselines; SiamDiff further enhances +the model performance and achieves new state-of-the-art in +terms of mean benchmark rank on both structure levels. +Overall, our contributions are summarized as: +1. We propose DiffPreT to use multimodal denoising dif- +fusion models for pre-training protein encoders, which +models the joint energy landscape of protein sequences +and structures. +2. We propose SiamDiff to maximize the mutual informa- +tion between representations of siamsese diffusion trajec- +tories, which captures the structural correlation between +different conformers or between structural homologs. +3. Both pre-training methods are effective on various +residue- and atom-level tasks. SiamDiff achieves the +state-of-the-art performance w.r.t. mean rank. +2. Related Work +Pre-training Methods on Proteins. Self-supervised pre- +training methods have been widely used to acquire co- +evolutionary information from large-scale protein sequence +corpus, inducing performant protein language models +(PLMs) (Elnaggar et al., 2021; Lu et al., 2020; Rives et al., +2021; Lin et al., 2022). Typical sequence pre-training meth- +ods include masked protein modeling (Elnaggar et al., 2021; +Rives et al., 2021; Lin et al., 2022) and contrastive learn- +ing (Lu et al., 2020). The pre-trained PLMs have achieved +impressive performance on a variety of downstream tasks +for structure and function prediction (Rao et al., 2019; Xu +et al., 2022c). Recent works have also studied pre-training +on unlabeled protein structures for generalizable represen- +tations, covering contrastive learning (Zhang et al., 2022; +Hermosilla & Ropinski, 2022), self-prediction of geometric +quantities (Zhang et al., 2022; Chen et al., 2022) and de- +noising score matching (Guo et al., 2022; Wu et al., 2022a). +Compared with existing works, our methods model the joint +distribution of sequences and structures via multimodal dif- +fusion models, which captures both co-evolutionary infor- +mation and detailed structural characteristics. +Diffusion Probabilistic Models (DPMs). DPM was first +proposed in Sohl-Dickstein et al. (2015) and has been re- +cently rekindled for its strong performance on image and +waveform generation (Ho et al., 2020; Chen et al., 2020a). +Besides the DPMs for continuous data, some works study +discrete DPMs and achieve impressive results on generating +texts (Austin et al., 2021; Li et al., 2022), graphs (Vignac +et al., 2022) and images (Hoogeboom et al., 2021). In- +spired by these progresses, DPMs have been adopted to +solve problems in chemistry and biology domain, includ- +ing molecule generation (Xu et al., 2022b; Hoogeboom +et al., 2022; Wu et al., 2022c; Jing et al., 2022), molecular +representation learning (Liu et al., 2022), protein structure +prediction (Wu et al., 2022b), protein-ligand binding (Corso +et al., 2022), protein design (Anand & Achim, 2022; Luo +et al., 2022; Ingraham et al., 2022; Watson et al., 2022) and +motif-scaffolding (Trippe et al., 2022). In this work, we +novelly study how DPMs can help protein representation +learning, which aligns with a recent effort (Abstreiter et al., +2021) on diffusion-based image representation learning. +3. Preliminary +Notation. A protein with nr residues (amino acids) and +na atoms can be represented as a sequence-structure tuple +P = (S, R). We use S = [s1, s2, · · · , snr] to denote its +sequence with si as the type of the i-th residue, while R = +[r1, r2..., rna] ∈ Rna×3 denotes its structure with ri as +the Cartesian coordinates of the i-th atom. We construct a +graph for each protein with edges connecting atoms with +the Euclidean distance lower than a threshold and then use +a graph neural network to model the structure. +Equivariance. Equivariance is ubiquitous in machine learn- +ing for modeling the symmetry in physical systems (Thomas +et al., 2018; Weiler et al., 2018) and is shown to be critical +for successful design and better generalization of 3D net- +works (K¨ohler et al., 2020). Formally, a function F : X → +Y is equivariant w.r.t. a group G if F ◦ ρX (x) = ρY ◦ F(x), + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +where ρX and ρY are transformations corresponding to an +element g ∈ G acting on the space X and Y, respectively. +The function is invariant w.r.t G if the transformations ρY +is identity. In this paper, we consider SE(3) group, i.e., +rotations and translations in 3D space. +Problem Definition. Given a set of unlabeled proteins P = +{P1, P2, ...}, our goal is to train a protein encoder φ(S, R) +to extract informative d-dimensional residue representations +h ∈ Rnr×d and atom representations a ∈ Rna×d that are +SE(3)-invariant w.r.t. protein structures R. +4. DiffPreT: Diffusion Models for +Pre-Training +Recently, there have been promising progress on applying +denoising diffusion models for protein structure-sequence +co-design (Luo et al., 2022; Anand & Achim, 2022). The +effectiveness of the multimodal diffusion model on mod- +eling the distribution of proteins suggests that the process +may reflect physical and chemical principles underlying pro- +tein formation (Anfinsen, 1972; Dill & MacCallum, 2012), +which could be beneficial for learning informative repre- +sentations. Based on this intuition, in this section, we ex- +plore the application of multimodal diffusion models on +pre-training protein encoders. We first introduce the defini- +tion of diffusion models on proteins in Sec. 4.1, then discuss +the parameterization of forward and reverse processes for +diffusion on protein structures in Sec. 4.2 and sequences +in Sec. 4.3 and derive the final objective for pre-training in +Sec. 4.4. Fig. 1 illustrates the high-level idea. +4.1. Diffusion Models on Proteins +Diffusion models are a class of deep generative models with +latent variables encoded by a forward diffusion process and +decoded by a reverse generative process (Sohl-Dickstein +et al., 2015). We use P0 to denote the ground-truth protein +and Pt for t = 1, · · · , T to be the latent variables over T +diffusion steps. Modeling the protein as an evolving thermo- +dynamic system, the forward process gradually injects small +noise to the data P0 until reaching a random noise distri- +bution at time T. The reverse process learns to denoise the +latent variable towards the data distribution. Both processes +are defined as Markov chains: +q(P1:T |P0) = �T +t=1 q(Pt|Pt−1), +(1) +pθ(P0:T −1|PT ) = �T +t=1 pθ(Pt−1|Pt), +(2) +where q(Pt|Pt−1) defines the forward process at step t +and pθ(Pt−1|Pt) with learnable parameters θ defines the +reverse process at step t. +Here we model the joint distribution of protein sequences +and structures via multimodal diffusion models. Follow- +ing Luo et al. (2022), we assume 1) the separate definition +of the forward process on structures and sequences and 2) +the conditional independence of sequences and structures in +the reverse process: +q(Pt|Pt−1) = q(Rt|Rt−1) · q(St|St−1), +(3) +pθ(Pt−1|Pt) = pθ(Rt−1|Pt) · pθ(St−1|Pt). +(4) +Next, we discuss how to define the diffusion models on +protein structures and sequences, respectively. +4.2. Diffusion models on 3D structures +We first introduce the definition of forward process on pro- +tein 3D structures and then discuss how to parameterize the +reverse process with atom representations a extracted by the +encoder. Since the coordinates of atoms are continuous vari- +ables in the 3D space, the forward process can be defined +by adding random Gaussian noise (Ho et al., 2020). Then, +the reverse process can be parameterized as a Gaussian with +a learnable mean and user-defined variance. That is, +q(Rt|Rt−1) = N(Rt; +� +1 − βtRt−1, βtI), +(5) +pθ(Rt−1|Pt) = N(Rt−1; µθ(Pt, t), σ2 +t I), +(6) +where β1, ..., βT are a series of fixed variances and σt can +be any user-defined variance. Since Rt is available as an +input, following Ho et al. (2020), we reparameterize the +mean µθ(Pt, t) as: +µθ(Pt, t) = +1 +√αt +� +Rt − +βt +√1 − ¯αt +ϵθ(Pt, t) +� +, +(7) +where αt = 1 − βt, ¯αt = �t +s=1 αs and the network +ϵθ(·) learns to decorrupt the data and should be translation- +invariant and rotation-equivariant w.r.t. the structure Rt. +Parameterization of ϵθ. We define our noise prediction +network with atom representations at (which is guaranteed +to be SE(3)-invariant w.r.t. Rt by the encoder) and atom +coordinates rt (which is SE(3)-equivariant w.r.t. Rt). We +draw inspirations from recent works (Satorras et al., 2021) to +build an equivariant output based on normalized directional +vectors between adjacent atom pairs. Each edge (i, j) is +encoded by its length ∥rt +i − rt +j∥2 and the representations of +two end nodes at +i, at +j, and the encoded score mi,j will be +used for aggregating directional vectors. Specifically, +[ϵθ(Pt, t)]i = +� +j∈N t(i) mi,j · +rt +i−rt +j +∥rt +i−rt +j∥2 , +with mi,j = MLP(at +i, at +j, MLP(∥rt +i − rt +j∥2)), +where N t(i) denotes the neighbors of the atom i in the +corresponding graph of Pt. Note that ϵθ(Pt, t) achieves +the equivariance requirement, as mi,j is SE(3)-invariant +w.r.t. Rt while rt +i − rt +j is translation-invariant and rotation- +equivariant w.r.t. Rt. +Since ϵθ is designed to be rotation-equivariant w.r.t. Rt, to +make the loss function invariant w.r.t. Rt, the supervision ϵ + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +!! +!"#$ +!" +!% +· · · +· · · +…YLVCGERGFF… +…Y*VC*ER*FF… +…**********… +"(!"|!"#$) +&&(!"#$|!") +(a) DiffPreT +(b) SiamDiff +!! +" +!! +#$! +!! +# +!! +% +· · · +· · · +!& +" +!& +#$! +!& +# +!& +% +· · · +· · · +!& +Torsional +Perturbation +! = ($, &) +Identity +!! +…YLVCGERGFF… +…Y*VC*ER*FF… +…**********… +…YLVCGERGFF… +…Y*VC*ER*FF… +…**********… +Figure 1: (a) High-level illustration of DiffPreT. Denoising diffusion modeling is performed on both protein structures +and sequences. (b) High-level illustration of SiamDiff. Two structurally-correlated conformers P1 and P2 are constructed, +mutual denoising of sequence-structure diffusion trajectory is performed across the correlated conformer pair. +is also supposed to achieve such equivariance. Therefore, we +adopt the chain-rule approach proposed in Xu et al. (2022b), +which decomposes the noise on pairwise distances to obtain +the modified noise vector ˆϵ as supervision. We refer readers +to Xu et al. (2022b) for more details. +4.3. Diffusion models on sequences +Different from those on 3D structures, diffusion models +on sequences require a definition of diffusion processes on +discrete attributes, which is typically defined as a Markov +chain. The key is to define the transition probability be- +tween different discrete states at each step for the Markov +chain and should guarantee its convergence to the stationary +distribution. Here we adopt a simple formulation in Austin +et al. (2021). We add an absorbing state [MASK] for the +Markov chain, and then each residue either stays the same or +transits to [MASK] with some probability at each step. For +the reverse process, a neural network ˜pθ is defined to predict +the probability of S0 and then parameterize the diffusion +trajectory with the probability q(St−1|St, ˜S0). That is, +q(St|St−1) = Cat +� +St; p = st−1Qt� +, +(8) +pθ(St−1|Pt) ∝ � +˜ +S0 q(St−1|St, ˜S0) · ˜pθ( ˜S0|Pt), +(9) +where st ∈ Rnr×21 denotes the one-hot feature for the +sequence St, Qt ∈ R21×21 denotes the corresponding tran- +sition matrix at step t, Cat(·) is the categorical distribution +and ˜S0 enumerates all possible residue types. +Parameterization of ˜pθ. We define the predictor ˜pθ with +residue representations ht. For each masked residue i in +St, we feed its representation ht +i to an MLP and predict the +type of the corresponding residue type s0 +i in S0: +˜pθ( ˜S0|Pt) = � +i ˜pθ(˜s0 +i |Pt) = � +i Softmax(˜s0 +i |MLP(ht +i)), +where the softmax function is applied over all residue types. +4.4. Pre-Training Objective +Finally, we derive the pre-training objective of DiffPreT by +optimizing the multimodal diffusion model above with the +ELBO loss (Ho et al., 2020): +L := E +��T +t=1 DKL +� +q(Pt−1|Pt, P0)||pθ(Pt−1|Pt) +�� +. +(10) +It can be proved that with the independence assumptions in +(3)(4), the objective can be decomposed into a structure loss +L(R) and a sequence loss L(S) (see proof in App. C.2): +L(R) :=E +��T +t=1 DKL +� +q(Rt−1|Rt, R0)||pθ(Rt−1|Pt) +�� +, +L(S) :=E +��T +t=1 DKL +� +q(St−1|St, S0)||pθ(St−1|Pt) +�� +. +Both loss functions can be simplified as follows. +Structure loss L(R). It has been shown in Ho et al. (2020) +that the loss function can be simplified under our parameter- +ization by calculating KL divergence between Gaussians as +weighted L2 distances between means ϵθ and ϵ (see details +in App. C.3): +L(R) = �T +t=1 γtEϵ∼N (0,I) +� +∥ϵ − ϵθ(Pt, t)∥2 +2 +�, +(11) +where the coefficients γt are determined by the variances +β1, ..., βt. In practice, we follow Ho et al. (2020) to set all +weights γt = 1 for the simplified loss L(R) +simple. +Sequence loss L(S). Since we parameterize pθ(St−1|Pt) +with ˜pθ( ˜S0|Pt) and q(St−1|St, ˜S0) as in (9), it can be +proved that the t-th KL divergence term in L(S) reaches +zero when ˜pθ( ˜S0|Pt) puts all mass on the ground truth S0 +(see proof in App. C.4). Therefore, for pre-training, we can +simplify the KL divergence to the cross entropy between +the correct residue type s0 +i and the prediction: +L(S) +simple = �T +t=1 +� +i CE +� +s0 +i , ˜pθ(s0 +i |Pt) +�, +(12) +where CE(·, ·) denotes the cross entropy loss. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +The ultimate training objective for our method is the sum of +simplified structure and sequence diffusion losses: +Ltotal = L(R) +simple + L(S) +simple. +(13) +5. Siamese Diffusion Trajectory Prediction +Through diffusion models on both protein sequences and +structures, the self-supervised learning approach proposed +in Sec. 4 tries to make representations capture (1) the atom- +and residue-level spatial interactions and (2) the statistical +dependencies of residue types within a single protein. Nev- +ertheless, no constraints or supervision have been added +for modeling relations between different protein structures, +especially different conformers of the same protein. Gener- +ated under different environmental factors, these conformers +typically share the same protein sequence but different struc- +tures, where side chain rotation is an important driver un- +derlying conformational variations. The properties of these +conformers are highly correlated (O’Connor et al., 2010), +the representations of which should reflect this correlation. +To incorporate these conformer-related information into +DiffPreT, in this section, we introduce a new approach called +Siamese Diffusion Trajectory Prediction (SiamDiff). The +high-level idea is to maximize the mutual information (MI) +between representations of diffusion trajectories of a pair +of correlated conformers generated from one protein. In +this way, SiamDiff will be able to keep the advantages of +DiffPreT and inject conformer-related information into rep- +resentations as well. We first propose a scheme to generate +randomly simulated conformers for each protein in Sec. 5.1. +Following DiffPreT, we generate the multimodal diffusion +trajectories for the conformers in Sec. 5.2. Then, in Sec. 5.3, +we show that the MI maximization can be transformed to +the problem of mutual prediction between diffusion trajec- +tories, which shares a similar loss function with DiffPreT. +The graphical illustration of this method is shown in Fig. 1. +5.1. Conformer Simulation Scheme +Given a protein, the first step of our method is to generate its +different conformers. However, directly sampling requires +an accurate characterization of the energy landscape of pro- +tein conformations, which is difficult and time-consuming. +To address this issue, we adopt a commonly used scheme +for sampling randomly simulated conformers by adding +torsional perturbations to side-chains (Ho & Agard, 2009). +Specifically, given the original protein P = (S, R), we +deem it as the native state P1 = (S1, R1) and generate a cor- +related conformer P2 = (S2, R2) by randomly perturbing +the protein structure, i.e., S2 = S1, R2 = perturb(R1, ϵ), +where ϵ ∈ [0, 2π)nr×4 is the noise drawn from a wrapped +normal distribution (Corso et al., 2022). The perturb(·, ·) +function will rotate the side-chain of each residue according +to the sampled torsional noises. In practice, atom clashes +can be avoided by adjusting the variance of added noises. +It should be noted that the scheme can be adapted flexibly +when considering different granularities of structures. For +instance, instead of considering side chain rotation on a fixed +backbone, one can consider a flexible backbone by rotating +backbone angles, to generate approximate conformers. +The current torsional perturbation scheme is simple yet ef- +fective, and is the first step toward including physics-driven +structural changes of a protein, while there remains room +for substantial improvement. For example, one can enhance +SiamDiff with available rotamer libraries (Shapovalov & +Dunbrack Jr, 2011), with the introduction of backbone flexi- +bility, or both, for which empirical force fields (Wang et al., +2004) can be used, which will be investigated in future. +5.2. Siamese Diffusion Trajectory Generation +Next, to keep the rich information in multimodal diffusion +trajectories as in DiffPreT, we generate the trajectories for +the pair of conformers, a.k.a., siamese trajectories. For- +mally, for conformers P1 and P2, we sample their diffu- +sion trajectories P0:T +1 +and P0:T +2 +via the mutlimodal diffu- +sion process defined in Sec. 4. Taking P1 = (S1, R1) +for example, we start the diffusion process from the state +P0 +1 = P1. The diffusion process is defined by the joint +diffusion on structures and sequences, i.e., q(P1:T +1 +|P0 +1) = +q(R1:T +1 +|R0 +1)q(S1:T +1 +|S0 +1). We utilize the conditional Gaus- +sian distributions in (5) to derive trajectories on structures +R1:T +1 +and the discrete Markov chain in (8) to derive the +sequence diffusion processes S1:T +1 +. In this way, we define +the trajectory P0:T +1 += {(St +1, Rt +1)}T +t=0 for P1 and can derive +the siamese trajectory P0:T +2 += {(St +2, Rt +2)}T +t=0 similarly. +5.3. Mutual Information Maximization between +Representations of Siamese Trajectories +To make representations reflect the correlation between dif- +ferent conformers of the same protein, we propose to maxi- +mize the MI between representations of siamese trajectories +constructed as above (see App. A for related works about +MI maximization). For notations, we use the bold sym- +bol to denote the representation of an object and use P 0:T +1 +and P 0:T +2 +to denote the corresponding random variables of +representations of the siamese trajectories P0:T +1 +and P0:T +2 +. +Because directly optimizing MI is intractable, we instead +maximize an approximate lower bound of MI described in +the following proposition (see proof in App. C.1). +Proposition 1 With some approximations, the mutual infor- +mation between representations of two siamese trajectories +is lower bounded by: +I(P 0:T +1 +; P 0:T +2 +) ≥ −1 +2(L(2→1) + L(1→2)) + C, + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +where C is a constant independent of our encoder and the +term from trajectory P0:T +b +to P0:T +a +is defined as +L(b→a) := EP0:T +a +,P0:T +b +��T +t=1 DKL +� +q(Pt−1 +a +|Pt +a, P0 +a)||p(Pt−1 +a +|Pt +a, P0:T +b +) +�� +, +with b → a being either 2 → 1 or 1 → 2. +The two terms share the similar formula as the ELBO loss +in (10). Take L(2→1) for example. Here q(Pt−1 +1 +|Pt +1, P0 +1) +is a posterior analytically tractable with our definition of +each diffusion step q(Pt +1|Pt−1 +1 +) in (5) and (8). The reverse +process is learnt to generate a less noisy state Pt−1 +1 +given the +current state Pt +1 and representations of the siamese trajectory +P0:T +2 +, which are extracted by the protein encoder to be +pre-trained. The parameterization of the reverse process +is similar as in Sec. 4.2 and 4.3, with the representations +replaced by those of P0:T +2 +(see App. B for details). +Essentially, we perform mutual prediction between two +siamese trajectories, which is similar to the idea of mu- +tual representation reconstruction in Grill et al. (2020); +Chen & He (2021). In practice, though with different struc- +tures, P1 and P2 share information about the same protein +and thus the whole trajectory of P2 may give too many +clues for the denoising towards Pt−1 +1 +, which makes the pre- +training task trivial. To address this issue, we parameterize +p(Pt−1 +1 +|Pt +1, P0:T +2 +) with pθ(Pt−1 +1 +|Pt +1, Pt +2). For diffusion +on sequences, we further guarantee that the same set of +residues are masked in St +1 and St +2 to avoid the leakage of +ground-truth residue types across correlated trajectories. +Final pre-training objective. Given the similarity between +the one-side objective and the ELBO loss in (10), we can use +a similar way to decompose the objective into structure and +sequence losses and then derive simplified loss functions +for each side. To summarize, the ultimate training objective +for our method is +Ltotal = 1 +2(L(R,2→1) +simple ++ L(S,2→1) +simple ++ L(R,1→2) +simple ++ L(S,1→2) +simple +), +where L(·,b→a) +simple +is the loss term defined by predicting trajec- +tory P0:T +a +from trajectory P0:T +b +(See App. B for derivation). +5.4. Discussion +Now we discuss the advantages of our method over previ- +ous works. The benefits of these critical designs will be +empirically demonstrated by experiments in Sec. 6.3. +Advantages of multimodal denoising. Compared with +previous diffusion models focusing on either protein se- +quences (Yang et al., 2022) or structures (Guo et al., 2022), +in this work, we perform joint diffusion on both modalities. +Note that given a sequence S and a structure R that exist +in the nature with high probability, the sequence-structure +tuple P = (S, R) may not be a valid state of this protein. +Consequently, instead of modeling the marginal distribution, +we are supposed to model the joint distribution of protein +sequences and structures. +Connection with diffusion models. Diffusion models have +achieved outstanding performance on image and text gen- +eration tasks (Dhariwal & Nichol, 2021; Li et al., 2022) +and recently been applied on unsupervised representation +learning (Abstreiter et al., 2021). The key to its success +is the denoising objective at different noise levels, which +has also been used for self-supervised learning in earlier +works of scheduled denoising autoencoders (Geras & Sut- +ton, 2014; Chandra & Sharma, 2014). These existing works +only denoise a protein sampled from the diffusion trajectory +from its native structure and sequence to random distribu- +tion, which adds no explicit supervision on representations +of different conformers. In contrast, our method incorpo- +rates the idea of mutual prediction of two siamese diffusion +trajectories so as to capture the correlation between different +conformers of one protein, which regularizes the manifold +underlying protein structures. +6. Experiments +We conduct experiments on both residue and atom levels to +prove the effectiveness of our method. +6.1. Experimental Setups +Pre-training datasets. Following Zhang et al. (2022), we +pre-train our models with the AlphaFold protein structure +database v1 (Jumper et al., 2021; Varadi et al., 2021), in- +cluding 365K proteome-wide predicted structures. +Downstream benchmark tasks. For downstream evalua- +tion, we adopt the EC prediction task (Gligorijevi´c et al., +2021) and four ATOM3D tasks (Townshend et al., 2020). +1. Enzyme Commission (EC) number prediction task +aims to predict EC numbers of proteins which describe +their catalysis behavior in biochemical reactions. This +task is formalized as 538 binary classification problems. +We adopt the dataset splits from Gligorijevi´c et al. (2021) +and use the test split with 95% sequence identity cutoff +following Zhang et al. (2022). +2. Protein Interface Prediction (PIP) requires the model +to predict whether two amino acids from two proteins +come into contact when the proteins bind (binary classi- +fication). The protein complexes of this benchmark are +split with 30% sequence identity cutoff. +3. Mutation Stability Prediction (MSP) task seeks to pre- +dict whether a mutation will increase the stability of a +protein complex or not (binary classification). The bench- +mark dataset is split upon a 30% sequence identity cutoff + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Table 1: Atom-level results on Atom3D tasks. +Method +PIP +MSP +RES +PSR +Mean +Rank +AUROC +AUROC +Accuracy +Global ρ +Mean ρ +GearNet-Edge +0.868±0.002 +0.633±0.067 +0.441±0.001 +0.782±0.021 +0.488 ±0.012 +7.4 +w/ pre-training +Denoising Score Matching +0.877±0.002 +0.629±0.040 +0.448±0.001 +0.813±0.003 +0.518±0.020 +5.0 +Residue Type Prediction +0.879±0.004 +0.620±0.027 +0.449±0.001 +0.826±0.020 +0.518±0.018 +3.2 +Distance Prediction +0.872±0.001 +0.677±0.020 +0.422±0.001 +0.840±0.020 +0.522±0.004 +3.8 +Angle Prediction +0.878±0.001 +0.642±0.013 +0.419±0.001 +0.813±0.007 +0.503±0.012 +5.4 +Dihedral Prediction +0.878±0.004 +0.591±0.008 +0.414±0.001 +0.821±0.002 +0.497±0.004 +6.4 +Multiview Contrast +0.871±0.003 +0.646±0.006 +0.368±0.001 +0.805±0.005 +0.502±0.009 +7.0 +DiffPreT +0.879±0.005 +0.605±0.036 +0.447±0.001 +0.821±0.007 +0.533±0.006 +3.6 +SiamDiff +0.880±0.002 +0.698±0.020 +0.449±0.001 +0.821±0.008 +0.523±0.007 +1.6 +Table 2: Residue-level results on EC and Atom3D tasks. +Method +EC +MSP +PSR +Mean +Rank +AUPR +Fmax +AUROC +Global ρ +Mean ρ +GearNet-Edge +0.837±0.002 +0.811±0.001 +0.644±0.023 +0.763±0.012 +0.373±0.021 +7.8 +w/ pre-training +Denoising Score Matching +0.859±0.003 +0.840±0.001 +0.645±0.028 +0.795±0.027 +0.429±0.017 +5.0 +Residue Type Prediction +0.851±0.002 +0.826±0.005 +0.636±0.003 +0.828±0.005 +0.480±0.031 +5.4 +Distance Prediction +0.858±0.003 +0.836±0.001 +0.623±0.007 +0.796±0.017 +0.416±0.021 +6.4 +Angle Prediction +0.873±0.003 +0.849±0.001 +0.631±0.041 +0.802±0.015 +0.446±0.009 +4.2 +Dihedral Prediction +0.858±0.001 +0.840±0.001 +0.568±0.022 +0.732±0.021 +0.398±0.022 +7.2 +Multiview Contrast +0.875±0.003 +0.857±0.003 +0.713±0.036 +0.752±0.012 +0.388±0.015 +4.0 +DiffPreT +0.864±0.002 +0.844±0.001 +0.673±0.042 +0.815±0.008 +0.505±0.007 +3.0 +SiamDiff +0.878±0.003 +0.857±0.003 +0.700±0.043 +0.832±0.007 +0.482±0.016 +1.4 +among different splits. +4. Residue Identity (RES) task studies the structural role +of an amino acid under its local environment. A model +predicts the type of the center amino acid based on its sur- +rounding atomic structure. The environments in different +splits are with different protein topology classes. +5. Protein Structure Ranking (PSR) predicts global dis- +tance test scores of structure predictions submitted +to the Critical Assessment of Structure Prediction +(CASP) (Kryshtafovych et al., 2019) competition. This +dataset is split according to the competition year. +Baseline methods. We evaluate our method on both atom- +and residue-level structures with GearNet-Edge (Zhang +et al., 2022) as the backbone model. GearNet-Edge mod- +els protein structures with different types of edges and +edge-type-specific convolutions, which is further enhanced +by message passing between edges. +Based on the en- +coder, we compare the proposed methods with previous +protein structural pre-training algorithms including multi- +view contrastive learning (Zhang et al., 2022), denoising +score matching (Guo et al., 2022) and four self-prediction +methods (Zhang et al., 2022), i.e., residue type, distance, an- +gle and dihedral prediction. Details can be found in App. D. +Since PIP and RES datasets are processed specifically for +atom-level models, we only consider EC, MSP and PSR as +residue-level tasks. Besides, we discard EC for atom-level +evaluation, because most proteins in the dataset only contain +backbone atoms when downloaded from PDB. +Training and evaluation. For fair comparison, we pre- +train our model for 50 epochs on the AlphaFold protein +structure database, following Zhang et al. (2022). For down- +stream evaluation, we fine-tune the pre-trained models for +50 epochs on EC, MSP and PSR. Due to the large size of the +RES and PIP dataset, we set the time limit as 24 hours and +thus only fine-tine each model for 10 epochs. More training +hyperparameters are stated in App. D. +We report the mean and standard deviation of each exper- +imental result on seeds 0, 1 and 2. For EC prediction, we +employ Fmax and AUPR as evaluation metrics, following +the original benchmark (Gligorijevi´c et al., 2021). We use +AUROC to measure the binary classification performance +of PIP and MSP. For PSR prediction, we utilize the global +and mean Spearman’s ρ to assess the ranking performance. +The micro-averaged accuracy serves as the evaluation met- +ric of RES. Detailed definitions of Fmax, global and mean +Spearman’s ρ are stated in App. D. +6.2. Experimental Results +Tables 1 and 2 report the results of GearNet-Edge on atom- +and residue-level benchmark tasks, respectively. We analyze +and discuss these results below. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Table 3: Ablation study on atom-level Atom3D tasks. +Method +PIP +MSP +RES +PSR +AUROC +AUROC +Accuracy +Global ρ +GearNet-Edge +0.868±0.002 +0.633±0.067 +0.441±0.001 +0.782±0.021 +SiamDiff +0.880±0.002 +0.698±0.020 +0.449±0.001 +0.821±0.008 +w/o MI max. +0.879±0.005 +0.605±0.036 +0.447±0.001 +0.821±0.007 +w/o seq. diff. +0.873±0.004 +0.695±0.002 +0.443±0.001 +0.803±0.010 +w/o struct. diff. +0.878±0.003 +0.652±0.021 +0.456±0.001 +0.805±0.005 +Overall, DiffPreT performs competitively against base- +line methods, and SiamDiff gains superior performance +over baselines. DiffPreT achieves the top-3 performance in +terms of mean rank on both structure levels. On all bench- +mark metrics, the performance of SiamDiff is among the top +three, which is not attained by any previous method, and it +ranks the first in terms of mean rank on both structure levels. +The strongest competitors on atom level (i.e., Residue Type +Prediction) and residue level (i.e., Multiview Contrast) both +perform poorly on the other structure level. These results +demonstrate the strength of DiffPreT and SiamDiff on boost- +ing diverse types of structure-based protein understanding +tasks, including function prediction (EC), protein-protein +interaction prediction (PIP), mutation stability prediction +(MSP), structural role understanding (RES) and protein +model quality assessment (PSR). +On PIP, SiamDiff and DiffPreT rank the first and sec- +ond among all methods, respectively. The superior per- +formance demonstrates that our approaches can well model +protein complex interfaces and further benefit the under- +standing of how different proteins interact. This can be +attributed to the denoising diffusion models on protein struc- +tures, with which the local structural patterns can be cap- +tured. +On MSP, SiamDiff performs best on two structure lev- +els on average, while the performance of DiffPreT is not +satisfactory. This task classifies a set of similar mutant +structures into two groups according to their ability to stabi- +lize protein-protein interactions. Such a task can be better +solved if the pre-trained model is aware of the correlation +between different mutant structures, so that more correlated +mutant structures are more likely to be assigned to the same +group. Compared to DiffPreT, SiamDiff can endow the +model with such capability by modeling the dependencies +between different conformers of one protein, and it thus +performs better in this task. +On RES, SiamDiff performs best, and DiffPreT ranks +the fourth place. In the RES task, the type of a masked +residue can be well implied by the types of its surrounding +residues. This explains the decent performance of Residue +Type Prediction, the pre-training objective of which exactly +matches the task. In our methods, the denoising diffusion of +protein sequences can model such residue type dependen- +cies, and thus lead to the superior performance. +On PSR, both DiffPreT and SiamDiff can achieve top- +3 best global ρ and mean ρ on both structure levels. +Among all baselines, only Residue Type Prediction can +achieve consistently good performance on both structure +levels. It is worth noticing that our methods and Residue +Type Prediction can all acquire the compatibility of protein +sequence and structure. Specifically, Residue Type Predic- +tion models the conditional distribution of sequences given +structures, while DiffPreT and SiamDiff model the joint dis- +tribution of sequences and structures. Such cross-modality +modeling mechanisms benefit their performance on the task +that assesses predicted structures for a protein sequence. +On EC, SiamDiff gains the best performance, and Diff- +PreT is among top 4 for both metrics. This task seeks +to annotate proteins with their catalyzed reactions, which +are mainly determined by the structures of active/binding +sites on protein surfaces. Experimental results verify that +DiffPreT can well capture such structural patterns through +denoising multimodal diffusion trajectories, and SiamDiff +further improves the performance on this task by modeling +the dependencies between correlated conformers. +6.3. Ablation Study +We study the effect of different parts of SiamDiff in Tbl. 3. +Effect of multimodal diffusion. We study two degenerated +settings of multimodal diffusion, i.e., w/o sequence diffusion +and w/o structure diffusion. Performance decay is observed +on 7 out of 8 benchmark metrics for these two degenerated +settings. These results prove the necessity of both structure +and sequence diffusion for SiamDiff to learn structure- and +sequence-aware protein representations. +Effect of MI maximization. Compared with SiamDiff, +the performance of DiffPreT drops on 3 out of 4 bench- +mark tasks. Therefore, modeling dependencies of correlated +conformers is beneficial for many downstream tasks, e.g., +mutation stability prediction on MSP (AUROC w/ and w/o +this component: 0.698±0.020 v.s. 0.605±0.036). +7. Conclusions and Future Work +In this work, we propose the DiffPreT approach to pre-train +a protein encoder by sequence-structure multimodal diffu- +sion modeling, which captures the inter-atomic interactions +within structure and the residue type dependencies along +sequence. We further propose the SiamDiff method to en- +hance DiffPreT by additionally modeling the correlation +between different conformers of one protein. Extensive ex- +periments on diverse types of tasks and on both atom- and +residue-level structures verify the competitive performance +of DiffPreT and the superior performance of SiamDiff. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +The current torsional perturbation scheme in SiamDiff is +simple yet effective, while it less explores the physical rules +hidden in side chain conformations. Therefore, our future +works will further enhance SiamDiff with physics-inspired +rotamer libraries (Shapovalov & Dunbrack Jr, 2011) and +physics-based force fields (Wang et al., 2004). +Acknowledgments +The authors would like to thank Meng Qu, Zhaocheng Zhu, +Shengchao Liu, Chence Shi, Jiarui Lu, Huiyu Cai, Xinyu +Yuan and Bozitao Zhong for their helpful discussions and +comments. +This project is supported by AIHN IBM-MILA partnership +program, the Natural Sciences and Engineering Research +Council (NSERC) Discovery Grant, the Canada CIFAR +AI Chair Program, collaboration grants between Microsoft +Research and Mila, Samsung Electronics Co., Ltd., Amazon +Faculty Research Award, Tencent AI Lab Rhino-Bird Gift +Fund, a NRC Collaborative R&D Project (AI4D-CORE-06) +as well as the IVADO Fundamental Research Project grant +PRF-2019-3583139727. +References +Abstreiter, K., Bauer, S., Sch¨olkopf, B., and Mehrjou, A. +Diffusion-based representation learning. arXiv preprint +arXiv:2105.14257, 2021. +Anand, N. and Achim, T. 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The encoders are designed to capture protein structural information on +different granularity, including residue-level structures (Gligorijevi´c et al., 2021; Zhang et al., 2022; Xu et al., 2022a), +atom-level structures (Hermosilla et al., 2021; Jing et al., 2021a; Wang et al., 2022) and protein surfaces (Gainza et al., +2020; Sverrisson et al., 2021; Somnath et al., 2021). In this work, we focus on pre-training a typical residue-level structure +encoder, i.e., GearNet-Edge (Zhang et al., 2022), and a typical atom-level structure encoder, i.e., GVP (Jing et al., 2021a). +Mutual Information (MI) Estimation and Maximization. MI can measure both the linear and non-linear dependency +between random variables. Some previous works (Belghazi et al., 2018; Hjelm et al., 2018) try to use neural networks to +estimate the lower bound of MI, including Donsker-Varadhan representation (Donsker & Varadhan, 1983), Jensen-Shannon +divergence (Fuglede & Topsoe, 2004) and Noise-Contrastive Estimation (NCE) (Gutmann & Hyv¨arinen, 2010; 2012). The +optimization with InfoNCE loss (Oord et al., 2018) maximizes a lower bound of MI and is broadly shown to be a superior +representation learning strategy (Chen et al., 2020b; Hassani & Khasahmadi, 2020; Xu et al., 2021; Liu et al., 2021; Zhang +et al., 2022). In this work, we adopt the MI lower bound proposed by Liu et al. (2022) with two conditional log-likelihoods, +and we formulate the learning objective by mutually denoising the multimodal diffusion processes of two correlated proteins. +B. Details of SiamDiff +In this section, we discuss details of our SiamDiff method. We first describe the parameterization of the generation process +pθ(Pt−1 +1 +|Pt +1, Pt +2) in Sec. B.1, derive the pre-training objective in Sec. B.2, and discuss some modifications when applied on +residue-level models in Sec. B.3. +B.1. Parameterization of Generation Process pθ(P0 +1|Pt +1, Pt +2) +Remember that we use P0:T +1 +and P0:T +2 +to denote the representation of the siamese trajectories P0:T +1 +and P0:T +2 +, respectively. +Different from the generation process in traditional diffusion models, the parameterization of pθ(Pt−1 +1 +|Pt +1, Pt +2) should +inject information from Pt +2. Therefore, we use the extracted residue and atom representations (denoted as at +2 and ht +2) of Pt +2 +for this denoising step. Given the conditional independence in (3)(4), this generation process can be decomposed into that +on protein structures and sequences similarly in Sec. 4.4. +Generation process on protein structures. As in (7), modeling the generation process of protein structures is to model +the noise on Rt +1 and gradually decorrupt the noisy structure. This can be parameterized with a noise prediction network +ϵθ(Pt +1, Pt +2, t) that is translation-invariant and rotation-equivariant w.r.t. Rt +1. Besides, the noise applied on Rt +1 should not +change with transformations on Rt +2, so ϵθ should be SE(3)-invariant w.r.t. Rt +2. +To achieve these goals, we build our noise prediction network with atom representations at +2 (which is SE(3)-invariant +w.r.t. Rt +2) and atom coordinates rt +1 (which is SE(3)-equivariant w.r.t. Rt +1). We define an equivariant output similarly as in +DiffPreT. Specifically, we have +[ϵθ(Pt +1, Pt +2, t)]i = +� +j∈N t +1(i) mi,j · +rt +1i−rt +1j +∥rt +1i−rt +1j∥2 , with mi,j = MLP(at +2i, at +2j, MLP(∥rt +1i − rt +1j∥2)), +where N t +1(i) denotes the neighbors of the atom i in the corresponding graph of Pt +1. Note that ϵθ(Pt +1, Pt +2, t) achieves +the equivariance requirement, as mi,j is SE(3)-invariant w.r.t. Rt +1 and Rt +2 while rt +1i − rt +1j is translation-invariant and +rotation-equivariant w.r.t. Rt +1. +Generation process on protein sequences. As in (8), the generation process on sequences aims to predict masked residue +types in S0 +1 with a predictor ˜pθ. In our setting of mutual prediction, we define the predictor based on representations of the +same residues in St +2, which are also masked. Hence, for each masked residue i in St +2, we feed its representation ht +2i to an +MLP and predict the type of the corresponding residue type s0 +1i in S0 +1: +˜pθ(S0 +1|Pt +1, Pt +2) = � +i ˜pθ(s0 +1i|Pt +1, Pt +2) = � +i Softmax(s0 +1i|MLP(ht +2i)), +where the softmax function is applied over all residue types. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +B.2. Pre-Training Objective +Given the defined forward and reverse process on two trajectories, we now derive the pre-training objective based on the +mutual diffusion loss in Prop. 1. We take the term L(2→1) for example and its counterpart can be derived in the same way. +The objective can be decomposed into a structure loss L(R,2→1) and a sequence loss L(S,2→1): +L(R,2→1) :=E +��T +t=1 DKL +� +q(Rt−1 +1 +|Rt +1, R0 +1)||pθ(Rt−1 +1 +|Pt +1, Pt +2) +�� +, +(14) +L(S,2→1) :=E +��T +t=1 DKL +� +q(St−1 +1 +|St +1, S0 +1)||pθ(St−1 +1 +|Pt +1, Pt +2) +�� +. +(15) +Based on the derivation in Sec. 4.4, the structure loss L(R,2→1) can be simplified as +L(R,2→1) +simple += �T +t=1 Eϵ∼N (0,I) +� +∥ϵ − ϵθ(Pt +1, Pt +2, t)∥2 +2 +�, +(16) +and the sequence loss L(S,2→1) can be simplified as +L(S,2→1) +simple += �T +t=1 +� +i CE +� +s0 +1i, ˜pθ(s0 +1i|Pt +1, Pt +2) +�. +(17) +Then, the final objective in Sec. 5.3 can be easily derived. +B.3. Residue-level model +Residue-level protein graphs can be seen as a concise version of atom graphs that enable efficient message passing between +nodes and edges. As in Zhang et al. (2022), we only keep the alpha carbon atom of each residue and add sequential, +radius and K-nearest neighbor edges as different types of edges. For SiamDiff, the residue-level model cannot discriminate +conformers generated by rotating side chains, since we only keep CA atoms. To solve this problem, we directly add Gaussian +noises to the coordinates instead to generate approximate conformers. Specifically, the correlated conformer P2 = (S2, R2) +is defined by S2 = S1, R2 = R1 + ϵ, where ϵ ∈ Rna×3 is the noise drawn from a normal distribution. +C. Proofs +In this section, we provide proofs for propositions in Sec. 4 and Sec. 5. Due to the similarity between the two methods, all +propositions are restated for SiamDiff. DiffPreT can be seen as a special case that two siamese trajectories collapse into one. +C.1. Proof of Proposition 1 +Proof. +First, the mutual information between representations of two trajectories is defined as: +I(P 0:T +1 +; P 0:T +2 +) = EP0:T +1 +,P0:T +2 +∼p(P 0:T +1 +,P 0:T +2 +) +� +log +p(P0:T +1 +,P0:T +2 +) +p(P0:T +1 +)p(P0:T +2 +) +� +, +(18) +where the joint distribution is defined as p(P 0:T +1 +, P 0:T +2 +) = p(P 0 +1 , P 0 +2 )q(P 1:T +1 +|P 0 +1 )q(P 1:T +2 +|P 0 +2 ). Next, we can derive a +lower bound with this definition: +I(P 0:T +1 +; P 0:T +2 +) = E +� +log +p(P0:T +1 +, P0:T +2 +) +p(P0 +1)q(P1:T +1 +|P0 +1)p(P0 +2)q(P1:T +2 +|P0 +2) +� +≥E +� +�log +p(P0:T +1 +, P0:T +2 +) +� +p(P0 +1)p(P0 +2)q(P1:T +1 +|P0 +1)q(P1:T +2 +|P0 +2) +� +� +=1 +2E +� +log +p(P0:T +1 +, P0:T +2 +)2 +p(P0 +1)p(P0 +2)q(P1:T +1 +|P0 +1)2q(P1:T +2 +|P0 +2)2 +� +=1 +2E +� +log +p(P0:T +1 +, P0:T +2 +) +p(P0 +2)q(P1:T +1 +|P0 +1)q(P1:T +2 +|P0 +2) ++ log +p(P0:T +1 +, P0:T +2 +) +p(P0 +1)q(P1:T +1 +|P0 +1)q(P1:T +2 +|P0 +2) +� +=1 +2E +� +log p(P0:T +1 +|P0:T +2 +) +q(P1:T +1 +|P0 +1) ++ log p(P0:T +2 +|P0:T +1 +) +q(P1:T +2 +|P0 +2) +� +. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +However, since the distribution of representations are intractable to sample for optimization, we instead sample the +trajectories P0:T +1 +and P0:T +2 +from our defined diffusion process, i.e., p(P0:T +1 +, P0:T +2 +) = p(P0 +1, P0 +2)q(P1:T +1 +|P0 +1)q(P1:T +2 +|P0 +2). +Besides, instead of predicting representations, we use the representations from one trajectory to recover the other trajectory, +which reflects more information than its representation. With these approximations, the lower bound above can be further +written as: +1 +2E +� +log p(P0:T +1 +|P0:T +2 +) +q(P1:T +1 +|P0 +1) + log p(P0:T +2 +|P0:T +1 +) +q(P1:T +2 +|P0 +2) +� +≈ 1 +2EP0:T +1 +,P0:T +2 +� +log p(P0:T +1 +|P0:T +2 +) +q(P1:T +1 +|P0 +1 ) + log p(P0:T +2 +|P0:T +1 +) +q(P1:T +2 +|P0 +2 ) +� +We now show the first term on the right hand side can be written as the loss defined in Proposition 1. The derivation is very +similar with the proof of Proposition 3 in Xu et al. (2022b). We include it here for completeness: +EP0:T +1 +,P0:T +2 +� +log p(P0:T +1 +|P0:T +2 +) +q(P1:T +1 +|P0 +1) +� +=EP0:T +1 +,P0:T +2 +� T +� +t=1 +log p(Pt−1 +1 +|Pt +1, P0:T +2 +) +q(Pt +1|Pt−1 +1 +) +� +=EP0:T +1 +,P0:T +2 +� +log (P0 +1|P1 +1, P0:T +2 +) +q(P1 +1|P0 +1) ++ +T +� +t=2 +log +� +p(Pt−1 +1 +|Pt +1, P0:T +2 +) +q(Pt−1 +1 +|Pt +1, P0 +1) +· q(Pt−1 +1 +|P0 +1) +q(Pt +1|P0 +1) +�� +=EP0:T +1 +,P0:T +2 +� +− log q(PT +1 |P0 +1) + log p(P0 +1|P1 +1, P0:T +2 +) + +T +� +t=2 +log p(Pt−1 +1 +|Pt +1, P0:T +2 +) +q(Pt−1 +1 +|Pt +1, P0 +1) +� += − EP0:T +1 +,P0:T +2 +��T +t=1 DKL +� +q(Pt−1 +1 +|Pt +1, P0 +1)||p(Pt−1 +1 +|Pt +1, P0:T +2 +) +�� ++ C(2→1) += − L(2→1) + C(2→1), +where we merge the term p(P0 +1|P1 +1, P0:T +2 +) into the sum of KL divergences for brevity and use C(2→1) to denote the constant +independent of our encoder. Note that the counterpart can be derived in the same way. Adding these two terms together +finishes the proof of Proposition 1. +□ +C.2. Proof of Pre-Training Loss Decomposition +We restate the proposition of pre-training loss decomposition rigorously as below. +Proposition 2 Given the assumptions 1) the separation of the diffusion process on protein structures and sequences +q(Pt +a|Pt−1 +a +) = q(Rt +a|Rt−1 +a +) · q(St +a|St−1 +a +), +(19) +and 2) the conditional independence of the generation process +pθ(Pt−1 +a +|Pt +a, Pt +b) = pθ(Rt−1 +a +|Pt +a, Pt +b) · pθ(St−1 +a +|Pt +a, Pt +b), +(20) +it can be proved that +L(b→a) = L(R,b→a) + L(S,b→a), +(21) +where the three loss terms are defined as +L(b→a) :=E +��T +t=1 DKL +� +q(Pt−1 +a +|Pt +a, P0 +a)||pθ(Pt−1 +a +|Pt +a, Pt +b) +�� +, +L(R,b→a) :=E +��T +t=1 DKL +� +q(Rt−1 +a +|Rt +a, R0 +a)||pθ(Rt−1 +a +|Pt +a, Pt +b) +�� +, +L(S,b→a) :=E +��T +t=1 DKL +� +q(St−1 +a +|St +a, S0 +a)||pθ(St−1 +a +|Pt +a, Pt +b) +�� +, +with b → a referring to the term from trajectory P0:T +b +to P0:T +a +. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Proof. +Let L(·) +t +to denote the t-th KL divergence term in L(·). Then, we have +L(b→a) +t += DKL +� +q(Pt−1 +a +|Pt +a, P0 +a)||pθ(Pt−1 +a +|Pt +a, P0:T +b +) +� += DKL +�� +q(Rt−1 +a +|Rt +a, R0 +a)q(St−1 +a +|St +a, S0 +a) +� +|| +� +pθ(Rt−1 +a +|Pt +a, P0:T +b +)pθ(St−1 +a +|Pt +a, P0:T +b +) +�� += DKL +� +q(Rt−1 +a +|Rt +a, R0 +a)||pθ(Rt−1 +a +|Pt +a, P0:T +b +) +� ++ DKL +� +q(St−1 +a +|St +a, S0 +a)||pθ(St−1 +a +|Pt +a, P0:T +b +) +� += L(R,b→a) +t ++ L(S,b→a) +t +, +where we use the assumptions (19) and (20) in the second equality. The third equality is due to the additive property of the +KL divergence for independent distributions. Adding T KL divergence terms together will lead to (21). +□ +C.3. Proof of Simplified Structure Loss +For completeness, we show how to derive the simplified structure loss. The proof is directly adapted from (Xu et al., 2022b). +Proposition 3 Given the definition of the forward process +q(Rt +a|Rt−1 +a +) = N(Rt +a; +� +1 − βtRt−1 +a +, βtI), +(22) +and the reverse process +pθ(Rt−1 +a +|Pt +a, Pt +b) = N(Rt−1; µθ(Pt +a, Pt +b, t), σ2 +t I), +(23) +µθ(Pt +a, Pt +b, t) = +1 +√αt +� +Rt +a − +βt +√1 − ¯αt +ϵθ(Pt +a, Pt +b, t) +� +, +(24) +the structure loss function +L(R,b→a) :=E +��T +t=1 DKL +� +q(Rt−1 +a +|Rt +a, R0 +a)||pθ(Rt−1 +a +|Pt +a, Pt +b) +�� +, +(25) +can be simplified to +L(R,b→a) = �T +t=1 γtEϵ∼N (0,I) +� +∥ϵ − ϵθ(Pt +a, Pt +b, t)∥2 +2 +�, +(26) +where γt = +βt +2αt(1−¯αt−1) with αt = 1 − βt, ¯αt = �t +s=1 αs and b → a is either 2 → 1 or 1 → 2. +Proof. +First, we prove q(Rt +a|R0 +a) = N(Rt +a; √ ¯αtR0 +a, (1 − ¯αt)I). Let ϵi be the standard Gaussian random variable at time +step i. Then, we have +Rt +a = √αtRt−1 +a ++ +� +βtϵt += √αt−1αtRt−2 +a ++ +� +αt−1βt−1ϵt−1 + +� +βtϵt += · · · += √¯αtR0 +a + +� +αtαt−1...α2β1ϵ1 + · · · + +� +αt−1βt−1ϵt−1 + +� +βtϵt, +which suggests that the mean of Rt +a is √¯αtR0 +a and the variance matrix is (αtαt−1...α2β1+· · ·+αt−1βt−1+βt)I = (1−¯α)I. +Next, we derive the posterior distribution as: +q(Rt−1 +a +|Rt +a, R0 +a) = q(Rt +a|Rt−1 +a +)q(Rt−1 +a +|R0 +a) +q(Rta|R0a) += N(Rt +a; √αtRt−1 +a +, βtI) · N(Rt−1 +a +; √¯αt−1R0 +a, (1 − ¯αt−1)I) +N(Rta; √¯αtR0a, (1 − ¯αt)I) += N(Rt−1; +√¯αt−1βt +1 − ¯αt +R0 +a + +√αt(1 − ¯αt−1) +1 − ¯αt +Rt +a, 1 − ¯αt−1 +1 − ¯αt +βtI). + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Let ˜βt = 1−¯αt−1 +1−¯αt βt, then the t-th KL divergence term can be written as: +DKL +� +q(Rt−1 +a +|Rt +a, R0 +a)||pθ(Rt−1 +a +|Pt +a, Pt +b) +� += 1 +2˜βt +���� +√¯αt−1βt +1 − ¯αt +R0 +a + +√αt(1 − ¯αt−1) +1 − ¯αt +Rt +a − +1 +√αt +� +Rt +a − +βt +√1 − ¯αt +ϵθ(Pt +a, Pt +b, t) +����� +2 += 1 +2˜βt +Eϵ +���� +√¯αt−1βt +1 − ¯αt +· Rt +a − √1 − ¯αtϵ +√¯αt ++ +√αt(1 − ¯αt−1) +1 − ¯αt +Rt +a − +1 +√αt +� +Rt +a − +βt +√1 − ¯αt +ϵθ(Pt +a, Pt +b, t) +����� +2 += 1 +2˜βt +· +β2 +t +αt(1 − ¯αt)Eϵ +��ϵ − ϵθ(Pt +a, Pt +b, t) +�� +=γtEϵ +� +∥ϵ − ϵθ(Pt +a, Pt +b, t)∥2 +2 +� +, +which completes the proof. +□ +C.4. Proof of Simplified Sequence Loss +Now we show the equivalence of optimizing sequence loss L(S,b→a) and the masked residue type prediction problem on S0 +a. +Proposition 4 Given the definition of reverse process on protein sequences +pθ(St−1 +a +|Pt +a, Pt +b) ∝ � +˜ +S0a q(St−1 +a +|St +a, ˜S0 +a) · ˜pθ( ˜S0 +a|Pt +a, Pt +b), +(27) +the sequence loss L(S,b→a) reaches zero when ˜pθ( ˜S0 +a|Pt +a, Pt +b) puts all mass on the ground truth S0 +a. +Proof. +The loss function can be written as: +L(S,b→a) := E +��T +t=1 DKL +� +q(St−1 +a +|St +a, S0 +a)||pθ(St−1 +a +|Pt +a, Pt +b) +�� += E +��T +t=1 DKL +� +q(St−1 +a +|St +a, S0 +a) +���� +���� +� +˜ +S0a q(St−1 +a +|St +a, ˜S0 +a) · ˜pθ( ˜S0 +a|Pt +1, Pt +2) +Z +�� +, +where Z is the normalization constant. Hence, when ˜pθ( ˜S0 +a|Pt +a, Pt +b) puts all mass on the ground truth S0 +a, the distribution +pθ(St−1 +a +|Pt +a, Pt +b) will be identical with q(St−1 +a +|St +a, S0 +a), which makes the KL divergence become zero. +□ +D. Experimental Details +In this section, we introduce the details of our experiments. All these methods are developed based on PyTorch and +TorchDrug (Zhu et al., 2022). +Graph construction. +For atom graphs, we connect atoms with Euclidean distance lower than a distance threshold. For +PSR and MSP tasks, we remove all hydrogen atoms following Jing et al. (2021b). For residue graphs, we discard all +non-alpha-carbon atoms and add three different types of directed edges: sequential edges, radius edges and K-nearest +neighbor edges. For sequential edges, two atoms are connected if their sequential distance is below a threshold and these +edges are divided into different types according to these distances. For two kinds of spatial edges, we connect atoms +according to Euclidean distance and k-nearest neighbors. We further apply a long range interaction filter that removes edges +with low sequential distances. We refer readers to Zhang et al. (2022) for more details. +Atom-level backbone models. +To adapt GearNet-Edge to atom-level structures with moderate computational cost, we +construct the atom graph by using only the spatial edge with the radius dradius = 4.5 ˚A. We concatenate one-hot features of +atom types and residue types as node features and concatenate (1) one-hot features of residue types of end nodes, (2) one-hot +features of edge types, (3) one-hot features of sequential distance, (4) spatial distance as edge features. The whole model +is composed of 6 message passing layers with 128 hidden dimensions and ReLU activation function. For edge message +passing, we employ the discretized angles to determine the edge types on the line graph. The final prediction is performed +upon the hidden representation concatenated across all layers. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Residue-level backbone models. +We directly borrow the best hyperparameters reported in the original paper of GearNet- +Edge (Zhang et al., 2022). We adopt the same configuration of relational graph construction, i.e., the sequential distance +threshold dseq = 3, the radius dradius = 10.0 ˚A, the number of neighbors k = 10 and the long range interaction cutoff +dlong = 5. We use one-hot features of residue types as node features and concatenate (1) features of end nodes, (2) one-hot +features of edge types, (3) one-hot features of sequential distance, (4) spatial distance as edge features. Then we use 6 +message passing layers with 512 hidden dimensions and ReLU as the activation function. For edge message passing, the +edge types on the line graph are determined by the discretized angles. The hidden representations in each layer of GearNet +will be concatenated for the final prediction. +Baseline pre-training methods. +Here we briefly introduce the considered baselines. Multiview Contrast aims to maximize +the mutual information between correlated views, which are extracted by randomly chosen augmentation functions to +capture protein sub-structures. Residue type, distance, angle and dihedral prediction masks single residues, single edges, +edge pairs and edge triplets, respectively, and then predict the corresponding properties. Denoising score matching performs +denoising on noised pairwise distance matrices based on the learnt representations. +For all baselines in (Zhang et al., 2022), we adopt the original configurations. For Multiview Contrast, we use subsequence +cropping that randomly extracts protein subsequences with no more than 50 residues and space cropping that takes all +residues within a 15 ˚A Euclidean ball with a random center residue. Then, either an identity function or a random edge +masking function with mask rate equal to 0.15 is applied for constructing views. The temperature τ in the InfoNCE loss +function is set as 0.07. We set the number of sampled items in each protein as 256 for Distance Prediction and as 512 for +Angle and Dihedral Prediction. The mask rate for Residue Type Prediction is set as 0.15. When masking a residue on atom +graphs, we discard all non-backbone atoms and set the residue features as zero. Since the backbone models and tasks in our +paper are quite different with those in Guo et al. (2022), we re-implement the method on our codebase. We consider 50 +different noise levels log-linearly ranging from 0.01 to 10.0. +In DiffPreT, for structure diffusion, we use a sigmoid schedule for variances βt with the lowest variance β1 = 1e − 3 and +the highest variance βT = 0.1. For sequence diffusion, we simply set the cumulative transition probability to [MASK] over +time steps as a linear interpolation between minimum mask rate 0.15 and maximum mask rate 1.0. The number of diffusion +steps is set as 100. In SiamDiff, we adopt the same hyperparameters for multimodal diffusion models. We set the variance +of torsional perturbation noises as 0.1π on the atom level and that of Gaussian perturbation noises as 0.3 on the residue level +when constructing the correlated conformer. +All other optimization configurations for these pre-training methods are reported in Table 4. All methods are pre-trained on +four Tesla A100 GPUs and Table 4 reports the batch sizes on each GPU. +Table 4: Optimization configurations for pre-training methods. Here max length denotes the maximum number of residues +kept in each protein and lr stands for learning rate. +Method +Max length +Batch size +Optimizer +lr +residue +atom +residue +atom +Residue Type Prediction +100 +100 +96 +64 +Adam +1e-3 +Distance Prediction +100 +100 +128 +64 +Adam +1e-3 +Angle Prediction +100 +100 +96 +64 +Adam +1e-3 +Dihedral Prediction +100 +100 +96 +64 +Adam +1e-3 +Multiview Contrast +- +- +96 +64 +Adam +1e-3 +Denoising Score Matching +200 +200 +12 +12 +Adam +1e-4 +DiffPreT +150 +100 +16 +64 +Adam +1e-4 +SiamDiff +150 +100 +16 +32 +Adam +1e-4 +Fine-tuning on downstream tasks. +For all models on all downstream tasks, we apply the a three-layer MLP head for +prediction, the hidden dimension of which is set to the dimension of model outputs. The number of used gpus and batch +sizes for each model are chosen according the memory limit. All residue-level tasks are run on 4 V100 GPUs while all +atom-level tasks are run on A100 GPUs. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Table 5: Optimization configurations for downstream evaluations. Here max length denotes the maximum number of +residues kept in each protein and lr stands for learning rate. +Task +# GPUS +Batch size +Optimizer +lr +residue +atom +residue +atom +EC +4 +N/A +2 +N/A +Adam +1e-4 +PIP +N/A +1 +N/A +8 +Adam +1e-4 +MSP +4 +1 +1 +8 +Adam +1e-4 +RES +N/A +4 +N/A +64 +Adam +1e-4 +PSR +4 +1 +8 +8 +Adam +1e-4 +Evaluation metrics. +We clarify the definitions of Fmax (used in EC), global Spearman’s ρ (used in PSR) and mean +Spearman’s ρ (used in PSR) as below: +• Fmax denotes the protein-centric maximum F-score. It first computes the precision and recall for each protein at a +decision threshold t ∈ [0, 1]: +precisioni(t) = +� +f 1[f ∈ Pi(t) ∩ Ti] +� +f 1[f ∈ Pi(t)] +, +recalli(t) = +� +f 1[f ∈ Pi(t) ∩ Ti] +� +f 1[f ∈ Ti] +, +(28) +where f denotes a functional term in the ontology, Ti is the set of experimentally determined functions for protein i, +Pi(t) is the set of predicted functions for protein i whose scores are greater or equal to t, and 1[·] represents the indicator +function. After that, the precision and recall are averaged over all proteins: +precision(t) = +1 +M(t) +� +i +precisioni(t), +recall(t) = 1 +N +� +i +recalli(t), +(29) +where N denotes the total number of proteins, and M(t) denotes the number of proteins which contain at least one +prediction above the threshold t, i.e., |Pi(t)| > 0. +Based on these two metrics, the Fmax score is defined as the maximum value of F-measure over all thresholds: +Fmax = max +t +�2 · precision(t) · recall(t) +precision(t) + recall(t) +� +. +(30) +• Global Spearman’s ρ for PSR measures the correlation between the predicted global distance test (GDT TS) score +and the ground truth. It computes the Spearman’s ρ between the prediction and the ground truth over all test proteins +without considering the different biopolymers that these proteins lie in. +• Mean Spearman’s ρ for PSR also measures the correlation between GDT TS predictions and the ground truth. +However, it first splits all test proteins into multiple groups based on their corresponding biopolymers, then computes +the Spearman’s ρ within each group, and finally reports the mean Spearman’s ρ over all groups. +E. Results of Different Diffusion Models for Pre-Training +In Sec. 4, we consider multimodal diffusion models on protein sequences and structures for pre-training. In this section, we +explore the performance of different diffusion models when applied for pre-training. The results are shown in Table 6. +First, we simply run diffusion models on protein sequences and structures for pre-training, both of which achieve im- +provement compared with the baseline GearNet-Edge. By combining two diffusion models, DiffPreT can achieve better +performance on PIP and PSR. This advantage will be further increased after using siamese diffusion trajectory prediction +as shown in Table 3. It can be observed that sequence diffusion achieves better performance than DiffPreT due to the +consistency of objectives between pre-training and RES tasks. +Besides, we also consider diffusion models on torsional angles of protein side chains for pre-training. This method has +shown its potential in the protein-ligand docking task (Corso et al., 2022). For each residue, we randomly select one + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Table 6: Results of different diffusion models for pre-training on atom-level Atom3D tasks. +Method +PIP +RES +PSR +AUROC +Accuracy +Global ρ +Mean ρ +GearNet-Edge +0.868 +0.441 +0.782 +0.488 +w/ pre-training +DiffPreT +0.879 +0.447 +0.821 +0.533 +SiamDiff +0.880 +0.449 +0.821 +0.523 +Sequence Diffusion +0.879 +0.456 +0.802 +0.508 +Structure Diffusion +0.877 +0.448 +0.813 +0.518 +Torsional Diffusion +0.877 +0.442 +0.819 +0.505 +Table 7: Atom-level results of GVP on Atom3D tasks. Accuracy is abbreviated as Acc. +Method +PSR +MSP +PIP +RES +Mean +Rank +Global ρ +Mean ρ +AUROC +AUROC +Acc. +GVP +0.809 +0.486 +0.610 +0.846 +0.481 +8.6 +w/ pre-training +Denoising Score Matching +0.849 +0.535 +0.625 +0.851 +0.529 +5.4 +Residue Type Prediction +0.845 +0.527 +0.652 +0.847 +0.518 +5.8 +Distance Prediction +0.825 +0.513 +0.632 +0.836 +0.483 +7.6 +Angle Prediction +0.872 +0.545 +0.637 +0.881 +0.557 +2.0 +Dihedral Prediction +0.852 +0.538 +0.677 +0.881 +0.555 +2.2 +Multiview Contrast +0.848 +0.518 +0.656 +0.833 +0.490 +6.4 +DiffPreT +0.850 +0.540 +0.631 +0.851 +0.542 +4.4 +SiamDiff +0.854 +0.548 +0.673 +0.863 +0.554 +2.2 +side-chain torsional angle and add some noises drawn from wrapped Gaussian distribution during the diffusion process. +Then, we predict the added noises with the extracted atom representations corresponding to the torsional angle. In Table 6, +we can see clear improvements on PIP and PSR tasks compared with GearNet-Edge. This suggests that it would be a +promising direction to explore more different diffusion models for pre-training, e.g., diffusion models on backbone dihedral +angles. +F. Results of Pre-Training GVP +To study the effect of our proposed pre-training methods on different backbone models, we show the pre-training results on +GVP (Jing et al., 2021a;b) in this section. +Setup. GVP constructs atom graphs and adds a vector channels for modeling equivariant features. The original design only +includes atom types as node features, which makes pre-training tasks with residue type prediction very difficult to learn. To +address this issue, we slightly modify its architecture to add the embedding of atom and corresponding residue types as atom +features. Then, the default configurations in Jing et al. (2021b) are adopted. We construct an atom graph for each protein by +drawing edges between atoms closer than 4.5 ˚A. Each edge is featured with a 16-dimensional Gaussian RBF encoding of its +Euclidean distance. We use five GVP layers and hidden representations with 16 vector and 100 scalar channels and use +ReLU and identity for scalar and vector activation functions, respectively. The dropout rate is set as 0.1. The final atom +representations are followed by two mean pooling layers to obtain residue and protein representations, respectively. All +other hyperparameters for pre-training and downstream tasks are the same as those in App. D. +Experimental results. The results are shown in Table 7. Among all pre-training methods, SiamDiff, angle prediction and +dihedral prediction are the top three. This is different from what we observe in Table 1, where residue type and distance +prediction are more competitive baselines. We hypothesize that this is because GearNet-Edge includes angle information in +the encoder while GVP does not. Therefore, GVP will benefit more from pre-training methods with supervision on angles. + +Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction +Nevertheless, we find that SiamDiff is the only method that performs well on different backbone models. + diff --git a/gdFLT4oBgHgl3EQfZC_Z/content/tmp_files/load_file.txt b/gdFLT4oBgHgl3EQfZC_Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2eeae905ded9139e56a24b22b73a81dcce51aa2c --- /dev/null +++ b/gdFLT4oBgHgl3EQfZC_Z/content/tmp_files/load_file.txt @@ -0,0 +1,1934 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf,len=1933 +page_content='Physics-Inspired Protein Encoder Pre-Training via Siamese Sequence-Structure Diffusion Trajectory Prediction Zuobai Zhang * 1 2 Minghao Xu * 1 2 Aur´elie Lozano 3 Vijil Chenthamarakshan 3 Payel Das 3 Jian Tang 1 4 5 Abstract Pre-training methods on proteins are recently gain- ing interest, leveraging either protein sequences or structures, while modeling their joint energy landscape is largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' In this work, in- spired by the success of denoising diffusion mod- els, we propose the DiffPreT approach to pre- train a protein encoder by sequence-structure mul- timodal diffusion modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' DiffPreT guides the encoder to recover the native protein sequences and structures from the perturbed ones along the multimodal diffusion trajectory, which acquires the joint distribution of sequences and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Considering the essential protein conformational variations, we enhance DiffPreT by a physics- inspired method called Siamese Diffusion Trajec- tory Prediction (SiamDiff) to capture the corre- lation between different conformers of a protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' SiamDiff attains this goal by maximizing the mu- tual information between representations of dif- fusion trajectories of structurally-correlated con- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' We study the effectiveness of DiffPreT and SiamDiff on both atom- and residue-level structure-based protein understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Ex- perimental results show that the performance of DiffPreT is consistently competitive on all tasks, and SiamDiff achieves new state-of-the-art perfor- mance, considering the mean ranks on all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' The source code will be released upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Introduction Machine learning based methods have achieved impressive results in predicting protein structures (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Baek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=', 2022) and functionality (Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Gligorijevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Among these meth- ods, various pre-training approaches (Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Equal contribution 1Mila - Qu´ebec AI Institute 2Universit´e de Montr´eal 3IBM Research 4HEC Montr´eal 5CIFAR AI Chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFLT4oBgHgl3EQfZC_Z/content/2301.12068v1.pdf'} +page_content=' Correspondence to: Payel Das 90 b were identified [63] and +compared against the SILVA SSU database 128 (blastn, +min. length 90, min. identity 80%, e value 1e-5). To ver- +ify that the microbial community composition was in- +deed +mostly +prokaryotic, +we +did +a +more +general +screening of the metagenomics reads that identified also +candidate 18S rRNA fragments > 90 b (see Additional +file 1: Tables S4-S5). The complete trimmed read sets +were assembled into contigs ≥ 1 kb with MEGAHIT [64] +(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21, +k max = 131, k step = 10. Genes were predicted using +Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66] +and tRNAscan-SE [67]. Assembled 16S rRNA sequences +were compared to a manually curated version from the +SILVA SSU database (e value ≥ 1e-5). Predicted protein +sequences +were +annotated +against +KEGG +with +GhostKOALA (genus_prokaryotes + family_eukaryotes ++ viruses) [68]. Marker genes for central metabolic +pathways and key environmental element transforma- +tions were identified based on K number assignments +[15, 69–71]. +Contigs ≥ 2.5 kb were binned with METABAT [72] +(superspecific mode) based on differential coverage +Vavourakis et al. Microbiome (2018) 6:168 +Page 13 of 18 + +values obtained by mapping all five trimmed readsets to +all five contig sets with Bowtie2 [73]. The bins were sub- +jected to post-binning (an overview of the workflow is +given in Additional file 2: Figure S13). Bins were +assessed with lineage-specific single copy genes using +CheckM [74] and further processed with the metage- +nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate +Phyla Radiant (CPR) is not included in the CheckM ref- +erence trees and are likely to have low-genome com- +pleteness, we used an existing training file of 797 CPR +genomes to identify putative CPR bins [76]. Bins with +CheckM-completeness ≥ 50% (884 out of 1778) and an +additional four CPR bins were further processed. Coding +sequences +were +annotated +for +taxonomy +against +NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to +verify that most hits in each bin were to prokaryotic ref- +erences. Phage or viral contigs were manually removed. +Genome +contamination (redundancy) +was estimated +based on marker sets of universal single copy genes +identified for Bacteria [30] and Archaea [78] as imple- +mented in Anvi’o. Genome coverage was obtained by +mapping trimmed reads with BBMap [79] v36.x (kfilter +31, subfilter 15, maxindel 80). Bins with ≥ 5% redun- +dancy were further refined with Anvi’o using circle phy- +lograms +(guide +trees +tnf-cov: +euclidian +ward) +and +scanned again for CPR. Post-binning resulted in a total +of 2499 metagenome-assembled genomes (MAGs), of +which 871 were either medium-quality genome drafts +(CheckM estimated completeness ≥ 50% and contamin- +ation ≤ 10% [80], Additional file 4) or lower quality draft +genomes from CPR. +Phylogeny of the MAGs was assessed based on 16 +single-copy ribosomal proteins and representative refer- +ence genomes of major prokaryote lineages across the +tree of life [17]. Individual ribosomal proteins in our +MAGs were identified by K number assignments. Only +ribosomal proteins ≥ 80 aa were considered. Initial +maximum-likelihood (ML) trees were constructed to de- +termine which organisms belonged to the Archaea, Bac- +teria, or CPR with FastTree 2 [81] (WAG + CAT). Final +separate trees for the three distant evolutionary groups +were constructed in the same manner. Each ribosomal +protein set was aligned separately with MAFFT [82] +(v7.055b, − auto) and concatenated only if a MAG +encoded at least 8 out of 16 proteins. For all trees, a +100× posterior bootstraps +analysis +was +performed. +Phylogenetic trees were visualized together with gen- +ome statistics and abundance information using iTOL +[83]. We cross-checked the taxonomic assignments +based on the phylogeny of the ribosomal protein cas- +sette +with +the +top +hit +contig annotations +against +NCBI-nr and with the reference lineage obtained with +CheckM. Lastly, we manually corrected the MAGs for +misplaced 16S rRNA genes. The final trees presented +in the manuscript were redrawn using FigTree v1.4.3 +[84]. +Detailed genome analyses +CPR +MAGs +were +re-annotated +more +thoroughly: +genes were predicted with Prokka [85], and functional +predictions were performed by running InterProScan +5 locally on the supplied COG, CDD, TIGRFAMs, +HAMAP, Pfam, and SMART databases [86]. BLAST +Koala was used for KEGG pathway predictions [68]. +To find putative carbohydrate-active enzymes in all +final MAGs, we used the web-resource dbCAN [87] +to annotate all predicted proteins ≥ 80 aa against +CAZy [88]. +To identify the top ten abundant MAGs from each re- +spective dataset, ten million randomly sampled single- +tons were mapped onto each MAG with a cut-off of 95% +identity in minimum of 50 bases. Coverage values were +additionally normalized for genome size and expressed +as reads per kilobase of sequence per gigabase of +mapped reads (RPKG) [89]. A positive score (from 871 +to 1) was assigned to each MAG according to the rank- +ing of the summed RPKG of MAGs in the high-salinity +datasets (B1Sed10 and T1Sed) and a negative score ac- +cording to the ranking of the summed RPKGs in the +moderate salinity datasets (CSSed10, CSSed11, T3Se +d10). Both scores were summed to get a “salinity prefer- +ence score” with MAGs recruiting preferably from high +salinity datasets on the positive end, moderate salinity +datasets in the negative end, and those without prefer- +ence in the middle. +We determined species delineation for the most +abundant MAGs and their closest reference genomes +(NCBI-nr) by Average Nucleotide Identity (ANI) and +conserved DNA-matrices, as follows [90]: ANI ≥ 95%, +conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA +< 69% = might be same species, ANI < 95%, condDNA +< 69% = different species. Single gene trees based on +maximum +likelihood +were +constructed +with +un- +trimmed alignments (MAFFT, L-INS-i model) and +FastTree 2 (WAG + CAT, increased accuracy, -spr4 +-mlacc 2 -slownni) using 100× bootstraps. References +were pulled from eggNOG (v4.5.1) [91] and supple- +mented +with +sequences +from +NCBI-nr +or +refined +according to [7, 33, 46, 92–94]. The curated MAGs +were +scanned +for +the +presence +of +rhodopsin +sequences with the hmmsearch software [95] and a +profile +hidden +Markov +model +(HMM) +of +the +bacteriorhodopsin-like protein family (Pfam accession +number +PF01036). +The +identified +sequences +with +significant similarity were aligned together with a +curated database composed of a collection of type-1 +rhodopsins, using MAFFT (L-INS-i accuracy model) +[82]. This protein alignment was further utilized to +Vavourakis et al. Microbiome (2018) 6:168 +Page 14 of 18 + +construct a maximum likelihood tree with 100× boot- +strap with FastTree 2 [81]. All other genes were +identified using the KEGG annotation. +Additional files +Additional file 1: Table S1. General features of the four sampled soda +lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA +gene sequences found in all ≥1 kb contigs of five soda sediment +metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide +biosynthesis found among different members of the CPR. Table S4. +Sub-kingdom classification of candidate SSU rRNA gene fragments +found in subsamples of 10 million random forward reads from the +five soda sediment metagenomes. Table S5. Top-level taxonomic +classification of the 18S rRNA gene fragments found in subsamples +of 10 million random forward reads from the five soda sediment +metagenomes. Table S6. Description of the metagenomic datasets, +NCBI Sequence Read Archive (SRA) accession numbers and general +statistics of the assembled contigs. (PDF 740 kb) +Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S +rRNA gene amplicon sequencing. Figure S2. Genome statistics of the +871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus +Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of +MAGs related to “Candidatus Acetothermia”, candidate division WS1 and +“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to +candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6. +Multiple sequence alignment of the V-type ATPase subunits K. Figure S7. +Multiple sequence alignment of the F-type ATPase subunits c. Figure S8. +Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- +like proteins. Figure S9. Maximum likelihood tree of the putative +rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all +MAGs from CPR members. Figure S11. Predicted isoelectric points +profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”. +Figure S12. Multiple sequence alignment of the dissimilatory +cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13. +Overview of the post-binning workflow used for genome recovery. +(PDF 6548 kb) +Additional file 3: Dataset S1. Relative abundance of the OTUs assigned +to the genus-level within the Archaea, Bacteria and organelles from +Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs +with less than 0.1% abundance accross all five datasets are not shown. +The names of highly abundant genera (≥1% in at least one of the data- +sets) are shown in bold. (XLSX 24 kb) +Additional file 4: Dataset S2. Organism names, statistics and general +description incl. Completeness and contamination estimates, phylogeny +and DDBJ/EMBL/Genbank accession numbers of the metagenome +assembled genomes (MAGs) described in this paper. All submitted +versions described in this paper are version XXXX01000000. Size = +recovered genome size, Completeness (Compl1), contamination (Cont), +strain heterogenity (Str het) and Taxon CheckM were inferred from +lineage-specific marker sets and a reference tree build with CheckM [74]. +Additional completeness (compl2) and redundancy (red) estimates were +inferred based on the presence of universal single copy genes for Bacteria +and Archaea [75]. Decision and confidence intervals from the Candidate +Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the +besthit in SILVA when 16S rRNA genes were present. Phylum/class 16 +ribosomal proteins is the taxonomy derived from our ribosomal protein +trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a +population genome with our 16S rRNA gene amplicon dataset +(Additional file 3). (XLSX 253 kb) +Additional file 5: Dataset S3. Estimated abundance and derived +salinity preference from each MAG in each metagenomic dataset +expressed as Reads per Kilobase of MAG per Gigabase of mapped reads +(RPKG) and “salinity preference score” (see Methods section), basis for +Fig. 4. (XLSX 143 kb) +Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and +conserved DNA (condna) matrices to determine species delineation +between the most abundant MAGs shown in Fig. 4, closely related +(less abundant) MAGs and NCBI reference genomes. Decision matrix +shows: 1 = same species, − 1 = might be same species, 0 = different +species (see Methods section). (XLSX 1161 kb) +Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker +genes and putative carbohydrate-active enzymes in the MAGs to infer putative +roles in C, N and S element cycles based on K-number assignments and CAZy +annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb) +Additional file 8: Information S1. More detailed description of the +main metabolisms encoded by Thioalkalivibrio-related MAGs. +Information S2 More detailed description of the main metabolisms +encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb) +Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the +marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The +basis for Fig. 6, namely presence and absence of key genes involved in +the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis +and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows +the MAGs positive for the marker gene cdhC (K00193) encoding for the +beta subunit of an acetyl-CoA decarboxylase synthase complex. While +acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- +type (methanogens) enzymes with the same function, we found few +discrepancies between marker gene and genome phylogeny within the +Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb) +Acknowledgments +We thank Dr. Nikolai Chernych for his technical assistance during the +isolation and purification of metagenomics DNA. We also thank the +Department of Energy Joint Genome Institute for sequencing the +metagenomes. +Funding +CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551). +A-SA and RG were supported by the research grant 17-04828S from the Grant +Agency of the Czech Republic. MM was supported by the Czech Academy of +Sciences (Postdoc program PPPLZ application number L200961651). DYS was +supported by the SIAM/Gravitation Program (Dutch Ministry of Education and +Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- +00121). Sequencing was performed by the U.S. Department of Energy Joint +Genome Institute, a DOE Office of Science User Facility, as part of the Community +Sequencing Program (contract no. DE-AC02- 05CH11231). +Availability of data and materials +The raw sequence reads of the five metagenomes have been deposited to +the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession +numbers and read and contig statistics). The final 871 MAGs described in this +paper have been deposited as Whole Genome Shotgun projects at DDBJ/ +EMBL/GenBank, and accession numbers are listed in Additional file 4 +(BioProject ID PRJNA434545). All versions described in this paper are version +XXXX01000000. The cleaned and dereplicated amplicon sequence datasets +are available in FigShare (https://figshare.com/s/7684627445e3621aba24). +Maximum likelihood trees based on the concatenated alignment of 16 +ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and +complementary datasets (used to plot completeness, contamination, +genome recovery size, G + C mol% and RPKG in iTOL), as well as K number +assignments for the predicted proteins of all MAGs (KEGG-orthologues, +Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions +of this article are also available in FigShare (https://figshare.com/s/ +7684627445e3621aba24). +Authors’ contributions +GM and DYS initiated this study and were responsible for the fieldwork, +sample preparation, and sequencing effort. CDV conceptualized the research +goals under supervision of DYS and GM, and performed the bioinformatics +analysis under close guidance of A-SA and RG. CDV is the primary author of +this manuscript. MM, RG, and CDV prepared the main figures. All authors +read and approved the final manuscript. +Ethics approval and consent to participate +Not applicable. +Vavourakis et al. Microbiome (2018) 6:168 +Page 15 of 18 + +Consent for publication +Not applicable. +Competing interests +The authors declare that they have no competing interests. +Publisher’s Note +Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Author details +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands. 2Department of Aquatic Microbial Ecology, Institute of +Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, +Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of +Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2, +Moscow, Russian Federation117312. 4Environmental Biotechnology, +Department of Biotechnology, Delft University of Technology, Van der +Maasweg 9, 2629, HZ, Delft, the Netherlands. +Received: 23 June 2018 Accepted: 3 September 2018 +References +1. +Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G. +Microbial diversity and biogeochemical cycling in soda lakes. 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Microbiome (2018) 6:168 +Page 18 of 18 + diff --git a/kb_39/content/tmp_files/load_file.txt b/kb_39/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e231246877edeece4465a22a0962746f9ca07a71 --- /dev/null +++ b/kb_39/content/tmp_files/load_file.txt @@ -0,0 +1,1147 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf,len=1146 +page_content='RESEARCH Open Access A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments Charlotte D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Vavourakis1 , Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sorokin3,4 and Gerard Muyzer1* Abstract Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Despite the high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but the microbiome of soda lake sediments received much less attention of microbiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Here, we performed metagenomic sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and a salt content between 70 and 400 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The recovered 16S rRNA gene sequences were mostly from Bacteria, even in the salt-saturated lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Most OTUs were assigned to uncultured families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We reconstructed 871 metagenome-assembled genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla Radiation (CPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Five new species of CPR were among the most dominant community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Novel dominant lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the Actinobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR and several hundred other novel prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The soda lake CPR is a functionally diverse group, but the most abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Second, we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those encompassing known homo-acetogens, sulfate-reducers, and methanogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Since only few environmental metagenomics studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine and freshwater sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl pathway Correspondence: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='Muijzer@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='nl †Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands Full list of author information is available at the end of the article © The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='0 International License (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The Creative Commons Public Domain Dedication waiver (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='org/publicdomain/zero/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='0/) applies to the data made available in this article, unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='1186/s40168-018-0548-7 MicrobiomeBackground Soda lakes are evaporative, athallasic salt lakes with low cal- cium and magnesium concentrations and a high-alkaline pH up to 11 buffered by dissolved (bi-) carbonate ions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' They are constrained to arid regions across the globe, mainly the tropical East African Rift Valley [2], the Libyan Desert [3], the deserts in California and Nevada [4], and the dry steppe belt of Central Asia that spans to southern Si- beria, north-eastern Mongolia, and Inner Mongolia in China [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' On top of the extreme salinity and alkaline pH, the Eurasian soda lakes experience extreme seasonal temperature differences, causing highly unstable water re- gimes and fluctuating salinities [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Yet, soda lakes harbor diverse communities of haloalkaliphilic microbes, mostly prokaryotes that are well adapted to survive and grow in these extreme environments and consist of similar func- tional groups in soda lakes around the world [1, 2, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The relative abundance of different groups is typically governed by the salinity of the brine [1, 7, 8], and microbial-mediated nutrient cycles become partially hampered only at salt-saturating conditions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' So far, all characterized prokaryotic lineages cultured from soda lakes comprise over 70 different species within more than 30 genera [1, 6, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' From these, only a lim- ited number of genomes have been sequenced today, mostly from chemolithoautotrophic sulfur-oxidizing bac- teria belonging to the genus Thioalkalivibrio (class Gam- maproteobacteria) [1, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It is well established that metagenomics enables the recovery of genomes and the identification of novel genetic diversity where culturing ef- forts fail [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In recent years, next-generation sequen- cing has recovered a massive number of genomes from previously unknown groups of prokaryotes [15, 16], including a strikingly large and diverse group called “Candidate Phyla Radiation” (CPR), only distantly related to other cultured bacterial lineages [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Previously, we conducted a metagenomics study on soda lakes and re- constructed novel genomes from uncultured Bacteroidetes and “Candidatus Nanohaloarchaeaota” living in hypersa- line Siberian soda brines [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Here, we turned our atten- tion to the far more complex prokaryotic communities living in the sediments of the hypersaline soda lakes from the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We give a broad overview of all the taxonomic groups sequenced and focus on the metabolic diversity found in the reconstructed genomes of the most abundant, uncultured organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Results Overall prokaryote community structure The salinities from the studied soda lakes ranged from moderately hypersaline (between 70 and 110 g L−1) to salt-saturated (400 g L−1 salt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The soluble carbonate al- kalinity was in the molar range, and the pH in all lakes was around ten (see Additional file 1: Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' To give an overview of the overall prokaryotic community com- position in each of the samples, we looked at the taxo- nomic classification of 16S rRNA genes recovered both by amplicon sequencing and direct metagenomics se- quencing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 1, see also Additional file 2: Figure S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The prokaryotic communities of all five sediment samples were highly diverse and consisted mostly of uncultured taxonomic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Bacteria were more abundant than Archaea, regardless of the salinity of the overlaying brine [7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Euryarchaeota were the second and third largest group in the sediments of the two salt-saturated lakes comprising ~ 10 and ~ 20% of the 16S rRNA genes in the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Most Euryarchaeota-related OTUs detected by amplicon se- quencing belonged either to the uncultured Thermoplas- mata group KTK 4A (SILVA classification) or the genera Halohasta and Halorubrum (class Halobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In ac- cordance with cultivation-dependent studies [6], most OTUs assigned to methanogens were from the class Methanomicrobia, especially the lithotrophic genus Methanocalculus (up to ~ 3%) and the methylotrophic genus Methanosalsum (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The varying ratio of the three dominant bacterial groups, Firmicutes, Bacteroidetes (including the newly proposed phyla Rhodothermaeota and Balneolaeota [18]), and Gammaproteobacteria, showed no clear trend in relation to the salinity in the lakes, but when Firmicutes were domin- ant, Bacteroidetes were less abundant and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Most Firmicutes belonged to the order Clostridales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Uncultured members from the family Syntrophomonadaceae had a relative abundance of more than 5% in all five metagen- omes and comprised in two lakes even ~ 11–20% of the recovered amplicon sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The second most abundant Firmicutes order was Halanaerobiales, particularly the genus Halanaerobium (family Halanaerobiaceae) and un- cultured members of the Halobacteroidaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The majority of Bacteroidetes-related OTUs could not be assigned down to the genus level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The uncultured ML635J-40 aquatic group (order Bacteroidales) comprised at least 5% of all five prokaryotic communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This group has been previously found to be abundant in Mono Lake [4] (a soda lake) and in an anoxic bioreactor degrading cyanobacterial biomass under haloalkaline conditions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Two other highly abun- dant (up to ~ 8%) uncultured groups from the class Balneo- lia (proposed new phylum Balneolaeota [18]) were also detected in other soda lakes before [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Within the Gam- maproteobacteria, the genus Thioalkalivibrio was abundant (~ 3% of the total community), but also uncultured members of HOC36 were prevailing at moderate salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Members of the Deltaproteobacteria, Alphaproteobacteria, and Chloroflexi comprised up to ~ 10% of the detected 16S rRNA gene in some of the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The GIF9 family of the class Dehalococcoidia was among the top three most abundant OTUs in two lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The extremely salt-tolerant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 2 of 18 and alkaliphilic genera Desulfonatronobacter (order Desulfo- bacterales) and Desulfonatronospira (order Desulfovibrio- nales) were the dominant Deltaproteobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Highly abundant OTUs, within the Actinobacteria belonged to the class Nitriliruptoria and within the Alphaproteobacteria to the family Rhodobacteraceae and the genus Roseibaca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The important nitrifying genus Nitrobacter (Alphaproteobacteria) was present in only one of the lakes with moderate salinity (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Some bacterial top-level taxa appeared less dominant (< 5%) from the 16S rRNA genes recovered from the metagenomes but were represented mainly by a single highly abundant OTU in the amplicon sequences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' in- cluding the haloalkaliphilic genus Truepera within the phylum Deinococcus-Thermus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' the genus Spirochaeata within the phylum Spirochaetes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' the family BSN166 within the phylum Ignavibacteriae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' the BD2–11 terres- trial group within the Gemmatimonadetes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' and the WCHB1–41 order within the Verrucomicrobia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All OTUs within the Thermotogae and Lentisphaerae belonged to uncultured genera from the family Kosmoto- gaceae and Oligosphaeraceae, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All Tenericu- tes-related OTUs belonged to the class Mollicutes, and especially the order NB1-n was dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In contrast, the phylum Planctomycetes was relatively diverse, with at least 11 different genus-level OTUs spread over four class-level groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' High-throughput genome recovery We obtained 717 medium-quality (≥ 50% complete, < 10% contamination) and 154 near-complete (≥ 90% complete, < 5% contamination) metagenome-assembled genomes (MAGs) across three major prokaryote groups: Archaea, Bacteria, and CPR (see Additional file 4 and Additional file 2: Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figures 2 and 3 show the top-level phylogeny of all MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The reference database used contains a repre- sentative for each major prokaryote lineage [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 1 Abundant prokaryotic groups in five hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' a Relative abundance of the top-level taxa (those with ≥ 1% abundance in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' b Relative abundance of the 16S rRNA OTUs (those with sum of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Three of the assessed soda lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 3 of 18 colored the different phyla from which we obtained a MAG in alternate blue and orange colors, and highlighted the MAGs obtained here in a darker shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Many MAGs belonged to uncultured groups commonly detected in soda lake 16S rRNA gene surveys, over 100 MAGs still belonged to candidate prokaryote phyla and divisions that to our knowledge were never detected be- fore in soda lakes, including CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Although only few MAGs had near-complete 16S rRNA genes, in most cases we were able to link available taxonomic gene an- notations and ribosomal protein phylogeny to the SILVA taxonomy of the OTUs assigned to the amplicon se- quences, while cross-checking the abundance profiles of both MAGs (Additional file 5) and OTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The soda lake CPR recovered from the metagenomes was restricted to a few distinct phyla within the Parcubacteria group, mostly affiliating with “Candidatus Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Zambryskibacteria” [15] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The first group of MAGs encompassed four different branches in our riboso- mal protein tree, suggesting a high-phylogenetic diversity, with 33 putative new species sampled here (ANI and con- DNA matrices given in Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Zambrys- kibacteria-”related MAGs consisted of at least five new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Few MAGs were recovered from CPR groups also detected by amplicon sequencing (see Additional file 2: Figure S1), namely the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Dojkabacteria” (former WS6), “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Saccharibacteria” (former TM7), CPR2, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Katanobacteria” (former WWE3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The archaeal tree is unrooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The CPR tree is rooted to the Wirthbacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 4 of 18 Most archaeal MAGs belonged to the phylum Euryarch- aeota and the abundant classes Halobacteria, Methanomi- crobia, and Thermoplasmata (related to OTU KTK 4A) within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In addition, three Thermoplasmata-related MAGs that encoded for the key enzyme for methanogenesis (methyl-coenzyme M reductase, mcr) affiliated with refer- ence genomes from Methanomassilicoccales, the seventh order of methanogens have been recovered [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Another MCR-encoding MAG was closely related to the latest discovered group of poly-extremophilic, methyl-reducing methanogens from hypersaline lakes from the class Methanonatronarchaeia [9] (related to OTU ST-12K10A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We recovered also one MAG from the class Methanobacteria and a high-quality MAG from the WCHA1–57 group (“Candidatus Methanofastidiosa” [22]) in the candidate division WSA2 (Arc I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Several MAGs were recovered from the DPANN archaeal groups “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Diapherotrites,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Aenigmarchaeota,” (see Additional file 2: Figure S3) and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Woesearch- aeota” (former Deep Sea Hydrothermal Vent Group 6, DHVEG-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Although we did not reconstruct any reasonable-sized MAGs from the TACK superphylum, we found several 16S rRNA genes on the assembled contigs that affiliated to the Thaumarchaeota (see Additional file 1: Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nearly every known bacterial phylum had an extremo- philic lineage sampled from our hypersaline soda lake sediments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In most cases, the soda lake lineages clearly formed separate branches appearing as sister groups to known reference lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The highest genome recovery was from the same top-level taxonomic groups that were also abundant in our 16S rRNA gene analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' From the Verrucomicrobia, most MAGs belonged to the order WCHB1-41 (16S rRNA gene identity 92–100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' However, in our ribosomal protein tree, they branched within the phylum Lentisphaerae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sixteen Tenericutes MAGs from at least 12 different species (Additional file 6) were closely related to the NB1-n group of Mollicutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Based on the recovered genome size and encoded meta- bolic potential, these organisms are free-living anaerobic fermenters of simple sugars, similar to what has recently been proposed for “Candidatus Izimaplasma” [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Alternate orange and blue colors show phyla/ classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 5 of 18 Several MAGs belonged to the candidate phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Omnitrophica,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Atribacteria,” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Acetother- mia” (former OP1), which were moderately abundant also in some sediment (see Additional file 2: Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' For the latter phylum, we suspect that four MAGs were more closely related to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' WS1 and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Lindow- bacteria” for which only few reference genomes are currently available in NCBI (see Additional file 2: Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Due to a high-sequencing coverage, we also managed to reconstruct several MAGs from rare Bacteria (< 100 amplicon sequences detected, see Additional file 2: Figure S1), including the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Hydrogenedentes,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Cloacimonetes,” ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' BRC1, Elusimicrobia, Caldi- serica, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Latescibacteria.” The MAGs from the latter phylum were more closely related to the recently proposed phylum “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Handelsmanbacteria” [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Two additional MAGs with 16S rRNA gene fragments with 94–95% identity to the class MD2898-B26 (Nitrospinae) were more likely members of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' KSB3 (proposed “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Moduliflexus” [24], see Additional file 2: Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Draft genomes of haloalkaliphilic CPR Strikingly, members of the CPR related to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nealson- bacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Vogelbacteria” were among the top 5% of abundant organisms in the surface sediments of the soda lakes, especially those with moderate salinity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Like most members of the CPR, the MAGs of the four most abundant “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nealsonbacteria” seem to be anaerobic fermenters [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' They lacked a complete TCA cycle and most complexes from the oxidative elec- tron transfer chain, except for the subunit F of a NADH-quinone oxidoreductase (complex I, nuoF, nuoG, nuoA) and coxB genes (complex II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All CPR MAGs had a near-complete glycolysis pathway (Embden-Meyerhof- Parnas) encoded, but pentose phosphate pathways were severely truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The commonly encoded F- and V-type ATPase can establish a membrane potential for symporter-antiporters by utilizing the ATP formed by substrate-level phosphorylation during fermentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All CPR have V-type ATPases that can translocate Na+ in addition to H+ (see Additional file 2: Figure S6), while only two members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Falkowbacteria” had puta- tive Na+-coupled F-type ATPases (see Additional file 2: Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The coupling of ATP hydrolysis to sodium translocation is advantageous to maintain pH homeosta- sis in alkaline environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Interestingly, with only two exceptions [26, 27], all CPR genomes recovered from other environments with neutral pH were reported to encode only F-type ATPases [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' One low-abundant MAG affiliated to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Peregrinibacteria” contained both the large subunit of a RuBisCO (type II/III, see Additional file 2: Figure S8) and a putative phosphoribu- lokinase (PRK, K00855) encoded in the same contig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This is remarkable because PRK homologs were not previously identified among CPR, and RuBisCo form II/ III was inferred to function in a nucleoside salvage path- way [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' One “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Saccharibacteria” MAG encoded for a putative channelrhodopsin (see Additional file 2: Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This is the first rhodopsin found among the CPR and suggests that this enigmatic group of organ- isms may have acquired evolutionary adaptations to a life in sunlit surface environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' A previous study showed that most CPR has coccoid cell morphotypes with a monoderm cell envelope resem- bling those from Gram-positives and Archaea but with a distinct S-layer [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Thick peptidoglycans coated with acidic surface polymers such as teichoic acids help pro- tect the cells of Gram-positives against reactive hydroxyl ions in highly alkaline environments [35] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All soda lake CPR had indeed the capability for peptidogly- can biosynthesis, but we found proteins typical for Gram-negatives for the biosynthesis of lipopolysaccha- rides (see Additional file 1: Table S3), homologous to the inner membrane proteins of type II secretion systems and to several proteins associated to the outer membrane and peptidoglycan, including OmpA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It remains to be determined whether the soda lake CPR also lacks an outer membrane and perhaps anchor lipopolysaccharides, S-layer proteins, and lipoproteins to the inner cell membrane or peptidoglycan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We also found gene encoding cardiolipin and squalene synthases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Increased levels of cardiolipin and the presence of squa- lene make the cytoplasmic membrane less leaky for protons [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In addition, cation/proton exchangers are known to play a crucial role for pH homeostasis in alka- liphilic prokaryotes as they help acidify the cytoplasm during the extrusion of cations [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Putative Na+/H+ exchangers of the Nha-type and multi-subunit Mnh-type were found only within a few soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Secondary active transport of K+ might be mediated in most soda lake CPR by KefB (COG0475)/kch Kef-type, glutathione- dependent K+ transport systems, with or without H+ antiport (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Various soda lake CPR had an acidic proteome, with pI curves resembling those found in extremely halophilic Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Intracellular proteins enriched in acidic amino acids might be an adaptation to a “salt-in” strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=', maintaining high intracellular potassium (K+) concentra- tions to keep osmotic balance [7, 37] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 5b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' see Additional file 2: Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Such a strategy is energet- ically favorable over de novo synthesis or import of osmolytes such as ectoine and glycine betaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We did not find genes for the synthesis of organic osmolytes and homologs of ABC-type transporters for primary active uptake of proline/glycine betaine which were encoded only in one MAG (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' For the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nealsonbac- teria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Vogelbacteria,” the salt-in strategy might be a unique feature for the soda lake species explaining Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 6 of 18 their high abundance in the hypersaline soda lake sedi- ments, as we did not found an acidic proteome pre- dicted from genomes obtained from other non-saline environments (See Additional file 2: Figure S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The uptake of K+ ions remains enigmatic for most soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Low-affinity Trk-type K+ uptake transporters (gen- erally with symport of H+) (67,68) were encoded only by a limited number of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We found three MAGs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4 Relative abundance and metabolic potential of the dominant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Abundance values, expressed as reads per kilobase of MAG per gigabase of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets (cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The metabolic potential derived from functional marker genes (Additional file 7) is depicted by the colored symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' = fixation, red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' = reduction, ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' = oxidation, dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' = disproportionation Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 7 of 18 a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 5 (See legend on next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=') Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 8 of 18 encoding for Kdp-type sensor kinases (kdpD) but no corresponding genes for the response regulator (kdpE) or for Kdp-ATPases that function as the inducible, high- affinity K+ transporters in other Bacteria (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Finally, mechanosensitive ion channels (mscS, mscL) and ABC- type multidrug transport systems (AcrAB, ccmA, EmrA, MdlB, NorM) and sodium efflux permeases (NatB) were encoded in almost every MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The first might rapidly restore the turgor pressure under fluctuating salinity conditions by releasing cytoplasmic ions [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Novel abundant groups involved in sulfur, nitrogen, and carbon cycles A new species of Thioalkalivibrio (family Ectothiorhodospir- aceae) was by far the most abundant in the sediments of the two salt-saturated lakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In the sediment of Bitter-1, also a purple sulfur bacterium from the same fam- ily was highly abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It was closely related to Halorho- dospira, a genus also frequently cultured from hypersaline soda lakes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' None of the abundant Ectothiorhodospira- ceae spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' had already a species-representative genome sequenced (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The potential of the Thioalk- alivibrio spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' for chemolithotrophic sulfur oxidation was evident (Additional file 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' see Additional file 8: Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Interestingly, the encoded nitrogen metabolisms were quite versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' While Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 1 had the poten- tial for nitrate reduction to nitrite, Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2 might perform dissimilatory nitrite reduction to ammonia (DNRA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' see Additional file 2: Figure S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Two deltaproteobacterial lineages of dissimilatory sulfate-reducing bacteria (SRB) were highly abundant in the soda lake sediment of Bitter-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' One MAG from the family Desulfobacteraceae (order Desulfobacterales) is the first genome from the genus Desulfonatronobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It encodes the genes for complete sulfate reduction to sul- fide using various electron donors, as well as for the complete oxidation of volatile fatty acids and alcohols, a unique feature for the genus Desulfonatronobacter among haloalkaliphilic SRB [10] (see Additional file 8: Information S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Fumarate and nitrite (DNRA, NrfAH) could be used as alternative electron acceptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The sec- ond dominant lineage was a new species from the genus Desulfonatronospira (family Desulfohalobiaceae, order Desulfovibrionales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Like other members of this genus, it had the potential to reduce or disproportionate partially reduced sulfur compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In addition, it could also use nitrite as an alternative electron acceptor (NrfAH) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' A novel lineage of gammaproteobacterial SOB was highly abundant in the sediments of the moderately hy- persaline Cock Soda Lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It appeared as a sister group of the family Xanthomonadaceae in the ribosomal protein tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This heterotrophic organism could conserve energy through aerobic respiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It might detoxify sulfide by oxidizing it to elemental sulfur (sqr) with subsequent re- duction or disproportionation of the polysulfides (psrA) chemically formed from the sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It also encoded the po- tential for DNRA (nrfA and napC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Genes likely involved in sulfide detoxification (sqr and psrA) were found also in several other abundant MAGs of heterotrophs, including one new abundant species from the family of Nitrilirup- toraceae (class Nitriliruptoria, phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We found a wide variety of carbohydrate-active enzymes in these MAGs, such as cellulases (GH1 family) in addition to genes for glycolysis and TCA cycle and a chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The latter was also found in other Actinobacteria from the genus Rubrobacter [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' No evidence was found for nitrile-degrading potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' A second novel, uncultured lineage of Gammaproteo- bacteria that was highly abundant at moderate salinities branched in our ribosomal protein tree as a sister group to the family Halothiobacillaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The MAGs encoded for a versatile metabolism typical for purple non-sulfur bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The MAGs contained puf genes, bch genes, genes for carotenoid biosynthesis (not shown), and a Calvin cycle for photoautotrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Alternatively, energy may be conserved through aerobic respiration, while acetate and proprionate could be taken up via an acetate permease (actP) and further used for acetyl-CoA biosynthesis and carbon assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Since the sqr gene was present, but no dsr or sox genes, the organism might oxidize sulfide only to elemental sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' One bin contained also nifDKH genes suggesting putative diazo- trophy, as well as a coenzyme F420 hydrogenase suggest- ing photoproduction of hydrogen [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The abundant Euryarchaeota organism showed a clear preference for higher salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We obtained one highly abundant MAG from the class Thermoplasmata that encoded a full-length 16S rRNA gene only distantly re- lated (91,2% identity, e value 0) to that of a member of the KTK 4A group found in a hypersaline endoevaporitic microbial mat [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The abundant soda lake organism is likely a new genus and species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All KTK 4A-related MAGs found here encoded for similar heterotrophic, fermentative metabolisms, with the potential for (See figure on previous page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' a Membrane transporters, channels, and lipids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Peptidoglycan is depicted in gray and S-layer proteins in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' b Predicted isoelectric points (bin width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='2) for the coding sequences of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also Additional file 2: Figure S11) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 9 of 18 anaerobic formate and CO oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The KTK 4A might be also primary degraders since they encoded pu- tative cellulases (CAZY-families GH1, GH5) and chiti- nases (GH18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Interestingly, half of the MAGs encoded a putative chlorophyll/bacteriochlorophyll a/b synthase (bchG), which is highly unusual for Archaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Although little can be inferred from the presence of only one marker gene, a functional bchG was previously also found in Crenarchaeota [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The remaining two highly abundant Euryarchaeota-related MAGs belonged to a new species of Halorubrum (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Key genes of the Wood-Ljungdahl pathway found in novel phylogenetic groups More than 50 MAGs were related to the family Syntro- phomonadaceae (class Clostridia, phylum Firmicutes) based on ribosomal protein phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All 16S rRNA gene sequences found in the MAGS had 86–95% iden- tity to sequences obtained from uncultured organisms related to the genus Dethiobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' While an isolated strain of Dethiobacter alkaliphilus is a facultative auto- troph that respires thiosulfate, elemental sulfur or polysulfides with hydrogen as an electron donor [42] or disproportionates sulfur [43], other haloalkaliphilic members of the Syntrophomonadaceae are reverse acetogens, oxidizing acetate in syntrophy with a hydro- genotrophic partner [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Two populations (different species, Additional file 6) were especially abundant in Cock Soda Lake (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' They encoded for a full CODH/ACS complex, the key enzyme for the reductive acetyl-CoA or Wood-Ljungdahl pathway (WL) and a complete Eastern branch for CO2 conversion to 5-methyl-tetrahydrofolate (Additional file 9) [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Acetogens use the WL to reduce CO2 to acetyl-CoA, which can be fixed into the cell or used to conserve en- ergy via acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Syntrophic acetate oxidizers, some sulfate reducing bacteria and aceticlastic methanogens run the WL in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Syntrophomonadaceae sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2 encoded for a putative thiosulfate/polysulfide reductase as well as a phosphotransacetylase (pta) and an acetate kinase (ack) for the ATP-dependent conversion of acet- ate to acetyl-CoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Although alternative pathways for the latter interconversion can exist, this second species has the complete potential for (reversed) acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Highly remarkable was the presence of a bacterial-type CODH/ACS complex and a near-complete eastern branch of the WL in a highly abundant species in Cock Soda Lake from the family Coriobacteriaceae (phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This prompted us to scan all 871 MAGs for the presence of acsB encoding for the beta-subunit of the oxido-reductase module of CODH/ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We con- firmed an encoded (near)-complete WL in several additional organisms belonging to phylogenetic groups not previously associated with this pathway [46] (Additional file 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We removed the Coriobacteriaceae acsB genes from the final dataset to construct a phylo- genetic tree since they were < 500 aa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 6) but found seven MAGs from the OPB41 class within the Actino- bacteria (16S rRNA gene fragment identity 94–96%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The eastern branch of WL can function independently in folate-dependent C1 metabolism [45], but the pres- ence of the Western-branch in a phylum that comprises mostly aerobic isolates is very surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The WL in combination with the potential for acetate to acetyl-CoA interconversion (pta/ack) and a glycolysis pathway were also present in the soda lake MAGs from the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Handelsmanbacteria,” “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Atribacteria” (latter branched within the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Acetothermia”), and the class LD1-PA32 (Chlamydiae), suggesting all these uncultured organisms might be heterotrophic acetogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' However, it should be noted that a PFOR typically connecting glycolysis to the WL was only encoded in the LD1-PA32 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' More- over, from the genetic make-up alone, it cannot be excluded that acetate is activated, and the WL run in reverse for syntrophic acetate oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Finally, the novel acsB genes from soda lake Halanaerobiaceae, Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) and from Brocadiaceae and Planctomycetaceae (Plancto- mycetes) disrupt the previously proposed dichotomy between Terrabacteria and Gracilicutes bacterial groups unifying 16S rRNA and acsB gene phylogenies [46] and suggest a far more complex evolutionary history of the WL pathway than previously anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Discussion Extensive classical microbiology efforts have been already undertaken to explore the unique extremophilic microbial communities inhabiting soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' These un- covered the presence of most of the functional groups participating in carbon, nitrogen, sulfur, and minor element cycling at haloalkaline conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The main re- sults of this work are summarized in several recent re- views [1, 6, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Since most microbes, including those living in soda lakes, still evade all cultivation ef- forts, a very effective way to discover new microbes and assess their physiology based on their genetic repertoire is either through single cell genomics or by directly se- quenced environmental DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This exploratory metage- nomics study performed on soda lake sediments effectively overcame the existing cultivation bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' First, we expanded the known diversity of CPR consider- ably with the first genomes of poly-extremophiles sam- pled from soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Although the presence of 16S rRNA genes from CPR in marine sediments and hy- persaline microbial mats was previously shown [34], until now, CPR MAGs were mainly obtained from deep, subsurface environments [15, 26, 29, 32, 49–52], and hu- man microbiota [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Despite being highly abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 10 of 18 100 % 90-100 % 70-90 % 50-70 % some MAGs all MAGs Bootstraps Genes present Glycolysis (EMP) PFOR WL-Eastern branch H4MPT TH4 WL-Western branch CODH/ACS Acetogenesis/ acetate activation (pta/ack) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='4 PVC group (Chlamydiae LD1-PA32) Syntrophorhabdus aromaticivorans PVC group bacterium CSSed11_184 Aerophobetes bacterium SCGC_AAA255-F10 Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Acetothermia Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Handelsmanbacteria Planctomycetaceae Anaerolineae Firmicutes Brocadiaceae Planctomycetes Methanomassiliicoccales Halobacteroidaceae Natranaerobiaceae Methanomicrobiales Desulfonatronospira Firmicutes Dehalococcoidia Armatimonadetes bacterium CSP1-3 Deltaproteobacteria Thermodesulfobacteria Desulfobulbaceae Halanaerobiaceae Nitrospirae Actinobacteria (OPB41) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Only sequences ≥ 500 aa were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see also Additional file 9: Dataset S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Additional lineages found in this study are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The three was rooted according to [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Circles at the nodes show confidence percentage of the bootstraps analysis (100×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' PVC group bacterium CSSed11_184 is likely a member of the WCHB1-41 class within the Verrucomicrobia Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 11 of 18 here, CPR went unnoticed in previous amplicon sequen- cing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This might be due to the fact that many CPR representatives have random inserts of various length in their 16S rRNA genes or due to primer mis- matches [29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This illustrates also that direct metage- nomics should not only be preferred over amplicon sequencing to infer functional potential, but the former is far more effective for the discovery of novel organ- isms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Second, we obtained many more genomes from “traditional” bacterial phyla such as the Planctomycetes and Chloroflexi, as well as candidate phyla, for which no soda lake isolates, hence no genomes were previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Third, even within the sulfur cycle, the most active and frequently studied element cycle in soda lakes [1], we found considerable metabolic novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Finally, we found the Wood-Ljungdahl pathway in several novel phyla, not closely related to any known acetogens, methanogens, or sulfate-reducing bacteria [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The lat- ter shows that our sequencing recovery effort has also significantly contributed to the discovery of metabolic novelty within various prokaryote phylogenetic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Salinity is often considered to be the major factor shaping prokaryote community composition in diverse habitats [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Extreme halophilic Euryarchaeota seem to be always the dominant group in salt-saturated hypersaline brines, both those with neutral or alkaline pH [1, 7, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Here, we found that although these haloarchaea are still relatively more abundant in the sed- iments exposed to brines with salt-saturating conditions, the clear majority of microbes in all investigated hyper- saline soda lake sediments are Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' It could be hypothesized that the sediment is a hide-out for the extreme alkalinity and salinity governing the water column, and that sediment stratification, especially in the anoxic part, offers plenty of opportunities for niche diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' On the other hand, it should no longer be a surprise that soda lakes are such productive and biodiverse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' First, it has been previously elaborated that soda lake organisms are exposed to approximately half the osmotic pressure in sodium carbonate-dominated brines compared to sodium chloride-dominated brines with the same Na+ molarity [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Second, nitrogen limitation in the community can be overcome when many members contribute to the fixation of atmospheric N2, and various forms of organic nitrogen are efficiently recycled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The soda lakes exam- ined in this study were also eutrophic, and sulfur com- pounds were abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sulfide is also far less toxic at high pH as it mostly occurs in the form of bisulfide (HS−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Besides the evident high metabolic and taxo- nomic diversity of dissimilatory sulfur-cycling bacteria, a diverse heterotrophic community can be sustained com- prising both generalist and very specialized carbon de- graders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Less eutrophic soda lakes might not suffer from carbon limitation either, due to a presence of high-bicarbonate concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' These effectively elim- inate the inorganic carbon limitation for primary pro- ducers who are highly active in soda lakes, especially Cyanobacteria [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Third, light that penetrates the surface of the sediment seems to stimulate oxygenic and anoxygenic phototrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Moreover, various het- erotrophs, such as the rhodopsin-containing haloarchaea and Bacteroidetes, have the option to tap into this un- limited energy source for example to help sustain the costly maintenance of osmotic balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Unexpectedly, we even found the first rhodopsin encoded by a member of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Fourth, tight syntrophic relations, as pro- posed for CPR members and Syntrophomonadaceae spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=', might be the solution to successful growth in an energetically challenging environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Since our metagenomes are snapshots in time and space, the failure to reconstruct specific MAGs gives no conclu- sive evidence for the absence of certain microbial-mediated element transformation in hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Additionally, technical limitations of the assembly and bin- ning of highly micro-diverse genome populations might hamper genome recovery [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' More importantly, the abundance of a specific microbe is not necessarily corre- lated to the importance of its performance in an ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Many metabolic capacities are redundant, and often key transformations are reserved for a few rare organisms that might proliferate for a short time-span when specific condi- tions allow for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' For example, although no MAGs were re- covered from chemolithoautotrophic nitrifiers [58], we did detect a Nitrobacter-related OTU by amplicon sequencing and assembled 16S rRNA genes from Thaumarchaeota, suggesting bacterial and archaeal nitrifiers are present in the surface sediments of soda lakes at very low abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Finally, the method of DNA isolation might impact the community composition apparent in the final metagenome sequenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Environmental samples contain complex mix- tures of different organisms, and it is impossible to find a protocol where the DNA from every single organism is ex- tracted as efficiently without compromising the final quality of the extracted DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' However, since we find all the im- portant taxonomic and functional groups known from pre- vious cultivation-dependent studies back in either our amplicon sequencing datasets or our directly sequenced metagenomes, we are confident that the community com- position and the MAGs presented here are representative for the microbiomes of the soda lake sediments in the Kulunda Steppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Conclusion Years of intensive microbiological research on soda lakes seem to have paid off, since many of the described gen- era we could detect here have a cultured representative from soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' However, as many of the abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 12 of 18 lineages and groups found in soda lake sediments are still uncultured, metagenomics proved to be a helpful tool to gain primary insights in the potential physiology and ecology of these poly-extremophilic prokaryotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We reconstructed the first genomes for many of such organisms and proposed new functional roles for the most abundant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Future studies should provide more in depth analyses of these genomes, especially from the less abundant organisms that might perform key ecological processes, such as methanogens and nitri- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In addition, they should focus on gaining physio- logical culture-based evidence or proof for in situ activity for the abundant organisms described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The key metabolic insights provided by this metagenomics study can lead to the design of new cultivation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' In general, sediment communities are far more complex than those found in the corresponding water column [53, 59] and are therefore often considered too complex for efficient metagenomic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Many of the novel lineages found here may therefore have related neutro- philic lineages in marine and freshwater sediments that await discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We demonstrate here that, by providing sufficient sequencing depth, the “state of the art metage- nomics toolbox” can effectively be used on sediments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Methods Site description and sample collection The top 10 cm sediments from four hypersaline, eutrophic soda lakes located in the Kulunda Steppe (south-western Siberia, Altai, Russia) were sampled in July of 2010 and 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' General features and exact location of the sampled soda lakes are summarized in Additional file 1: Table S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' a map of the area was published previously [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Cock Soda Lake (a stand-alone lake, sampled both in 2010 and 2011) and Tanatar-3 (Tanatar system) were moderately hypersa- line (~ 100 g L−1) with sandy sediment, while Tanatar-1 and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) with sulfide-rich sapropel sediments underlined by rock trona deposits [7, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Especially, Bitter-1 harbors a very active microbial community, probably due to its high- organic and -mineral content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Surface sediments were col- lected by a plastic corer into sterile glass containers and transported to the laboratory in a cooler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' DNA isolation, 16S rRNA amplicon, and metagenomic sequencing The colloidal fraction of each sediment sample (~ 10% of 50 g) was separated from the course sandy fraction by several short (30–60 s) low-speed (1–2,000 rpm in 50 mL Falcon tubes) centrifugation steps and washed with 1–2 M NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The pelleted colloidal sedi- ment fraction was first subjected to 3 cycles of freezing in liquid nitrogen/thawing, then re-suspended in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='1 M Tris (pH 8)/10 mM EDTA, and then subjected to harsh bead beating treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Next, the samples were incu- bated with lysozyme (15 mg/mL) for 2 h at 37 °C followed by a SDS (10% w/v) and proteinase K (10 μg/ mL) treatment for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' at 45 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' High molecular weight DNA was isolated using phenol/chloroform ex- traction, quality-checked, and sequenced as described previously [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Direct high-throughput sequencing of the DNA was performed on an Illumina HiSeq 2000 plat- form to generate 150 b paired-end reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Amplification of the V4-V6 region of prokaryote 16S rRNA genes using barcoded 926F-1392R primers, amplicon purifica- tion, quantification, and Roche (454)-sequencing was performed together in a batch with brine samples from the same sampling campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Barcodes and adapter se- quences were removed from de-multiplexed amplicon sequence reads and analyzed with the automated NGS analysis pipeline of the SILVA rRNA gene database pro- ject [61] (SILVAngs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='3, database release version 128) using default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The OTUs (97% identity) assigned down to the genus level were only considered when they had a relative abundance ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='1% in at least one of the five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Processing metagenomics reads, assembly, binning, and post-binning Metagenomic raw reads were quality trimmed using Sickle [62] (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='33), and only reads ≥ 21 b were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The prokaryotic community structure at taxo- nomic top levels was extrapolated from ten million ran- domly sampled singletons from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Candidate 16S rRNA fragments > 90 b were identified [63] and compared against the SILVA SSU database 128 (blastn, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' length 90, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' identity 80%, e value 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' To ver- ify that the microbial community composition was in- deed mostly prokaryotic, we did a more general screening of the metagenomics reads that identified also candidate 18S rRNA fragments > 90 b (see Additional file 1: Tables S4-S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The complete trimmed read sets were assembled into contigs ≥ 1 kb with MEGAHIT [64] (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='3–6-gc3983f9) using paired-end mode, k min = 21, k max = 131, k step = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Genes were predicted using Prodigal [65] (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='2) and RNAs with rna_hmm3 [66] and tRNAscan-SE [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Assembled 16S rRNA sequences were compared to a manually curated version from the SILVA SSU database (e value ≥ 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Predicted protein sequences were annotated against KEGG with GhostKOALA (genus_prokaryotes + family_eukaryotes + viruses) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Marker genes for central metabolic pathways and key environmental element transforma- tions were identified based on K number assignments [15, 69–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Contigs ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='5 kb were binned with METABAT [72] (superspecific mode) based on differential coverage Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 13 of 18 values obtained by mapping all five trimmed readsets to all five contig sets with Bowtie2 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The bins were sub- jected to post-binning (an overview of the workflow is given in Additional file 2: Figure S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Bins were assessed with lineage-specific single copy genes using CheckM [74] and further processed with the metage- nomics workflow in Anvi’o [75] (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Since Candidate Phyla Radiant (CPR) is not included in the CheckM ref- erence trees and are likely to have low-genome com- pleteness, we used an existing training file of 797 CPR genomes to identify putative CPR bins [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Bins with CheckM-completeness ≥ 50% (884 out of 1778) and an additional four CPR bins were further processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Coding sequences were annotated for taxonomy against NCBI-nr (July, 2017) with USEARCH [77] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='32) to verify that most hits in each bin were to prokaryotic ref- erences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phage or viral contigs were manually removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Genome contamination (redundancy) was estimated based on marker sets of universal single copy genes identified for Bacteria [30] and Archaea [78] as imple- mented in Anvi’o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Genome coverage was obtained by mapping trimmed reads with BBMap [79] v36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='x (kfilter 31, subfilter 15, maxindel 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Bins with ≥ 5% redun- dancy were further refined with Anvi’o using circle phy- lograms (guide trees tnf-cov: euclidian ward) and scanned again for CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Post-binning resulted in a total of 2499 metagenome-assembled genomes (MAGs), of which 871 were either medium-quality genome drafts (CheckM estimated completeness ≥ 50% and contamin- ation ≤ 10% [80], Additional file 4) or lower quality draft genomes from CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phylogeny of the MAGs was assessed based on 16 single-copy ribosomal proteins and representative refer- ence genomes of major prokaryote lineages across the tree of life [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Individual ribosomal proteins in our MAGs were identified by K number assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Only ribosomal proteins ≥ 80 aa were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Initial maximum-likelihood (ML) trees were constructed to de- termine which organisms belonged to the Archaea, Bac- teria, or CPR with FastTree 2 [81] (WAG + CAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Final separate trees for the three distant evolutionary groups were constructed in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Each ribosomal protein set was aligned separately with MAFFT [82] (v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='055b, − auto) and concatenated only if a MAG encoded at least 8 out of 16 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' For all trees, a 100× posterior bootstraps analysis was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phylogenetic trees were visualized together with gen- ome statistics and abundance information using iTOL [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We cross-checked the taxonomic assignments based on the phylogeny of the ribosomal protein cas- sette with the top hit contig annotations against NCBI-nr and with the reference lineage obtained with CheckM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Lastly, we manually corrected the MAGs for misplaced 16S rRNA genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The final trees presented in the manuscript were redrawn using FigTree v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='3 [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Detailed genome analyses CPR MAGs were re-annotated more thoroughly: genes were predicted with Prokka [85], and functional predictions were performed by running InterProScan 5 locally on the supplied COG, CDD, TIGRFAMs, HAMAP, Pfam, and SMART databases [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' BLAST Koala was used for KEGG pathway predictions [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' To find putative carbohydrate-active enzymes in all final MAGs, we used the web-resource dbCAN [87] to annotate all predicted proteins ≥ 80 aa against CAZy [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' To identify the top ten abundant MAGs from each re- spective dataset, ten million randomly sampled single- tons were mapped onto each MAG with a cut-off of 95% identity in minimum of 50 bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Coverage values were additionally normalized for genome size and expressed as reads per kilobase of sequence per gigabase of mapped reads (RPKG) [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' A positive score (from 871 to 1) was assigned to each MAG according to the rank- ing of the summed RPKG of MAGs in the high-salinity datasets (B1Sed10 and T1Sed) and a negative score ac- cording to the ranking of the summed RPKGs in the moderate salinity datasets (CSSed10, CSSed11, T3Se d10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Both scores were summed to get a “salinity prefer- ence score” with MAGs recruiting preferably from high salinity datasets on the positive end, moderate salinity datasets in the negative end, and those without prefer- ence in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We determined species delineation for the most abundant MAGs and their closest reference genomes (NCBI-nr) by Average Nucleotide Identity (ANI) and conserved DNA-matrices, as follows [90]: ANI ≥ 95%, conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA < 69% = might be same species, ANI < 95%, condDNA < 69% = different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Single gene trees based on maximum likelihood were constructed with un- trimmed alignments (MAFFT, L-INS-i model) and FastTree 2 (WAG + CAT, increased accuracy, -spr4 mlacc 2 -slownni) using 100× bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' References were pulled from eggNOG (v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='1) [91] and supple- mented with sequences from NCBI-nr or refined according to [7, 33, 46, 92–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The curated MAGs were scanned for the presence of rhodopsin sequences with the hmmsearch software [95] and a profile hidden Markov model (HMM) of the bacteriorhodopsin-like protein family (Pfam accession number PF01036).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The identified sequences with significant similarity were aligned together with a curated database composed of a collection of type-1 rhodopsins, using MAFFT (L-INS-i accuracy model) [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' This protein alignment was further utilized to Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 14 of 18 construct a maximum likelihood tree with 100× boot- strap with FastTree 2 [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All other genes were identified using the KEGG annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Additional files Additional file 1: Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' General features of the four sampled soda lakes at time of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' SILVA classification of the 16S rRNA gene sequences found in all ≥1 kb contigs of five soda sediment metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Enzymes involved in lipopolysaccharide biosynthesis found among different members of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sub-kingdom classification of candidate SSU rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Top-level taxonomic classification of the 18S rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Description of the metagenomic datasets, NCBI Sequence Read Archive (SRA) accession numbers and general statistics of the assembled contigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (PDF 740 kb) Additional file 2: Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Taxonomic fingerprints determined by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Genome statistics of the 871 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phylogeny of MAGs belonging to “Candidatus Aenigmarchaeota” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nanohaloarchaeota”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phylogeny of MAGs related to “Candidatus Acetothermia”, candidate division WS1 and “Candidatus Lindowbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phylogeny of MAGs related to candidate division KSB3 and “Candidatus Schekmanbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Multiple sequence alignment of the V-type ATPase subunits K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Multiple sequence alignment of the F-type ATPase subunits c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- like proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Maximum likelihood tree of the putative rhodopsins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Predicted isoelectric points (pI) profiles of all MAGs from CPR members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Predicted isoelectric points profiles for members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Vogelbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Multiple sequence alignment of the dissimilatory cytochrome c nitrite reductases (nrfA/TvNiR, K03385).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Overview of the post-binning workflow used for genome recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (PDF 6548 kb) Additional file 3: Dataset S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Relative abundance of the OTUs assigned to the genus-level within the Archaea, Bacteria and organelles from Eukaryota detected by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The OTUs with less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='1% abundance accross all five datasets are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The names of highly abundant genera (≥1% in at least one of the data- sets) are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (XLSX 24 kb) Additional file 4: Dataset S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Organism names, statistics and general description incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Completeness and contamination estimates, phylogeny and DDBJ/EMBL/Genbank accession numbers of the metagenome assembled genomes (MAGs) described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All submitted versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Size = recovered genome size, Completeness (Compl1), contamination (Cont), strain heterogenity (Str het) and Taxon CheckM were inferred from lineage-specific marker sets and a reference tree build with CheckM [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Additional completeness (compl2) and redundancy (red) estimates were inferred based on the presence of universal single copy genes for Bacteria and Archaea [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Decision and confidence intervals from the Candidate Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the besthit in SILVA when 16S rRNA genes were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Phylum/class 16 ribosomal proteins is the taxonomy derived from our ribosomal protein trees (see main text: Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' OTU gives the inferred link of a population genome with our 16S rRNA gene amplicon dataset (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (XLSX 253 kb) Additional file 5: Dataset S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Estimated abundance and derived salinity preference from each MAG in each metagenomic dataset expressed as Reads per Kilobase of MAG per Gigabase of mapped reads (RPKG) and “salinity preference score” (see Methods section), basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (XLSX 143 kb) Additional file 6: Dataset S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Average Nucleotide Identity (ANI) and conserved DNA (condna) matrices to determine species delineation between the most abundant MAGs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4, closely related (less abundant) MAGs and NCBI reference genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Decision matrix shows: 1 = same species, − 1 = might be same species, 0 = different species (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (XLSX 1161 kb) Additional file 7: Dataset S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sheet 1 Presence and absence of marker genes and putative carbohydrate-active enzymes in the MAGs to infer putative roles in C, N and S element cycles based on K-number assignments and CAZy annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sheet 2 Summary basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (XLSX 41 kb) Additional file 8: Information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' More detailed description of the main metabolisms encoded by Thioalkalivibrio-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Information S2 More detailed description of the main metabolisms encoded by Deltaproteobacterial-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (PDF 219 kb) Additional file 9: Dataset 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sheet 1 shows the MAGs positive for the marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 6, namely presence and absence of key genes involved in the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis and pyruvate to CO2 conversion is shown for each MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sheet 2 shows the MAGs positive for the marker gene cdhC (K00193) encoding for the beta subunit of an acetyl-CoA decarboxylase synthase complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' While acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- type (methanogens) enzymes with the same function, we found few discrepancies between marker gene and genome phylogeny within the Methanomassiliicoccaceae and Chloroflexi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' (XLSX 52 kb) Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Nikolai Chernych for his technical assistance during the isolation and purification of metagenomics DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' We also thank the Department of Energy Joint Genome Institute for sequencing the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Funding CDV and GM were supported by the ERC Advanced Grant PARASOL (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 322551).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' A-SA and RG were supported by the research grant 17-04828S from the Grant Agency of the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' MM was supported by the Czech Academy of Sciences (Postdoc program PPPLZ application number L200961651).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' DYS was supported by the SIAM/Gravitation Program (Dutch Ministry of Education and Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- 00121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sequencing was performed by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, as part of the Community Sequencing Program (contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' DE-AC02- 05CH11231).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Availability of data and materials The raw sequence reads of the five metagenomes have been deposited to the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession numbers and read and contig statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The final 871 MAGs described in this paper have been deposited as Whole Genome Shotgun projects at DDBJ/ EMBL/GenBank, and accession numbers are listed in Additional file 4 (BioProject ID PRJNA434545).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' The cleaned and dereplicated amplicon sequence datasets are available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='com/s/7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Maximum likelihood trees based on the concatenated alignment of 16 ribosomal proteins, basis for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2 and 3, in newick format (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='tre file) and complementary datasets (used to plot completeness, contamination, genome recovery size, G + C mol% and RPKG in iTOL), as well as K number assignments for the predicted proteins of all MAGs (KEGG-orthologues, Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions of this article are also available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content='com/s/ 7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Authors’ contributions GM and DYS initiated this study and were responsible for the fieldwork, sample preparation, and sequencing effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' CDV conceptualized the research goals under supervision of DYS and GM, and performed the bioinformatics analysis under close guidance of A-SA and RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' CDV is the primary author of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' MM, RG, and CDV prepared the main figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Ethics approval and consent to participate Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbiome (2018) 6:168 Page 15 of 18 Consent for publication Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Author details 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2Department of Aquatic Microbial Ecology, Institute of Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 3Winogradsky Institute of Microbiology, Research Centre of Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 2, Moscow, Russian Federation117312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' 4Environmental Biotechnology, Department of Biotechnology, Delft University of Technology, Van der Maasweg 9, 2629, HZ, Delft, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Received: 23 June 2018 Accepted: 3 September 2018 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_39/content/kb_39.pdf'} +page_content=' Microbial diversity and biogeochemical cycling in soda lakes.' metadata={'source': 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Programming +by Freezing Slow States +Yijia Wang and Daniel R. Jiang +University of Pittsburgh +January 4, 2023 +Abstract +We consider infinite horizon Markov decision processes (MDPs) with fast-slow structure, +meaning that certain parts of the state space move “fast” (and in a sense, are more influential) +while other parts transition more “slowly.” Such structure is common in real-world problems +where sequential decisions need to be made at high frequencies, yet information that varies at a +slower timescale also influences the optimal policy. Examples include: (1) service allocation for +a multi-class queue with (slowly varying) stochastic costs, (2) a restless multi-armed bandit with +an environmental state, and (3) energy demand response, where both day-ahead and real-time +prices play a role in the firm’s revenue. Models that fully capture these problems often result +in MDPs with large state spaces and large effective time horizons (due to frequent decisions), +rendering them computationally intractable. We propose an approximate dynamic programming +algorithmic framework based on the idea of “freezing” the slow states, solving a set of simpler +finite-horizon MDPs (the lower-level MDPs), and applying value iteration (VI) to an auxiliary +MDP that transitions on a slower timescale (the upper-level MDP). We also extend the technique +to a function approximation setting, where a feature-based linear architecture is used. On the +theoretical side, we analyze the regret incurred by each variant of our frozen-state approach. +Finally, we give empirical evidence that the frozen-state approach generates effective policies +using just a fraction of the computational cost, while illustrating that simply omitting slow +states from the decision modeling is often not a viable heuristic. +1 +arXiv:2301.00922v1 [cs.AI] 3 Jan 2023 + +1 +Introduction +We consider sequential decision problems, modeled as Markov decision processes (MDPs), that are +endowed with a new “fast-slow” structure: a fast-slow MDP has a state that can be divided into +two parts, a slow state and a fast state. At each time step, the transition of the slow state results +in a change that is relatively small compared to that of the fast state. An alternative view from +the perspective of the reward function (rather than the transition function) is that the reward is +less sensitive to changes in the slow state. Fast-slow structure is common in important real-world +problems where sequential decisions need to be made at high frequencies, yet information that varies +at a slower timescale also influences the optimal policy. The following examples illustrate this idea. +1. Service allocation in multi-class queues. The first example is a dynamic service allocation +problem for a multi-class queue (Ansell et al., 2003; Brown and Haugh, 2017), with the addition +of stochastic holding costs (i.e., the cost of leaving items in the queue) that vary slowly and +can be viewed as the slow state (Lee and Vojnovic, 2021). One prominent motivation is the +case of energy-aware job scheduling in data centers, where variations of electricity prices over +time can influence the holding cost (Ren et al., 2012; Zhou et al., 2013; Mao et al., 2019). +2. Restless multi-armed bandit with an environmental state. Our second example ap- +plication is the restless multi-armed bandit (Whittle, 1988; Weber and Weiss, 1990; Killian +et al., 2021; Zhang and Frazier, 2021) with an environmental state, a model that is applicable +to a problems ranging from machine maintenance (Smallwood and Sondik, 1973; Duan et al., +2018; Ruiz-Hernández et al., 2020) to dynamic assortment planning (Brown and Smith, 2020) +to public health and preventative healthcare (Mate et al., 2020; Lee et al., 2019; Biswas et al., +2021). The restless bandit model involves making intervention decisions (e.g., whether to per- +form maintenance) on “arms” (e.g., machines), each of which is associated with an evolving +internal state. Here, the environmental state can be viewed as the slow state, because it often +transitions slowly relative to the arms’ internal states. +3. Energy demand response. We can also apply the fast-slow framework in sequential deci- +sion problems from the realm of demand response in the electricity market. Specifically, we +consider the problem faced an energy aggregator who observes a day-ahead price and then +2 + +simultaneously bids a reduction quantity into the demand response market and sets the com- +pensation for demand reduction from consumers (Albadi and El-Saadany, 2008; Eid et al., +2016; Khezeli and Bitar, 2017; Khezeli et al., 2017; Wang et al., 2018). Essentially, the aggre- +gator hopes to generate profit from the difference between the contracted price for delivery +of demand reduction to the market and the price that offers customers for that reduction. +However, the aggregator has to consider the demand elasticity of its customers, along with +the stochasticity of day-ahead prices and real-time prices (which determine the “penalty” for +mistmatch between the promised and realized quantities of demand reduction). Since real- +time prices are much more volatile compared to the day-ahead prices, it is reasonable to view +day-ahead prices as the slow state. +Attempts to optimally solve a model that incorporates the full state space along with the true +decision-making frequency often encounter computational issues, due to the challenge of solving an +MDP with a large state space over a large number of periods. Anecdotal evidence suggests that +to improve tractability, both practitioners and academic researchers may elect to design simplified +decision models that ignore the effect of the slow state on components of their problems. In other +words, these states might be intentionally left out of the state variable by, e.g., fixing them to +constant values. Although such an approach results in policies that can be obtained in a compu- +tationally tractable manner, we see in Section 9 that they can incur significant regret compared to +the optimal policy. +1.1 +Main Contributions +In this paper, we propose somewhat of a compromise between the solving the full MDP and com- +pletely ignoring slow states, by designing a framework around periodically “freezing” and “releasing” +slow states, and re-using policies that are computed based on a frozen slow-state model. Specifically, +we make the following contributions: +1. We first consider a fast-slow MDP and provide an (exact) reformulation into an MDP with +hierarchical structure. +The upper level is a slow-timescale infinite horizon MDP and the +lower level is a fast-timescale finite horizon MDP with T periods. One period of the upper- +level problem is composed of a complete lower-level problem. +We propose a frozen-state +3 + +approximation to the reformulated MDP, along with an associated frozen-state value iteration +(FSVI) algorithm, where the slow state is frozen in the lower-level problem, while each period +in the upper-level problem “releases” the slow state. Computational benefits arise in several +ways: (1) re-use of the lower-level policy (which is computed once) when applying value +iteration in the upper level, (2) frozen states simplify the dynamics of the lower-level MDP +(dramatically fewer successor states), and (3) the lower-level MDP thus becomes separable +into independent MDPs, opening the door to speedups via parallel computation. Solving the +frozen-state approximation gives a policy that switches between the one action from upper- +level policy and T − 1 actions from the lower-level policy. We give a theoretical analysis that +upper bounds the expected regret from applying this policy compared to the optimal policy. +2. We then discuss an additional step of approximation that further reduces computational re- +quirements, called the nominal-state approximation, which takes advantage of a factored re- +ward function assumption and approximates the lower-level MDP using a fixed set of “nominal” +slow states. The consequence is that instead of solving the lower-level MDP for all slow states, +this approximation allows us to solve it only for the set of nominal slow states, which are then +used to approximate the lower-level value for other slow states. We also provide an upper +bounds on the expected regret of the policy obtained from the nominal state approximation. +3. Next, we show how the fast-slow framework can also be exploited in an approximate dynamic +programming (ADP) setting (Bertsekas and Tsitsiklis, 1996; Powell, 2007). Specifically, we +design a frozen-state approximate value iteration (FSAVI) algorithm that mimics FSVI but +uses a linear architecture to approximate the value function in both the lower and upper +levels. The linear architecture combines estimated values from a set of pre-selected states to +form approximations of the value function at other states, based on the technique introduced +in Tsitsiklis and Van Roy (1996). We provide an analysis of the expected regret for policies +generated by FSAVI. +4. Lastly, we perform a systematic empirical study on three problem settings (service alloca- +tion in multi-class queues, restless bandit with an environmental state, and energy demand +response). We show that the proposed algorithms based on the frozen-state approximation +quickly converge to good policies using significantly less computation compared to standard +4 + +methods (value iteration, approximate value iteration, Q-learning, deep Q-networks, and a +baseline that ignores slow states). Notably, our results show that ignoring the slow state leads +particularly poor results. We also give qualitative evidence that policies generated by the +frozen-state approach have structural features resembling those of the optimal policy. +2 +Related Work +In this section, we provide a brief review of related literature. First, there exists a stream of literature +focused on sequential decision making problems with exact hierarchical, multi-timescale structure. +Chang et al. (2003) study multi-timescale MDPs, which are composed of M different decisions that +are made on M different discrete timescales. The authors consider the impact of upper-level states +and actions on the transition of the lower levels, an idea is also present in our fast-slow framework. +Multi-timescale MDPs have often been applied in supply chain problems, including production plan- +ning in semiconductor fabrication (Panigrahi and Bhatnagar, 2004; Bhatnagar and Panigrahi, 2006), +hydropower portfolio management (Zhu et al., 2006), and strategic network growth for reverse supply +chains (Wongthatsanekorn et al., 2010). Wang et al. (2018) propose a row-generation-based algo- +rithm to solve a linear programming formulation of the multi-timescale MDP. Jacobson et al. (1999) +consider “piecewise stationary” MDPs, where the transition and reward functions are “renewed” ev- +ery N + 1 periods, motivated by problems where routine decisions are periodically interrupted by +higher-level decisions. For the case of large renewal periods, they propose a policy called the “initially +stationary policy” which uses a fixed decision rule for some number of initial periods in each renewal +cycle. Our fast-slow model focuses on a novel fast-slow structure present in many MDPs and unlike +the above work, does not assume any natural/exact hierarchical structure. Instead, we focus on +how a particular type of (frozen-state) hierarchical structure can be used as an approximation to the +true MDP. However, we note that many MDPs with natural two-timescale structure can also fit into +our framework, and therefore, given that perspective, our model can be viewed as a generalization. +Our proposed frozen-state algorithms are also related to literature on hierarchical reinforce- +ment learning, which are methods that artificially decompose a complex problem into smaller sub- +problems (Barto and Mahadevan, 2003). Approaches include the options framework (Sutton et al., +1999), the hierarchies of abstract machines (HAMs) approach (Parr and Russell, 1998), and MAXQ +5 + +value function decomposition (Dietterich, 2000). +Out of these three approaches, the options framework is most closely related to this paper. A +Markov option (also called a macro-action or temporally extended action) is composed of a policy, a +termination condition, and an initiation set (Sutton et al., 1999; Precup, 2000). One of the biggest +challenges is to automatically construct options that can effectively speed up reinforcement learning. +A large portion of work in this direction is based on subgoals, states that might be beneficial to +reach (Digney, 1998; McGovern and Barto, 2001; Jonsson and Barto, 2005; Ciosek and Silver, 2015; +Wang et al., 2022). The subgoals are identified by utilizing the learned model of the environment +(Menache et al., 2002; Mannor et al., 2004; Şimşek and Barto, 2004; Şimşek et al., 2005), or through +trajectories without learning a model (McGovern and Barto, 2001; Stolle and Precup, 2002). The +options (and subgoals) framework is largely motivated by robotics and navigation-related tasks, +while we are particularly interested in solving problems that arise in the operations research and +operations management domains. The problems that we study do not decompose naturally into +“subgoals” — leading us to identify and focus on the fast-slow structure, which does indeed arise +naturally for many problems of interest. +Another work that is related to the options framework is Song and Xu (2020), who divide +finite-horizon MDPs into two sub-problems along the time horizon, and concatenate their optimal +solutions to generate an overall solution. Our paper also, in a sense, divides MDPs along the time +horizon, but our work is quite different from Song and Xu (2020) in that we work on infinite horizon +problems and convert them into auxiliary problems that operate on a slower timescale, which takes +advantage of reusable lower-level policies. More importantly, the various methods we propose all +build on the idea of freezing certain states to reduce computational cost, which is unique to our +approach and to our knowledge, this is a novel direction that has not been proposed before. +3 +Fast-Slow MDPs +In this section, we introduce the base model, the original MDP to be solved and formally introduce +the notion of a fast-slow MDP. We then provide a hierarchical reformulation of the base model using +fixed-horizon policies, and show the equivalence (in optimal value) between the two models. +6 + +3.1 +Base Model +Consider a discrete-time MDP ⟨S, A, W, f, r, γ⟩, where S is the finite state space, A is the finite +action space, W is the space of possible realizations of an exogenous, independent and identically +distributed (i.i.d.) noise process {wt} defined on a discrete probability space (Ω, F, P), f : S × +A × W → S is the transition function, r : S × A → [0, rmax] is the bounded reward function, and +γ ∈ [0, 1) is the discount factor for future rewards (Puterman, 2014). The objective is +U ∗(s) = max +{νt} E +� ∞ +� +t=0 +γt r +� +st, νt(st) +� ��� s0 = s +� +, +(1) +where states transition according to st+1 = f(st, at, wt+1) and we optimize over sequences of policies +νt : S → A, which are deterministic mappings from states to actions. The expectation is taken over +exogenous noise process {wt}∞ +t=1. We assume throughout that S, A, X, Y, S × A, and X × Y are +equipped with the Euclidean metric,1 which is naturally the case for many applications. +Assumption 1 (Separability and the Fast-Slow Property). Suppose the following hold: +(i) The state space S is separable and can be written as S = X × Y. We call X the “slow state +space” and Y the “fast state space.” +(ii) Let st = (xt, yt) ∈ S, where xt ∈ X is the slow state and yt ∈ Y the fast state, at ∈ A, +and wt+1 ∈ W. The transition dynamics st+1 = f(st, at, wt+1) ∈ S can be written with the +notation: +xt+1 = fX (xt, yt, at, wt+1) ∈ X +and +yt+1 = fY(xt, yt, at, wt+1) ∈ Y, +for some fX : S × A × W → X and fY : S × A × W → Y. +(iii) For any state (x, y) ∈ S, action a ∈ A, and exogenous noise w ∈ W, suppose the one-step +transitions of x and y satisfy: +��y − fY(x, y, a, w) +�� +2 ≤ dY +and +��x − fX (x, y, a, w) +�� +2 ≤ αdY, +1However, as long as the relevant spaces are metric spaces, the framework continues to hold. We choose Euclidean +metrics as they are natural for our applications. +7 + +for some dY < ∞ and α ∈ [0, 1]. +Remark 1. Note that one particularly instructive example is the case of exogenous slow states, +where xt+1 = fX (xt, wt+1). +Here, the transition does not depend on the action at, nor does it +depend on the fast state yt. Such a model is common in practice: examples of exogenous slow states +include prices, weather conditions, and other environmental variables that are not influenced by the +decision maker’s actions or the states of the primary system. See, e.g., Yu and Mannor (2009), who +study a related model called the “arbitrarily modulated MDP.” +Assumption 2 (Lipschitz Properties). Suppose that the reward function r, transition function f, +and optimal value function U ∗ are Lipschitz with respect to ∥ · ∥2: +|r(s, a) − r(s′, a′)| ≤ Lr +��(s, a) − (s′, a′) +�� +2, +(2) +��f(s, a, w) − f(s′, a′, w) +�� +2 ≤ Lf +��(s, a) − (s′, a′) +�� +2, +(3) +��U ∗(s) − U ∗(s′) +�� +2 ≤ LU +��s − s′�� +2, +(4) +for some Lipschitz constants Lr, Lf, and LU. Lipschitz assumptions are common in the literature; +see, for example, Ok et al. (2018), Domingues et al. (2021), Sinclair et al. (2020), and Sinclair +et al. (2022). In Appendix F, we give bounds on LU in terms of Lr and Lf. While we could have +used those results directly and omitted the assumption on LU, we opt to include (4) to increase the +clarity of our results. +Definition 1 (Fast-Slow MDP). An MDP ⟨S, A, W, f, r, γ⟩ is called a (α, dY)-fast-slow MDP if +Assumptions 1 and 2 are satisfied. +Given any state s = (x, y), noise w, and policy ν, we use the notation fν(s, w) = f(s, ν(s), w), +fν +X (x, y, w) = fX +� +x, y, ν(x, y), w +� +, fν +Y(x, y, w) = fY +� +x, y, ν(x, y), w +� +, and r(x, y, ν) = r(x, y, ν(x, y)) +throughout the paper. The value of a stationary policy2 ν at state (x, y) is the expected cumulative +reward starting from state (x, y) following policy ν, i.e., +U ν(x, y) = E +� ∞ +� +t=0 +γtr +� +xt, yt, ν +� ��� (x0, y0) = (x, y) +� += r +� +x, y, ν +� ++ γ E +� +U ν(x′, y′) +� +, +2It is well-known that there exists an optimal policy to (1) that is both stationary and deterministic. See Puterman +(2014). +8 + +where (x′, y′) = fν(x, y, w) and (xt+1, yt+1) = fν(xt, yt, wt) for all t. The optimal value function at +state U ∗(x, y), as defined in (1), satisfies the Bellman equation, i.e., +U ∗(x, y) = max +a +r(x, y, a) + γ E +� +U ∗(x′, y′) +� +. +(5) +Denote by H the Bellman operator of the base model; for any state (x, y) and value function U, +(HU)(x, y) = max +a +r(x, y, a) + γ E +� +U(f(x, y, a, w)) +� +. +(6) +A policy that is greedy with respect to the optimal value function, i.e., +ν∗(x, y) = arg max +a +r(x, y, a) + γ E +� +U ∗(x′, y′) +� +. +is an optimal policy, and the optimal value U ∗ and the value of the optimal policy U ν∗ are the same. +3.2 +Hierarchical Reformulation using Fixed-Horizon Policies +In this section, we derive an exact hierarchical reformulation with the original timescale broken up +into groups of T periods each. The reformulation holds for a general MDP ⟨S, A, W, f, r, γ⟩, but +the concepts that we introduce in this section will serve as the basis for developing our frozen-state +computational approach for fast-slow MDPs. +Denote (µ, π) a T-horizon policy, which is a sequence of T policies (µ, π1, . . . , πT−1), µ : S → A, +πt : S → A and π = (π1, . . . , πT−1). Following (µ, π) means that we take the first action according +to µ and then next T − 1 actions according to π. Given any state s0, the T-period reward function +(of the base model) associated with (µ, π) is written as: +R(s0, µ(s0), π) = r(s0, µ) + +T−1 +� +t=1 +γt r(st, πt), +(7) +where s1 = fµ(s0, w1) and st+1 = fπt(st, wt+1) for t > 0. +A T-periodic policy (µ, π) refers to the infinite sequence that repeatedly applies the T-horizon +policy (µ, π), i.e., (µ, π, µ, π, . . .). Note that despite it not being a stationary policy, the T-periodic +policy (µ, π) can be implemented in the infinite horizon problem defined in (1). The value of the +9 + +T-periodic policy (µ, π) at state s0 is +¯U µ(s0, π) = E +� ∞ +� +k=0 +γkT R(sk, µ(sk), π) +��� s0 = s +� += E +� +R(s0, µ(s0), π) + γT ¯U µ(sT , π) +� +, +where, again, s1 = fµ(s0, w1) and st+1 = fπt(st, wt+1) for t > 0 within each cycle of T periods. +Figure 1 compares stationary policy ν and a T-periodic policy (µ, π) for the case of T = 4. In the +figure, we also illustrate how rewards can be written in an “aggregated” fashion over the T periods +using (7). +ν +ν +ν +ν +ν +ν +ν +⋯ +ν +π1 +π2 +π3 +⋯ +μ +π1 +π2 +π3 +μ +R(s0, μ(s0), π) +R(s4, μ(s4), π) +Figure 1: Illustration of a stationary policy µ (upper timeline) and a T-periodic policy (µ, π) (lower timeline) +for T = 4. The periods covered by the T-period reward associated with (µ, π) is shown in the lower timeline. +The optimal value function satisfies the following Bellman equation: +¯U ∗(s0) = max +(µ,π) E +� +R(s0, µ(s0), π) + γT ¯U ∗(sT ) +� +, +(8) +where the “action” now involves selecting the π as well. Denote (µ∗, π∗) an optimal T-periodic +policy, which solves (8). In Proposition 3.1, we prove that the base model (5) and the hierarchical +reformulation (8) are equivalent in a certain sense. +Proposition 3.1. Given an MDP ⟨S, A, W, f, r, γ⟩, the following hold: +(i) The optimal value of the base model (5) is equal to the optimal value of the hierarchical refor- +mulation (8), i.e., U ∗ = ¯U ∗. +(ii) An optimal stationary policy ν∗ with respect to the base model (5) is also an optimal policy for +the hierarchical reformulation (8), i.e., ¯U ∗ = ¯U ν∗. +Proof. See Appendix A.2. +10 + +Part (i) of Proposition 3.1 is most relevant to our situation in the sense that the optimal T- +periodic policy (µ∗, π∗) is no better than the stationary optimal policy ν∗. Therefore, solving the +hierarchical reformulation (8) allows us to achieve the same value as the ν∗, the optimal policy to +the original base model (5). +Note that, at this point, we have simply reformulated the problem, but (8) is no easier to solve +that (5). Despite the more favorable discount factor γT in (8), its action space is now effectively +the space of T-horizon policies, rather than a single action a. In the next section, we propose an +approximation that allows us to fix a lower-level policy π and only optimize µ. This allows us to +enjoy the γT discount factor while maintaining the same action space. +4 +The Frozen-State Approximation +We propose a frozen-state approximation, where we make two simplifications to the T-period finite- +horizon problem with terminal value U ∗ that is embedded in each T-period “time step” of (8), +termed the lower-level problem. First, motivated by the slow transitions of x given in Assumption +1, we “freeze” slow states for all T periods of the lower-level problem, and second, we decouple the +problem from the main MDP by solving an approximation with zero terminal value instead of U ∗. +The first simplification reduces the computation needed to solve the finite-horizon MDP, while +the second simplification, due to the decoupling from the main problem, allows us to pre-compute an +approximation to π∗, which we denote ˜π∗. By then fixing ˜π∗, we are able to construct an auxiliary +problem that proceeds at a timescale that is a factor of T slower than the MDP of the base model +(equivalently, the discount factor becomes γT instead of γ), yet optimizing over the same action +space. This naturally leads to ADP algorithms with computational benefits (see Sections 6, 7, and +8). The number of periods T to freeze the state is a parameter to the approach. See Figure 2 for a +high-level illustration; we provide a detailed description of the approach in the next few sections. +Remark 2. It is important to note that the freezing of states only occurs “within the algorithm” as +a step toward more efficient computation of policies. Our resulting policies are then implemented +in the underlying base model MDP, which proceeds naturally according to its true dynamics. Our +theoretical and empirical results always attempt to answer the question: how well does a approximate +policy, which is computed by pretending certain states are frozen, perform in the true model? +11 + +π*1 +π*2 +π*3 +x1, y1 +x2, y2 +x3, y3 +U* +(a) The lower-level problem (i.e., optimizing over π) em- +bedded in (8). +˜π*1 +˜π*2 +˜π*3 +x1, y1 +x1, y2 +x1, y3 +J*T ≡ 0 +(b) The lower-level problem of the frozen-state approxi- +mation, with frozen x1 and J∗ +T instead of U ∗. +Figure 2: A comparison of the lower-level problem of the hierarchical reformulation vs the lower-level problem +of the frozen-state approximation. +4.1 +The Lower-Level MDP (Frozen Slow States) +We view the problem from period 1 to period T as the “lower level” of the frozen-state approxima- +tion.3 To form the lower-level problem of the frozen-state approximation, we consider this T − 1 +period problem in isolation: +J ˜π +1 (x, y) = E +�T−1 +� +t=1 +γt−1 r(x1, yt, ˜πt) +��� (x1, y1) = (x, y) +� +and +J∗ +1(x, y) = max +˜π +J ˜π +t (x, y) +(9) +where xt+1 = xt = x remains frozen, yt+1 = f ˜πt +Y (x, yt, wt+1), and ˜π = (˜π1, . . . , ˜πT−1). The problem +(9) can be solved using finite-horizon dynamic programming: accordingly, let the terminal J∗ +T ≡ 0 +and for t = 1, 2, . . . , T − 1, let +J∗ +t (x, y) = max +a +r(x, y, a) + γ E +� +J∗ +t+1(x, y′) +� +, +(10) +where y′ = fY(x, y, a, w). We also have the standard recursion for the performance of a policy: +J ˜π +t (x, y) = r(x, y, ˜πt(x, y)) + γ E +� +J ˜π +t (x, f ˜πt +Y (x, y, wt+1)) +� +, +(11) +with J ˜π +T ≡ 0. We denote by ˜H the Bellman operator of the lower-level problem, which is on the +same timescale as the base model (hence, the discount factor is γ) and looks similar to the Bellman +operator H defined in (6), but the transition of the slow-state x is frozen. For any state (x, y) and +3This corresponds to the periods relevant to π from (µ, π) in the hierarchical reformulation (8), whose structure the +frozen-state approximation mimics. +12 + +lower-level value function Jt+1,4 define: +� ˜HJt+1 +� +(x, y) = max +a +r(x, y, a) + γ E +� +Jt+1(x, fY(x, y, a, w)) +� +. +(12) +Note that (12) can be viewed as an approximation to (6). Analogously, let ¯H ˜π be the Bellman +operator associated with (11). +Also, let ˜π∗ = (˜π∗ +1, . . . , ˜π∗ +T−1) be the finite-horizon policy that is greedy with respect to J∗ +t : +˜π∗ +t (x, y) = arg max +a +r(x, y, a) + γ E +� +J∗ +t+1(x, y′) +� +. +It may not immediately be clear why freezing slow states is desired. There are two main computa- +tional benefits to solving (10) instead of an analogous version of (10) without freezing x: +• In algorithms like value iteration (Puterman, 2014), each update requires computing expecta- +tions over successor states, and therefore the number of successor states impacts the number +of operations for each step of value iteration. When x is frozen, the number of successor states +is much smaller since we only have successor fast states (y′): in other words, we only need to +compute E +� +J∗ +t+1(x, y′) +� +instead of E +� +J∗ +t+1(x′, y′) +� +.5 +• Second, (10) can effectively be viewed as |X| independent MDPs, one for each x ∈ X, allowing +for the possibility of computing the policy with additional parallelism. In the nominal-state +approximation discussed Section 7, we analyze the error of an approach that solves only a +small number out of the |X| independent MDPs. +4.2 +The Upper-Level MDP (True State Dynamics) +Let us now consider the upper-level problem of the frozen-state approximation, which is an infinite +horizon problem with groups of T periods aggregated. Denote the stationary upper-level policy by +µ : S → A, which is the policy that we are attempting to optimize in the upper-level problem. The +upper-level problem takes two “inputs” related to the lower-level problem: (1) J1, an approximation +4We include time indexing on the value function to emphasize that this Bellman operator is used in a finite-horizon +(i.e., non-stationary) setting, but the definition of ˜H itself does not depend on t. +5Even in the case that the expectation is approximated via sampling, the former requires sampling from a lower- +dimensional successor state distribution. +13 + +of the optimal lower-level value J∗ +1, (2) π, a lower-level finite-horizon policy. Fixing these inputs, +the value at state s0 = (x0, y0) by executing policy µ is +V µ(s0, J1, π) = E +� ˜R(s0, µ(s0), J1) + γT V µ(sT (µ, π), J1, π) +� +, +where sT (µ, π) is the state reached according to the true system dynamics by following (µ, π), +starting from s0 and +˜R(s0, a, J1) = r(s0, a) + γ J1 +� +f(s0, a, w) +� +(13) +is a one-step approximation to the T-period reward function R, defined in (7). Figure 3 helps to +visualize the upper-level MDP. +π1 +π2 +π3 +⋯ +μ +π1 +π2 +π3 +μ +x1 +x1 +x1 +x0 +x4 +x5 +x5 +x5 +γT +J1 +J1 +Figure 3: Illustration of the upper-level problem. Notably, the discount factor is γT and the reward function, +from the point of view of µ, depends on the lower-level value function J1. This value function is computed +by freezing states, as visualized by the grey box. +The optimal value (for this approximation) at state s0 can be written as +V ∗(s0, J1, π) = max +a +E +� ˜R(s0, a, J1) + γT V ∗(sT (a, π), J1, π) +� +, +(14) +where sT (a, π) is the state reached according to the true system dynamics by first taking action a +and then following π, starting from s0. +Throughout the paper, we use the notation V µ(J1, π) and V ∗(J1, π) to refer to the value function +(i.e., S → R) obtained when the MDP is evaluated or solved for a fixed J1 and π. We also define +the Bellman operator associated with (14): +� +FJ1,πV +� +(s0) = max +a +E +� ˜R(s0, a, J1) + γT V (sT (a, π)) +� +, +(15) +14 + +which will become useful later on. +Recall that the optimal lower-level policy (that solves the frozen-state model) is denoted ˜π∗ and +its optimal value is J∗ +1. Let ˜µ∗ be the optimal upper-level policy corresponding to these inputs, i.e., +the policy greedy with respect to V ∗(s0, J∗ +1, ˜π∗). Thus, (˜µ∗, ˜π∗) is the resulting T-periodic policy +from the frozen-state hierarchical approximation; we refer to it as the T-periodic frozen-state policy. +4.3 +Characterizing the Exact and Frozen-State Reward Functions +Recall that (µ∗, π∗) is an optimal T-periodic policy of the base model’s hierarchical reformulation +(8). Suppose π∗ is available. Then, the Bellman equation of the base model reformulation is +U ∗(x0, y0) = ¯U ∗(x0, y0) += max +a +E +� +R(x0, y0, a, π∗) + γT ¯U ∗(xT , yT ) +� += max +a +E +� +r(x0, y0, a) + +T−1 +� +t=1 +γt r(xt, yt, π∗ +t ) + γT U ∗(xT , yT ) +� += max +a +E +� +r(x0, y0, a) + γ +� +HT−1U ∗� +(x1, y1) +� +, +(16) +where the notation Hk is shorthand for k applications of the operator H, i.e., HkU = H(Hk−1U) +and H1U = HU. Therefore, the expected T-horizon reward can be written as +E +� +R(x0, y0, a, π∗) +� += E +� +r(x0, y0, a) + γ +� +HT−1U ∗� +(x1, y1) − γT U ∗(xT , yT ) +� +. +(17) +Given the optimal value J∗ +1 of the lower level (10), the T-horizon reward of the upper level (14) +can be written as +E +� ˜R(x0, y0, a, J∗ +1) +� += r(x0, y0, a) + γ E +� +J∗ +1(x1, y1) +� += r(x0, y0, a) + γ +� ˜HT−1J∗ +T +� +(x1, y1), += r(x0, y0, a) + γ +� ˜HT−1 0 +� +(x1, y1), +(18) +where 0 is the all-zero value function. The difference between (17) and (18) can be interpreted +as follows: in the former, we follow a lower-level policy that is aware of a terminal value U ∗ (but +15 + +exclude that value when defining the T-horizon reward), while in the latter, we follow a lower-level +policy that sees zero terminal reward at the end of the T − 1 periods. +The first step to understanding the performance of the frozen-state policy is to analyze the +reward approximation E[ ˜R(s0, a, J∗ +1)] compared to the true reward E[R(s0, a, π∗)]. Proposition 4.1 +shows how the difference between two reward functions is dependent on the number of frozen periods +T, along with the problem parameters. +Proposition 4.1 (Reward Approximation Error). Let ⟨S, A, W, f, r, γ⟩ be a (α, dY)-fast-slow MDP +satisfying Assumption 2. Let π∗ be the optimal lower-level policy for the base model reformulation (8) +and J∗ +1 be the optimal (first-stage) value of the lower-level problem in the frozen-state approximation +(10). For any state s0 = (x0, y0) and action a, the approximation error between the T-horizon reward +of hierarchical reformulation and the frozen-state approximation, i.e., the discrepancy between (17) +and (18), can be bounded as: +��E +� +R(s0, a, π∗) +� +− E +� ˜R(s0, a, J∗ +1) +��� +≤ αdY +� +Lr +T−2 +� +i=1 +γi +i−1 +� +j=0 +Lj +f +� ++ γT−1LU +� +αdY +T−2 +� +j=0 +Lj +f + γdY(α + 2)(T − 1) +� +, +(19) +Proof. The detailed proof is in Appendix A.3. +For more convenient notation, we define ϵr(γ, α, dY, Lr, Lf, T) to be the right-hand-side of (19): +ϵr(γ, α, dY, L, T) = αdY +� +Lr +T−2 +� +i=1 +γi +i−1 +� +j=0 +Lj +f +� ++ γT−1LU +� +αdY +T−2 +� +j=0 +Lj +f + γdY(α + 2)(T − 1) +� +, +where L = (Lr, Lf, LU) emphasizes the dependence on the various Lipschitz constants. In subse- +quent sections, we use ϵr as an ingredient in analyzing the regret of various frozen-state policies. +5 +Regret of the Frozen-State Policy (˜µ∗, ˜π∗) +In this section, we will show an upper bound on the regret from applying the T-periodic policy +(˜µ∗, ˜π∗) instead of the optimal policy ν∗ in the base model. Note that this is the policy obtained +if we were able to perfectly solve the frozen-state approximation. We start with definitions of the +16 + +regret of both stationary and T-periodic policies. +Definition 2 (Regret). Consider a fast-slow MDP with initial state s0 and optimal policy ν∗. The +regret of a stationary policy ν is defined as +R(s0, ν) = U ν∗(s0) − U ν(s0) +and +R(ν) = max +s0 +R(s0, ν). +The regret of the T-periodic policy (µ, π) is defined as: +R(s0, µ, π) = U ν∗(s0) − ¯U µ(s0, π) = ¯U ∗(s0) − ¯U µ(s0, π) +and +R(µ, π) = max +s0 +R(s0, µ, π). +The second equality in the definition of R(s0, µ, π) uses the value equivalence between the base model +and its hierarchical reformulation (Proposition 3.1). +Remark 3. As a follow-up comment to Remark 2, notice that V ∗(s0, J∗ +1, ˜π∗) does not directly enter +the regret definition, as V ∗(s0, J∗ +1, ˜π∗) is just the optimal value of the frozen-state approximation, +not the value of its implied greedy policy ˜µ∗ when evaluated in the base model. However, the regret +of course depends on V ∗(s0, J∗ +1, ˜π∗) indirectly, because ˜µ∗ depends on V ∗(s0, J∗ +1, ˜π∗). +In this section, we derive a bound on R(˜µ∗, ˜π∗), the regret of applying T-periodic policy (˜µ∗, ˜π∗) +to the base model. First, we start with a general lemma, that will be used throughout the paper as +a tool to analyze variants of FSVI. +Lemma 5.1. Suppose we have an approximation (π, J1) to the lower-level solution (π∗, U∗). Fur- +ther, suppose we have an approximation V to the upper-level solution V ∗(J1, π). Consider a T- +periodic policy (µ, π), where +µ(s0) = arg max +a∈A +E +� ˜R(s0, a, J1) + γT V (sT (a, π)) +� +. +(20) +Then, the regret of (µ, π) can be bounded as follows: +R(µ, π) ≤ +� +2γT +(1 − γT )2 + +2 +1 − γT +� +ϵr(π∗, J1) ++ +� +2γ2T +(1 − γT )2 + +2γT +1 − γT +� +LU d(α, dY, T) + +2γT +1 − γT +��V ∗(J1, π) − V +�� +∞, +17 + +where ϵr(π∗, J1) = maxs,a |E[R(s, a, π∗)] − E[ ˜R(s, a, J1)]| and d(α, dY, T) = 2dY(α + 1)(T − 1). +Proof. See Appendix B.2. +This result above can be interpreted as the regret being bounded by +reward error + end-of-horizon error + V -approximation error, +which directly corresponds to the three terms in the bound. The reward error is due to freezing the +slow state; the end-of-horizon error is due to using zero terminal value; and the V -approximation +error is due to not solving the upper-level problem exactly. +The main result of this section follows directly from Lemma 5.1 and is given in Theorem 5.1, +which shows the expected regret R(˜µ∗, ˜π∗) of applying the policy learned from the frozen-state +hierarchical approximation to the base model. +Theorem 5.1. Let ⟨S, A, W, f, r, γ⟩ be a (α, dY)-fast-slow MDP satisfying Assumption 2. +The +regret of applying the T-periodic policy (˜µ∗, ˜π∗) in the base model is bounded by +R(˜µ∗, ˜π∗) ≤ +� +2γT +(1 − γT )2 + +2 +1 − γT +� +ϵr(γ, α, dY, L, T) + +� +2γ2T +(1 − γT )2 + +2γT +1 − γT +� +LU d(α, dY, T) +where d(α, dY, T) = 2dY(α + 1)(T − 1). +Proof. We apply Lemma 5.1 with π = ˜π∗, J1 = J∗ +1, and V = V ∗(J∗ +1, ˜π∗), while noting that by +Proposition 4.1, ϵr(π∗, J1) ≤ ϵr(γ, α, dY, L, T). +6 +Frozen-State Value Iteration +In this section, we introduce the our new approach: the frozen-state value iteration (FSVI) algo- +rithm. The main ideas of our approach are: +1. Solve the lower-level MDP with frozen states to obtain a policy ˜π∗ and its value J∗ +1. Since the +lower-level problem is a finite horizon MDP, it can be solved exactly using T − 1 steps of VI. +2. Apply value iteration (VI) to the upper-level problem starting with some initial value function +V 0, while using J∗ +1 to approximate the T-horizon reward and ˜π∗ for T-step transitions. Note +18 + +that this is an infinite-horizon MDP that operates at a slower timescale and enjoys a much +more favorable discount factor of γT . +Before we dive into the details and analysis of FSVI, we start by mentioning that the naive approach +to solving the base model MDP (5) is to directly apply standard VI (see, e.g., Bertsekas and Tsitsiklis +(1996)). For completeness, we provide the full description in Algorithm 1. +Algorithm 1: Exact VI for the Base Model +Input: Initial values U 0, number of iterations k. +Output: Approximation to the optimal policy νk. +1 for i = 1, 2, . . . , k do +2 +for s in the state space S do +3 +U i(s) = maxa r(s, a) + γ E +� +U i−1(f(s, a, w)) +� +. +4 +end +5 end +6 for s in the state space S do +7 +νk(s) = arg maxa r(s, a) + γ E +� +U k(f(s, a, w)) +� +. +8 end +Proposition 6.1 is a well-known property that gives the required number of iterations of exact +VI on the base model needed for the resulting policy to achieve a desired level of regret. +Proposition 6.1. Let νk be the result of running Algorithm 1 on the base model (5). Then, +R(νk) = ∥U νk − U ∗∥∞ ≤ 2rmaxγk+1 +(1 − γ)2 . +Proof. See Appendix C.2. +6.1 +Analysis of FSVI +A detailed specification is given in Algorithm 2. We denote the resulting value function approxima- +tion after k iterations of value iteration as V k, from which we obtain a policy ˜µk. The T-periodic +policy output by FSVI is formed by combining ˜µk with the optimal finite-horizon policy ˜π∗ from +the lower-level MDP: (˜µk, ˜π∗). +19 + +Algorithm 2: Frozen-State Value Iteration (FSVI) +Input: Initial values J∗ +T ≡ 0 and V 0, number of iterations k. +Output: Approximation of the T-periodic frozen-state policy (˜µk, ˜π∗) and J∗ +1. +1 for t = T − 1, T − 2, . . . , 1 do +2 +for each slow state x ∈ X do +3 +for each fast state y ∈ Y do +4 +J∗ +t (x, y) = maxa r(x, y, a) + γ E +� +J∗ +t+1(x, fY(x, y, a, w)) +� +. +5 +˜π∗ +t (x, y) = arg maxa r(x, y, a) + γ E +� +J∗ +t+1(x, fY(x, y, a, w)) +� +. +6 +end +7 +end +8 end +9 for i = 1, 2, . . . , k do +10 +for s0 = (x0, y0) in the state space X × Y do +11 +V i(x0, y0, J∗ +1, ˜π∗) = maxa E +� ˜R(s0, a, J∗ +1) + γT V i−1(xT , yT , J∗ +1, ˜π∗) +� +. +12 +end +13 end +14 for s0 = (x0, y0) in the state space X × Y do +15 +˜µk(x0, y0) = arg maxa E +� ˜R(s0, a, J∗ +1) + γT V k(xT , yT , J∗ +1, ˜π∗) +� +. +16 end +20 + +An instance of FSVI is associated with two primary quantities: k, the number of VI iterations, +and T, the number of periods the slow state is frozen in the frozen-state approximation. The next +theorem makes use of this lemma to provide a bound on the regret of the policy obtained for a +particular k and T. +Theorem 6.1. Let (˜µk, ˜π∗) be the resulting T-periodic policy after running FSVI for k iterations. +The regret incurred when running (˜µk, ˜π∗) in the base model satisfies +R(˜µk, ˜π∗) ≤ +� +2γT +(1 − γT )2 + +2 +1 − γT +� +ϵr(γ, α, dY, L, T) ++ +� +2γ2T +(1 − γT )2 + +2γT +1 − γT +� +LU d(α, dY, T) + +2rmaxγ(k+1)T +(1 − γ)(1 − γT ), +where the last term, which depends on k, accounts for the error due to value iteration. +Proof. See Appendix C.3. +6.2 +Running Time of FSVI +It is well-known that each iteration of standard VI has time complexity O +� +|S|2|A| +� +, which provides +a contraction factor of γ (Littman et al., 1995). The upper level of FSVI, on the other hand, enjoys +an improved contraction factor γT with the same per-iteration running time of O +� +|S|2|A| +� +, given +that we pay a one-time fixed cost of solving the lower level. The O +� +|S|2|A| +� +consists of |S||A| due +to the number of state-action pairs at which to compute the Bellman update and another factor of +|S| due to the number of successor states. Since freezing slow states restricts the successor states to +Y, each iteration of the lower-level VI (Lines 2-7 of Algorithm 2) has running time O +� +|X||Y|2|A| +� +. +An additional O +� +|S|2 T +� +is required to compute the T-step transition probabilities of following ˜π∗, +to be used in the upper-level VI, resulting in a one-time fixed cost of O(|X||Y|2|A| T + |S|2 T). +Particularly when |X| is large, this can be a reasonable fixed cost to pay in order to get the much +improved discount factor of γT going forward (as we will show in the numerical results of Section +9). In Sections 7 and 8, we propose two extensions to FSVI that further reduce its computational +requirements. +21 + +7 +An Approximation for Nearly-Factored MDPs +One potential drawback of Algorithm 2 is that solving the lower-level problem requires solving an +MDP for each slow state x ∈ X. In this section, we consider the situation where the reward function +satisfies a certain nearly-factored assumption. In such a case, it is possible to design an extension of +FSVI that solves the lower-level problem for a nominal slow state (or a small number of slow states) +and then leverage the nearly-factored structure to approximate the lower-level values at other slow +states. Such a scheme would lower the one-time, fixed computational cost (i.e., the effort required +to solve the lower-level problem) of applying FSVI. +Assumption 3 (Nearly-Factored Reward). A fast-slow MDP ⟨S, A, W, f, r, γ⟩ has a “nearly-factored” +reward if there exists functions g, h, and ζ > 0 such that: +|g(x) + h(y, a) − r(x, y, a)| ≤ ζ, +for all x ∈ X, y ∈ Y, a ∈ A. +In other words, the reward r is a sum of slow and fast components with error at most ζ. +This terminology comes from the notion factored MDPs, a commonly-studied type of weakly- +connected structure that notably assumes an additive reward function (see, e.g., Boutilier et al. +(2000) and Osband and Van Roy (2014)). Although the analysis in this paper is based on additive +separability of the reward function (i.e., r(x, y, a) ≈ g(x) + h(y, a)), it is easy to extend the analysis +to other types of separable rewards, such as r(x, y, a) ≈ ⟨g(x), h(y, a)⟩. +7.1 +The Nominal-State Approximation in the Lower Level +We now seek to reduce the amount of computation needed to solve the lower-level MDP in the case +where Assumption 3 holds. The two main ingredients of our approach are: +1. An approximation the lower-level (frozen-state) MDP by another MDP with reward function +exactly equal to g(x) + h(y, a), instead of r(s, a). We call it the separable approximation of +the lower-level MDP. +2. A solution ¯J1(x∗, y) to the separable approximation at a particular nominal state6 x∗ ∈ X. +6In this section, we discuss the case with a single nominal slow state x∗ for simplicity. An extension to multiple +22 + +For any other x ∈ X, we can use Assumption 3 as a basis to approximate the value at x using +only ¯J1(x∗, y), without the need to solve an MDP for slow state x. +Fix a nominal state x∗ ∈ X. The Bellman recursion for the separable approximation at x∗ is +analogous to (10): let ¯JT ≡ 0 and for t = 1, 2, . . . , T − 1, let +¯Jt(x∗, y) = max +a +g(x∗) + h(y, a) + γ E +� ¯Jt+1(x∗, y′) +� +, +(21) +where y′ = fY(x∗, y, a, w). Note that x∗ continues to be frozen and we have simply replaced the +reward r with g + h. Let ¯π be (T − 1)-period policy associated with (21). Since ¯Jt is only defined +for the slow state x∗, we need to extend it to x ̸= x∗. Let ∆g(x) = g(x) − g(x∗). Leveraging the +separability of the reward function, we propose +¯Jt(x, y) = +T−t−1 +� +i=0 +γi∆g(x) + ¯Jt(x∗, y), +(22) +where we account for the reward error by applying a correction (but we do not account for error in +the transitions from using x∗ instead of x; see Lemma 7.1 for a full analysis). Such an approximation +allows for solving the frozen-state MDP only for x∗ and using the result to approximate the value +for other slow states, dramatically reducing the amount of overhead when using FSVI. The Nominal +FSVI algorithm makes use of this idea and is introduced in Algorithm 3. +Lemma 7.1. Consider ¯Jt(x, y), as defined by (21) and (22), which is an approximation to the true +frozen-state value J∗ +t (x, y), as defined in (10). Under Assumption 3, it holds that: +�� ¯Jt(x, y) − J∗ +t (˜x, ˜y) +�� ≤ +�T−t−1 +� +i=0 +γi +�� +ζ + Lr ∥x − ˜x∥2 +� ++ +�T−t−1 +� +i=0 +γiLi +f +� +Lr∥y − ˜y∥2 ++ +�T−t−1 +� +i=1 +Li +f +T−t−1 +� +j=i +γj +� +Lr ∥x∗ − ˜x∥2. +Proof. See Appendix D.1. +The next proposition is a simple consequence of Lemma 7.1. +nominal slow states is straightforward. +23 + +Algorithm 3: Nominal FSVI +Input: A nominal state x∗, initial values ¯JT ≡ 0 and V 0, number of iterations k. +Output: Approximation of the T-periodic frozen-state policy (¯µk +nom, ¯πnom) and ¯J1. +1 for t = T − 1, T − 2, . . . , 1 do +2 +for each fast state y ∈ Y do +3 +¯Jt(x∗, y) = maxa g(x∗) + h(y, a) + γ E +� ¯Jt+1(x∗, fY(x∗, y, a, w)) +� +. +4 +end +5 end +6 Define ¯Jt(x, y) using (22) and let ¯πnom be greedy with respect to ¯Jt. +7 for i = 1, 2, . . . , k do +8 +for s0 = (x0, y0) in the state space X × Y do +9 +V i(x0, y0, ¯J1, ¯πnom) = maxa E +� ˜R(s0, a, ¯J1) + γT V i−1(xT , yT , ¯J1, ¯πnom) +� +. +10 +end +11 end +12 for s0 = (x0, y0) in the state space X × Y do +13 +¯µk +nom(x0, y0) = arg maxa E +� ˜R(s0, a, ¯J1) + γT V k(xT , yT , ¯J1, ¯πnom) +� +. +14 end +Proposition 7.1. Recall from (13) the definition of ˜R(s0, a, J1), the approximation of the T-period +reward function. Under Assumption 3, the error between using a nominal state approximation versus +fully optimizing the frozen-state lower level is: +��E +� ˜R(s0, a, J∗ +1) +� +− E +� ˜R(s0, a, ¯J1) +��� ≤ +T−1 +� +i=1 +γiζ + +�T−2 +� +i=1 +Li +f +T−1 +� +j=i+1 +γj +� +Lr max +x +∥x∗ − x∥2, +where J∗ +t is computed as in Algorithm 2 and ¯Jt is computed as in Algorithm 3. +Proof. See Appendix D.2. +Theorem 7.1. Let (¯µk +nom, ¯πnom) be the result after running Nominal FSVI for k iterations with +nominal state x∗. The regret incurred when running (¯µk +nom, ¯πnom) in the base model satisfies +R(¯µk +nom, ¯πnom) ≤ +� +2γT +(1 − γT )2 + +2 +1 − γT +� +ϵr,nom(γ, α, dY, L, T, ζ, x∗) ++ +� +2γ2T +(1 − γT )2 + +2γT +1 − γT +� +LU d(α, dY, T) + +2rmaxγ(k+1)T +(1 − γ)(1 − γT ), +24 + +where the reward error is given by +ϵr,nom(γ, α,dY, L, T, ζ, x∗) += ϵr(γ, α, dY, L, T) + +T−1 +� +i=1 +γiζ + +�T−2 +� +i=1 +Li +f +T−1 +� +j=i+1 +γj +� +Lr max +x +∥x∗ − x∥2. +Proof. The proof is similar to the proof of Theorem 5.1, except we need to compute the ϵr(π∗, ¯J1) +term of Lemma 5.1. Combining Proposition 7.1 with the result in Proposition 4.1, we can see that +ϵr,nom(γ, α, dY, L, T, ζ, x∗) bounds ϵr(π∗, ¯J1). +8 +Value Function Approximation with a Linear Architecture +So far in this paper, we have proposed a frozen-state approximation that is able to exploit a cer- +tain fast-slow problem structure. We then proposed another level of approximation using nominal +slow states, which is valid when the MDP has a nearly-factored reward function. Both of these +approximations use a tabular value function representation and therefore, each iteration of VI in +the upper level requires looping through the entire state space X × Y (although the nominal-state +approximation does reduce the computational burden in the lower-level problem). +In this section, we explore the use of a linear architecture for a more compact representation +of the value function. We show how approximate VI (AVI) can be combined with the frozen-state +approximation, resulting in an algorithm that can scale to fast-slow MDPs with much larger state +spaces. The form of AVI that we use is based on the technique first proposed in Tsitsiklis and +Van Roy (1996) and later also used in Zanette et al. (2019). A technical contribution we make here +is to prove error bounds when this approximation architecture is used in a hierarchical setting that +combines finite-horizon and infinite-horizon components (i.e., our frozen-state VI). +8.1 +The Approximation Architecture +Let φ(s) = +� +φ1(s), φ2(s), . . . , φM(s) +�⊺ ∈ RM be an M-dimensional feature vector evaluated at state +s ∈ S. An approximation { ˆJt(ωt)}T +t=1 of the lower-level value functions {J∗ +t }T +t=1 of the frozen- +state approximation is given by a sequence of parameter vectors {ωt}T +t=1 with ωt ∈ RM, where the +25 + +component of ˆJt(ωt) associated with s is given by +ˆJt(s, ωt) = φ⊺(s) ωt, +for t = 1, 2, . . . , T. +(Note that since J∗ +T ≡ 0, we can set ωT = 0.) The approximation ˆV (β) to the upper-level value +function (of the frozen-state model) V ∗(J∗ +1, ˜π∗) is given by a parameter vector β ∈ RM, where +ˆV (s, β) = φ⊺(s) β. +For simplicity, we have used the same features in both the upper and lower levels. +It is well-known that naive specifications of approximate value iteration applied to linear architec- +tures can produce divergent behavior (Bertsekas and Tsitsiklis, 1996). To circumvent this potential +issue, our algorithmic approach depends on a set of pre-selected states ˜S = {s1, s2, . . . , sM}, an +idea popularized in Tsitsiklis and Van Roy (1996), who showed that if certain assumptions on these +states and the feature vectors are satisfied, then divergence is avoided. A similar algorithm is also +described more recently in Zanette et al. (2019). +In this section, we need an ordering of the state space, so without loss of generality, we assume +that S = {1, 2, . . . , N} and that the first M are the pre-selected states, i.e., sm = m for m = +1, 2, . . . , M. We make the following assumption on the feature vectors; this is essentially Assumption +2 of Tsitsiklis and Van Roy (1996), adapted to our setting. +Assumption 4. Let ˜S = {s1, s2, . . . , sM} be a set of pre-selected anchor states. Suppose the fol- +lowing conditions on the features φ are satisfied. +1. The vectors φ(s1), φ(s2), . . . , φ(sM) are linearly independent. +2. There exists some γ′ ∈ [γ, 1) such that for any state s ∈ S, there are coefficients θm(s) ∈ R +for m = 1, 2, . . . , M satisfying: +M +� +m=1 +|θm(s)| ≤ 1 +and +φ(s) = γ′ +γ +M +� +m=1 +θm(s) φ(sm). +The interpretation of this assumption is that the feature space {φ(s) | s ∈ S} lies in the convex hull +26 + +of the points defined by the pre-selected states: +� +±(γ′/γ) φ(sm) +�M +m=1. To reduce notional clutter, we +will define κ = γ′/γ to be the amplification factor induced by the features. +Following Tsitsiklis and Van Roy (1996), we let Φ ∈ RN×M be a matrix with the s-th row equal +to φ⊺(s) and let L ∈ RM×M be a matrix with the m-th row equal to φ⊺(sm). If we let G be the +remaining rows of Φ, then we see that Φ = +� +L; G +� +. Next, by Assumption 4, the matrix L has a +unique matrix inverse L−1 ∈ RM×M. We define Φ† ∈ RM×N as follows: for m ∈ {1, 2, . . . , M}, +suppose the m-th column of Φ† is equal to the m-th column of L−1, and let all the other entries of +Φ† be zero. In other words, Φ† = [L−1 0]. Therefore, we see that Φ† is a left inverse of Φ: +Φ†Φ = [L−1 0] +� +�� +L +G +� +�� = L−1L = I, +where I ∈ RM×M is the identity matrix. +8.2 +Frozen-State Approximate Value Iteration +Recall the lower-level Bellman operator ¯H from (12) and the upper-level Bellman operator F ˆJ1,ˆπ de- +fined in (15). The high-level idea behind our new approach, frozen-state approximate value iteration +(FSAVI) is as follows: +• Lower-level AVI. We first run approximate value iteration (under basis functions Φ) for +the lower-level problem. +Letting ω∗ +T = 0, the parameter ω∗ +t is estimated by first evalu- +ating ¯H ˆJt+1(ω∗ +t+1) at the pre-selected states, and then computing ω∗ +t so that ˆJt(s, ω∗ +t ) = +� ¯H ˆJt+1(ω∗ +t+1) +� +(s) for s ∈ ˜S. +• Upper-level AVI. Suppose that after solving the lower level, we have parameter vectors +ω∗ = (ω∗ +1, ω∗ +2, . . . , ω∗ +T ), implying lower-level value functions ˆJt(ω∗ +t ) = Φω∗ +t and an associated +greedy policy ˆπ(ω∗) = (ˆπ1(ω∗), . . . , ˆπT−1(ω∗)): +ˆπt(x, y, ω∗) = arg max +a +r(x, y, a) + γ E +� ˆJt+1(x, y′, ω∗ +t+1) +� +. +(23) +For the upper level, the parameter βk is updated to βk+1 in iteration k + 1 by first evaluating +F ˆJ1(ω∗ +1),ˆπ(ω∗) ˆV (βk) at the pre-selected states, then computing βk+1 so that ˆV (s, βk+1) = +27 + +� +F ˆJ1(ω∗ +1),ˆπ(ω∗) ˆV (βk) +� +(s) for s ∈ ˜S. Note that, taking Φ as fixed, the dependence of the upper +level on the lower level can be represented succinctly through ω∗. Therefore, we will use the +simplified notation Fω := F ˆJ1(ω1),ˆπ(ω) going forward. +To start, we define two new Bellman operators for the parameter space: +¯H′ = Φ† ◦ ¯H ◦ Φ +and +F ′ +ω = Φ† ◦ Fω ◦ Φ. +To understand the definition of H′, consider the lower level. Suppose we start with a parameter +vector ω∗ +t+1, representing an approximate value function at time period t+1 given by ˆJt+1(ω∗ +t+1) = +Φω∗ +t+1. The update to the next parameter vector ω∗ +t is obtained by applying ¯H to ˆJt+1(ω∗ +t+1), as +we would normally do, and then applying Φ† to project back to the parameter space. For the upper +level, a similar logic holds to go from βi to βi+1: we first have the approximate upper-level value +function ˆV (βi) = Φβk and then apply the normal Bellman update Fω∗, before lastly obtaining the +updated parameter βi+1 using Φ†. Therefore, we have +ω∗ +t = ¯H′(ω∗ +t+1) +and +βi+1 = F ′ +ω∗(βi). +We show in the Lemma E.5 of the Appendix that F ′ +ω∗ is an (κγT )-contraction in the norm ∥ · ∥Φ on +RM defined by ∥β∥Φ = ∥Φβ∥∞ and therefore has a fixed point β∗. We now define two quantities +related to the approximation error of the linear architecture. +Definition 3. Define the linear architecture approximation error for the lower level as +εlow = maxt∈{1,2,...,T} infωt∈RM +��J∗ +t − ˆJt(ωt) +�� +∞. +(24) +Let V ∗ +ω be the fixed point of Fω. For the upper level, we define error ϵup as +εup = supω infβ∈RM +��V ∗ +ω − ˆV (β) +�� +∞. +(25) +Both (24) and (25) are related to the approximation errors defined in Tsitsiklis and Van Roy (1996); +moreover, taking a uniform bound over quantities that need to be approximated resembles the errors +defined in Munos and Szepesvári (2008). +28 + +Algorithm 4: Frozen-State Approximate Value Iteration (FSAVI) +Input: ˜S = {s1, s2, . . . , sM}, φ, initial weights ωT = β0 = 0, number of iterations k. +Output: Approximation of the T-periodic frozen-state policy +� +ˆµ(βk, ω∗), ˆπω∗� +and ˆJ1(ω∗) +1 for t = T − 1, T − 2, . . . , 1 do +2 +for each pre-selected state s = (x, y) ∈ ˜S do +3 +Jt(x, y) = maxa r(x, y, a) + γ E +� ˆJt+1(x, fY(x, y, a, w), ωt+1) +� +. +4 +end +5 +Set remaining entries of Jt to zero. Update parameter vector: ω∗ +t = Φ†Jt. +6 end +7 Let ˆπω∗ be greedy with respect to ˆJt(ω∗ +t ) = Φω∗ +t , similar to (23). +8 for i = 1, 2, . . . , k do +9 +for each pre-selected state s0 ∈ ˜S do +10 +V i(s0) = maxa E +� ˜R(s, a, ˆJ1(ω∗ +1)) + γT ˆV (sT (a, ˜πavi), βi−1) +� +. +11 +Set remaining entries of V i to zero. Update parameter vector: βi = Φ† V i. +12 +end +13 end +14 for s0 in the state space S do +15 +ˆµ(βk, ω∗)(s0) = arg maxa E +� ˜R(s0, a, ˆJ1(ω∗ +1)) + γT ˆV (sT (a, ˜πω∗), βk) +� +. +16 end +Theorem 8.1. Let (ˆµ(βk, ω∗), ˆπω∗) be the result after running FSAVI for k iterations for a given ˜S +and φ. The regret incurred when running (ˆµ(βk, ω∗), ˆπω∗) in the base model satisfies +R +� +ˆµ(βk, ω∗), ˆπω∗� +≤ +� +2γT +(1 − γT )2 + +2 +1 − γT +� +ϵr,avi(γ, α, dY, L, T, γ′, εlow) ++ +� +2γ2T +(1 − γT )2 + +2γT +1 − γT +� +LU d(α, dY, T) + +� 1 + κ +1 − κγT +� +εup + (κγT )k +� κ2 − κ2(κγ)T+1 +(1 − κγT )(1 − κγ) +� +rmax, +where the reward error is given by +ϵr,avi(γ, α, dY, L, T, γ′, εlow) = ϵr(γ, α, dY, L, T) + +� 1 + κ +1 − κγ − (κγ)T (1 + γ) +γ − κγ2 +� +εlow. +Proof. See Appendix E.2. +29 + +9 +Numerical Experiments +We now apply our algorithms to three applied examples: (1) dynamic service allocation for a multi- +class queue, (2) restless multi-armed bandit for asset maintenance optimization, and (3) energy +demand response. For the service allocation and asset maintenance problems, which have relatively +smaller state spaces, we compare the VI-based variants: Base VI, Slow-agnostic VI, Q-learning, +FSVI, and Nominal FSVI. For the energy demand response problem, which has a relatively larger +state space, we compare the feature-based variants: Base AVI, Slow-agnostic AVI, DQN, FSAVI, +and Nominal FSAVI. We study the performance of each method with respect to its running time. +Building off of the discussion in Section 6.2, we define the running time of each iteration of a given +method to be |Sobs||A||Ssucc|, where Sobs is the set of states at which we compute the Bellman +update and Ssucc is the set of successor states evaluated. +We now give more details about the +methods being compared. +• Base VI/AVI. We refer to Algorithm 1 as the “Base VI” approach, which is simply standard +VI applied to the base model with no changes (with discount factor γ). The “Base AVI” base- +line uses the VI with the linear architecture described in Section 8 directly on the base model. +More precisely, it iterates the approximate Bellman operator Φ† ◦ H ◦ Φ, where H is the Bell- +man operator for the base model defined in (6). For succinctness, we have omitted a detailed +algorithm specification. For Base VI, we use exact transition probabilities when computing +the expectation in the Bellman update. However, because Base AVI is used for problems with +larger state spaces, exact evaluation of the expectation is more difficult. Instead, we resort to +using a Monte Carlo-based sample average, with a sample of |Ssucc| = 40 successor states. +• Slow-agnostic VI/AVI. As we discussed in the introduction, simplified decision models that +ignore the slow state are often used in practice to improve tractability. To make this precise, +we propose the following slow-agnostic Bellman operator to test a particular instantiation of +this idea that averages over slow states and ignores it thereafter: +� +HfastW +� +(y) = max +a∈A |X|−1 � +x∈X +� +r(x, y, a) + γ E +� +W(fY(x, y, a, w) +�� +, +30 + +where W is a value function defined only over y ∈ Y. “Slow-agnostic VI” iterates the op- +erator Hfast, while “Slow-agnostic AVI” iterates Φ† ◦ Hfast ◦ Φ. For the AVI version, we use +|Ssucc| = 20 successor state samples to approximate the expectation (this is smaller than in +Base AVI because we only need to sample over Y, rather than X ×Y). Upon implementation, +the policy ignores the slow state and only uses the value of y to take actions. +• Q-learning/Deep Q-networks. We also compare our approaches to the well-known rein- +forcement learning method, Q-learning (QL) (Watkins, 1989), along with its deep reinforce- +ment learning variant, Deep Q-networks (DQN) (Mnih et al., 2013, 2015). +• Ours: FSVI/Nominal FSVI (multiple). Next, we have our VI-based approaches based +on the frozen-state approximation. The FSVI and Nominal FSVI methods are described in +detail in Algorithms 2 and 3. “Nominal FSVI (multiple)” refers to an extension to multiple +nominal states (as mentioned in Section 7). The extension uses 9 nominal states for the service +allocation problem (3 in each dimension of the 2-dimensional slow state) and 5 nominal states +for the other two problems. The nominal states are equally spaced within the bounds of each +slow state dimension. To apply Line 6 of Algorithm 3 (Nominal FSVI), we select the nearest +nominal state by Euclidean distance to (x, y). For both methods, we freeze slow states for +T = 10 periods. +• Ours: FSAVI/Nominal FSAVI (multiple) Lastly, we have our AVI-based frozen-state +methods, which also freeze slow states for T = 10 periods. FSAVI is described in Algorithm +4 and Nominal FSAVI (multiple) is the natural combination of Nominal FSVI (multiple) and +FSAVI; the computational benefit here is that since x∗ is fixed, the lower-level linear ap- +proximation only needs to be performed over y rather than (x, y). For both AVI methods, +we use M = 0.3 |S| pre-selected states with Gaussian radial basis functions for φ.7 We use +|Ssucc| = 250 successor state samples to approximate the expectation, spread over 25 simulated +sample paths of the lower-level policy of length T = 10.8 +7The M basis functions operate on normalized state space (with each state variable normalized to [0, 1]), with their +centers spaced evenly and width equal to 0.02. +8Note that each iteration of the upper level of FSAVI is more computationally intensive than the upper level of Base +AVI due to the need for simulating the lower level policy. We account for this accurately in the computational cost +calculation to provide a fair comparison. +31 + +To evaluate policies, we use a truncated horizon of 100 periods (10T) and each method is +evaluated over 10 independent runs. In order to create a more fine-grained performance plot, in +each Bellman update of the VI-based variants, we allow evaluating the performance of policies “in +between” complete iterations of VI, when the Bellman updates have only been executed for a subset +of states in the state space. For each run, the order of state updates is randomly shuffled. The +AVI-based variants, however, require all pre-selected states to be observed before the parameter +vector can be updated; hence, we plot the performance of the policy after each full AVI iteration is +complete. +0 +1 +2 +3 +4 +5 +Computational Cost +1e5 +60 +55 +50 +45 +40 +Test Reward +QL +Base VI +Slow-agnostic VI +FSVI +Nominal FSVI +(a) Multi-class service allocation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Computational Cost +1e5 +8 +10 +12 +14 +16 +18 +Test Reward +QL +Base VI +Slow-agnostic VI +FSVI +Nominal FSVI +(b) Restless two-armed bandit +0 +1 +2 +3 +4 +5 +Computational Cost +1e7 +1.1 +1.2 +1.3 +1.4 +Test Reward +1e4 +DQN +Base AVI +Slow-agnostic AVI +FSAVI +Nominal FSAVI +(c) Energy demand response +Figure 4: Performance of each algorithm on the three example applications: (a) multi-class service allocation +and queueing, (b) restless two-armed bandit for asset maintenance optimization, and (c) energy demand +response. The solid lines show median performance and the error bars represent the 10th-90th percentiles +across 10 random seeds. The x-axis shows the computational cost, defined by |Sobs||A||Ssucc|. +9.1 +Multi-Class Service Allocation with Stochastic Holding Costs +We study a version of the multi-class service system problem based on the model presented in +Ansell et al. (2003) and later Brown and Haugh (2017), extended to the case of stochastic holding +32 + +costs. Many variations of the original problem (without stochastic costs) have been studied in the +literature; to name a few, see Cox et al. (1961), Harrison (1975), Van Mieghem (1995), and Gittins +(1979). Our variant with stochastic holding costs is partially motivated by the model in Lee and +Vojnovic (2021), which proposes a learning algorithm for job scheduling. The main idea behind this +example is that when the holding cost stochastic process evolves slowly, it becomes a reasonable +candidate for the slow state in our frozen-state framework. +We consider a single server and two classes of customers. For class j ∈ {1, 2}, the arrival rate is +µj9, service rate is λj, and the queue capacity is Qj. Let yt,j be the number of customers in queue +j and let zt be the class of the customer that is currently being served (if zt = 0, this represents +when there is no customer and hence, no decision to be made). Assume λ0 = 0. The holding cost +of queue j, applied to each customer in the queue, is represented by an exogenous Markov process +{xt,j}t. The dynamics of the system obey the following: +1. With probability µj, a class zt = j customer arrives and yt+1,zt = min(yt,zt + 1, Qzt). +2. With probability λzt, the server completes serving the current customer and the queue length +transitions according to yt+1,zt = yt,zt − 1. +3. With probability 1 − λzt − � +j µt,j, no event happens and yt+1,j = yt,j for all j. +Each time the server completes serving the current customer, an action at ∈ {0, 1, 2} is taken +to decide the class of customer to be served next, with at = 0 representing the case where +no customer needs to be served. +The reward function is simply the negative of the total cost: +r(xt,1, xt,2, yt,1, yt,2, at) = − �2 +j=1 xt,jyt,j. For the frozen-state approximation, we let the slow state +be (xt,1, xt,2) and the fast state be (yt,1, yt,2, zt). +For the nominal state approximation, we take the following approach. +We solve the lower- +level problem by letting ¯Jt(x1, x2, y1, y2, z) = ¯Jt,1(x1, y1, z) + ¯Jt,2(x2, y2, z), where ¯Jt,j(xj, yj, z) +represents the reward of the optimal frozen-state policy associated with queue j. +The value of +¯Jt,j(xj, yj, z) is computed by setting ¯JT,j(xj, yj, z) = −xjyj and for other t < T, ¯Jt,j(xj, yj, z) = +9We abuse notation here by reusing µ, which was previously used in the paper to denote the upper-level policy. +33 + +rj(xj, yj, a∗) + γ E +� ¯Jt+1,j(xj, y′ +j, z′) +� +, where rj(xj, yj, a) = −xjyj, (y′ +1, y′ +2, z′) = fY(s, a, w), and +a∗ = arg max +a∈A +r(x1, x2, y1, y2, a) + γE +� ¯Jt+1(x1, x2, y′ +1, y′ +2, z′) +� +. +We then use a multiplicative decomposition rj(xj, yj, a) = gj(xj)hj(yj) with gj(xj) = −xj and +hj(yj) = yj, and apply a multiplicative correction gj(xj)/gj(x∗ +j) to get ¯Jt,j(xj, yj, z) from ¯Jt,j(x∗ +j, yj, z). +To make the results more easily interpretable as a function of the holding cost, we consider a +case where the two classes have the same arrival rates, service rates, and capacities: µ1 = µ2 = 0.2, +λ1 = λ2 = 0.3, and Q1 = Q2 = 3. For the cost process, let {ai}6 +i=1 be six equally spaced values +from the interval [0.01, 0.2]. Our cost process transitions on the set {ai}6 +i=1 such that if xt,j = ai, +then xt+1,j = xt,j with probability 0.9, xt+1,j = a(i−1)∨1 with probability 0.05, and xt+1,j = a(i+1)∧n +with probability 0.05. +9.1.1 +Results and Discussion for Service Allocation +Figure 4a shows the performance of the algorithms. As a function of the computational effort, +Nominal FSVI and FSVI quickly outperform the other baselines. Base VI and Q-learning converge +more slowly, but eventually they find policies with decent (but not superior) performance. Not +surprisingly, Slow-agnostic VI plateaus quickly. +These results illustrate that for the multi-class +service allocation problem, although the slow state is important enough that it should not be +ignored, there are drastic computational benefits of applying the frozen-state idea. +Figure 5 provides a qualitative comparison of the various policies by visualizing the actions +taken in two situations: when the cost of queue 1 is lower and when the cost of queue 2 is lower. +There are a few main takeaways. First, the upper level policies, along with the first 8 periods of +the lower-level policies, of FSVI and Nominal FSVI resemble the Base VI policy: that is, they tend +to serve customers in currently high cost queue, as long as that queue is nonempty. +Second, we observe deficiencies in the Slow-agnostic VI and Q-learning policies. By ignoring the +slow state, Slow-agnostic VI’s policy serves the nonempty queue when the other queue is empty, +and tends to serve the shorter queue when both queues are nonempty. +Q-learning also finds a +suboptimal policy within the computational budget, tending to focus on the longer queue (but not +the higher cost queue) in many cases. +34 + +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(a) Base VI +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(b) Slow-agnostic VI +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(c) Q-learning +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(d) FSVI, upper +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(e) FSVI, lower, t = 8 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(f) FSVI, lower, t = 9 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(g) Nominal FSVI, upper +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(h) Nominal FSVI, lower, t = 8 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 < xt, 2 +0 +1 +2 +3 +Queue 1 +0 +1 +2 +3 +Queue 2 +xt, 1 > xt, 2 +(i) Nominal FSVI, lower, t = 9 +Figure 5: Visualization of the policies learned by all methods for the multi-class service allocation and +queueing problem. In each plot, the x- and y-axes represent the length of the first and the second queues, +respectively. A red square indicates the policy choosing to serve customers in queue 1 for the majority of +runs (replications), while a blue square indicates the policy choosing to serving customers in queue 2 more +often than not. The shade of the color represents the frequency of taking that action over 10 runs. For each +policy, we show two situations: when the holding cost of queue 1 is lower (xt,1 ≤ xt,2, where we expect the +optimal policy to primarily serve queue 2), and when the holding cost of queue 2 is lower (xt,1 > xt,2, where +we expect the optimal policy to primarily serve queue 1). Note that the first row shows methods that learn +stationary policies, while the second and third rows show various snapshots of the non-stationary policies +learned by FSVI and Nominal FSVI. +Third, we see that in the final lower-level period (t = 9), FSVI and Nominal FSVI learn poor +policies due to the “end-of-horizon effect” of the zero terminal value approximation in the finite- +horizon problem in the lower level. When both queues are nonempty, regardless of which queue is +served, there will be no downstream impact (and the queue will not shorten). Although the frozen +state approximation introduces suboptimal decisions in some periods, we can see that the good +actions taken in the upper level and early stages of the lower level outweigh this issue. +35 + +9.2 +Restless Multi-Armed Bandit with Environmental States +We now move on to the case of a restless multi-armed bandit (Whittle, 1988; Weber and Weiss, +1990; Killian et al., 2021; Zhang and Frazier, 2021). +This problem arises in a wide variety of +settings, including from machine maintenance (Smallwood and Sondik, 1973; Duan et al., 2018; +Ruiz-Hernández et al., 2020), dynamic assortment planning (Brown and Smith, 2020), public health +intervention decisions (Bhattacharya, 2018; Mate et al., 2020), and preventative healthcare (Lee +et al., 2019; Biswas et al., 2021). +We construct a two-armed instance that features an exogenous environmental context, inspired +by maintenance problems, to illustrate our frozen-state algorithms. Suppose there are two arms +j ∈ {1, 2} and each arm can either be operational (yt,j = 1) or non-operational (yt,j = 0). At +the end of each period, the operator chooses whether each arm should receive an intervention (i.e., +maintain or repair): at = (at,1, at,2), with each at,j ∈ {0, 1}. Each intervention incurs a cost of +1. The state yt+1,j of arm j in the next period depends on three factors: its current state yt,j, +whether it is maintained at, and the exogenous environment state that describes the condition of +the overall system xt. We consider 25 possible values for the environment state: xt ∈ {0, 1, . . . , 24}), +with xt+1 equal to xt + 2, xt + 1, xt, xt − 1 and xt − 2 with probabilities 0.05, 0.15, 0.6, 0.15, and +0.05 respectively. Lower values of xt increase the probability of of an arm becoming (and staying) +non-operational; the precise values of the transition probabilities are described in Figure 6. The +immediate reward function is r(xt, yt,1, yt,2, at) = 2 � +j yt,j −� +j at,i. We view the environment state +xt as the slow state. We use the additive nominal state approximation proposed in Section 7, which +trivially applies here because the reward function does not depend on the slow state. +9.2.1 +Results and Discussion for Restless Bandit +Figure 4b shows the performance of the algorithms as a function of the computational cost. The +trends are similar to what we observed in Figure 4a, except here we see some notable instability of +the Slow-agnostic policy. A likely explanation is given at the end of this subsection. +We use Figure 7 to visualize the final policies learned by each method. The x-axes represent the +environment state, while the y-axis represents the operating state of both arms (machines). Darker +gray squares on the left (right) panels indicate a high frequency of intervening arm 1 (arm 2). We +36 + +1 +0 +0.7 +0.01 +0.99 +0.3 +1 +0 +0.2 +0.5 +0.5 +0.8 +at = 0 +at = 1 +xt = 0 +1 +0 +0.1 +0.05 +0.95 +0.9 +1 +0 +0.01 +0.99 +0.01 +0.99 +at = 0 +at = 1 +xt = 24 +⋯ +improving environment state +Figure 6: The transition probabilities for both arms in the two extreme environment states. When the +environment is in the poorest condition xt = 0, each arm stays in the non-operational state (yt,j = 0 to +yt+1,j = 0) with probabilities 0.99 (no intervention, at = 0) and 0.5 (with intervention, at = 1). They go +from operational to non-operational (yt,j = 1 to yt+1,j = 0) with probabilities 0.7 (no intervention, at = 0) +and 0.2 (with intervention, at = 1). When the environment is in the best condition xt = 24, the same +probabilities are 0.95, 0.01, 0.1, and 0.01. For other values of the environment xt, the probabilities are +equally spaced between the two extreme conditions. +observe that although the Base VI policy is not particularly stable across the 10 runs, Base VI, +FSVI, and Nominal FSVI all learn a policy with similar structure: intervene non-operating arms +and always intervene if the environment state is smaller than 5. Given the performance plot in +Figure 4b, this type of structure results in high performing policies. In addition, we observe that +the frozen-state variants are significantly more stable across runs. +The Slow-agnostic policy learns to focus on non-operating arms, but its inability to distinguish +between slow states hurts its performance. To understand the unstable behavior observed in Fig- +ure 4b, note that Slow-agnostic VI is only able to produce policies that apply the same action to +entire rows of the grid. Considering what a “good” policy looks like (i.e., the Base VI, FSVI, and +Nominal FSVI plots), it becomes clear that different Slow-agnostic VI policies can have dramati- +cally different performance. For the sake of illustration, let us suppose that the FSVI policy for +arm 1 is indeed optimal. Now, for Slow-agnostic VI, consider switching from the current policy that +intervenes in arm 1 for (0, 0) and (0, 1) to a policy that intervenes for (0, 0), (0, 1), and (1, 0) — +we suddenly go from having 5 states with suboptimal actions (xt ≤ 4 for yt = (1, 0)) to 20 states +37 + +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 1 +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 2 +(a) Base VI +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 1 +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 2 +(b) Slow-agnostic VI +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 1 +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 2 +(c) Q-learning +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 1 +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 2 +(d) FSVI, upper and FSVI, lower t = 5 +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 1 +0 +5 +10 +15 +(0,0) +(0,1) +(1,0) +(1,1) +Machine 2 +(e) Nominal FSVI, upper and Nominal FSVI, lower t = 5 +Figure 7: Visualization of the resulting policies across methods for the restless bandit. In each subplot, the +x-axis is the environment state and the y-axis show the operating state of each arm (machine). The left +panel shows whether to intervene machine 1, while the right panel shows whether to intervene machine 2. +Darker gray squares indicate a high frequency of intervention across the 10 runs (replications) of the method. +Note that in the last two subplots, the policy shown is the same policy for both the upper level and lower +level, period t = 5. +with suboptimal actions (xt ≥ 5 for yt = (1, 0)). In other words, the lack of flexibility of the policy +space can cause widely varying performance across policies. +Finally, we note that Q-learning does not seem to have learned a policy with any notable +structure, except that it more likely to intervene when both arms are in the non-operational state. +38 + +9.3 +Energy Demand Response +For our last example application, we consider the problem of an energy aggregator that provides +demand response to consumers, while simultaneously selling the demand reduction to the demand +response market, inspired by Khezeli et al. (2017). At the beginning of each period t, the aggregator +commits to delivering an amount of energy at in the real-time (RT) market using a forward contract, +settled at the day-ahead (DA) price xt. +The DA price process follows a discretized Ornstein- +Uhlenbeck process xt+1 − xt = c1(c2 − xt) + ϵda +t+1, where c1 = 0.2237, c2 = 21.4095, estimated using +data from the California Independent System Operator (CAISO). +To complete the promised delivery, the aggregator uses dynamic pricing and offers payment to +elicit demand reduction, or demand response, from its customers. Following the model of Khezeli +et al. (2017), if the demand reduction (which can be considered equivalent to delivering energy) falls +short of the forward contract at, the aggregator purchases the remaining energy at the RT shortage +price p− +t , and if the demand reduction exceeds at, the aggregator sells the additional energy at the +RT overage price p+ +t . We model p− +t and p+ +t using multiplicative adjustments to the DA price xt, +i.e., p− +t = xty− +t and p+ +t = xty+ +t , with yt+ < 1 < y− +t . +We consider a system with two (large) customers (e.g., a university or company) m ∈ {1, 2}. The +demand response is a function of the compensation provided by the aggregator. For convenience, we +represent the offered compensation as a fraction of the the DA price, i.e., qt,m = αt,m xt, where αt,m ∈ +{0.1, 0.275, 0.45, 0.625, 0.8} for each m. The vector αt = (αt,1, αt,2, . . . , αt,m) represents the pricing +decisions made by the aggregator. The demand response follows a linear model dm(xt, αt,m) = +bm,1 + bm,2 (αt,mxt) + ϵdr +t+1,m, where ϵdr +t+1,m is the noise and bm,1, bm,2 are known coefficients. The +state of the system is st = (xt, y− +t , y+ +t ). The reward function is +r(xt, y+ +t ,y− +t , at, αt) = xtat − +2 +� +m=1 +qt,m E +� +dm(xt, αt,m) +� ++ E +� +xty+ +t +� +2 +� +m=1 +dm(xt, αt,m) − at +�+ +− xty− +t +� +at − +2 +� +m=1 +dm(xt, αt,m) +�+� +. +The DA price process xt is rounded/clipped to values in 0.1 increments between 10 and 30, +and its noise term ϵda +t+1 follows discretized normal distribution with standard deviation 1. The RT +adjustment factors y− +t and y+ +t are each uniformly distributed over 10 equally spaced discrete values +39 + +in the ranges [0.5, 0.8] and [1.05, 1.25], respectively. The possible values of the pricing decisions +at,m are limited to the set {10, 12, . . . , 20} in our experiment. Finally, for the demand response +model, we set b1,1 = 1.72, b1,2 = 0.55, b2,1 = 5.87, b2,2 = 0.26, with ϵdr +t+1,m follows discretized normal +distribution with standard deviation 0.5. With this particular set of coefficients, the impact of an +additional unit of compensation offered is larger for customer 1 than for customer 2, but maximum +expected demand response of customer 2 is larger than that of customer 1. For the nominal state +approximation, we use a multiplicative decomposition, where the reward function is approximated +as g(x∗ +t ) h(x∗ +t , y+ +t , y− +t , at, αt) with g(x∗ +t ) = x∗ +t and +h(x∗ +t ,y+ +t , y− +t , at, αt) = at − +2 +� +m=1 +αt,m E +� +dm(x∗ +t , αt,m) +� ++ E +� +y+ +t +�� +m +dm(x∗ +t , αt,m) − at +�+ +− y− +t +� +at − +2 +� +m=1 +dm(x∗ +t , αt,m) +�+� +. +Instead of the additive correction term used in (22), we apply a multiplicative correction g(x)/g(x∗). +1200 +1400 +1600 +1800 +0 +50 +100 +150 +(a) Base AVI +1200 +1400 +1600 +1800 +0 +50 +100 +150 +(b) FSAVI +1200 +1400 +1600 +1800 +0 +50 +100 +150 +(c) Nominal FSAVI +1200 +1400 +1600 +1800 +0 +50 +100 +150 +(d) Slow-agnostic AVI +1200 +1400 +1600 +1800 +0 +50 +100 +150 +(e) DQN +Figure 8: Histograms of the cumulative amount bid by the aggregator over 100 periods (x-axis) for each +method. The y-axis are counts over 1000 simulations of the resulting policy, and the dotted vertical line +shows the mean. +40 + +0.4 +0.6 +0 +50 +100 +150 +Customer 1 +Customer 2 +(a) Base AVI +0.4 +0.6 +0 +50 +100 +150 +Customer 1 +Customer 2 +(b) FSAVI +0.4 +0.6 +0 +50 +100 +150 +Customer 1 +Customer 2 +(c) Nominal FSAVI +0.4 +0.6 +0 +50 +100 +150 +Customer 1 +Customer 2 +(d) Slow-agnostic AVI +0.4 +0.6 +0 +50 +100 +150 +Customer 1 +Customer 2 +(e) DQN +Figure 9: Histograms showing the proportion of the total amount bid that is satisfied by each customer +(x-axis) over 1000 simulations of 100 periods each. The y-axis are counts over 1000 simulations, and the +dotted vertical lines show the means for the customers. This is essentially an illustration of the aggregator’s +pricing behavior. +9.3.1 +Results and Discussion for Demand Response +The performance comparisons for the demand response problem are shown in Figure 4c. The results +confirm that the trends observed for the VI-based methods in Figures 4a and 4b hold up in the +AVI setting. The new methods, FSAVI and Nominal FSAVI continue to outperform the others. +Base AVI and DQN both improve slowly, but continually. Slow-agnostic AVI, however, plateaus +and displays a small yet still noticeable oscillation. +For a qualitative understanding of the differences between the policies, we show in Figure 8 +histograms of the aggregator’s cumulative bids. Base AVI, FSAVI and Nominal FSAVI show similar +bimodal bidding behavior and mean values, while the bidding behaviors of Slow-agnostic AVI and +DQN deviate noticeably. +The former has a much narrower distribution, while the latter shows +more uniform behavior. Figure 9 shows histograms of the proportions of the overall amount bid +that is satisfied by each customer. Once again, we observe Base AVI, FSAVI, and Nominal FSAVI +consolidating around a particular pricing behavior, with average amount satisfied by customer 1 +being slightly higher (DQN exhibits the reverse). +41 + +10 +Conclusion +In this paper, we studied a new class of MDPs with a type of structure called fast-slow structure, +motivated by practical applications where some states move slowly and are relatively less influential +than others, but still important enough not to ignore them during modeling. Based on this structure, +we propose a set of new algorithms based on the idea of freezing the slow state for several periods at a +time to ease the computational burden of approximate dynamic programming. 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Restless bandits with many arms: Beating the central limit theorem. arXiv +preprint arXiv:2107.11911, 2021. +47 + +Z. Zhou, Z. Lan, W. Tang, and N. Desai. Reducing energy costs for ibm blue gene/p via power-aware job +scheduling. In Workshop on Job Scheduling Strategies for Parallel Processing, pages 96–115. Springer, +2013. +C. Zhu, J. Zhou, W. Wu, and L. Mo. Hydropower portfolios management via Markov decision process. In +IECON 2006-32nd Annual Conference on IEEE Industrial Electronics, pages 2883–2888. IEEE, 2006. +48 + +A +Proofs from Sections 3 and 4 +A.1 +Technical Lemmas +Lemma A.1. Consider a (α, dY)-fast-slow MDP. For any states (x0, y0) and (˜x0, ˜y0), let (xt, yt) +and (˜xt, ˜yt) be the states reached after t transitions under a policy π = (π0, . . . , πt−1), i.e., (xt, yt) = +fπ(xt−1, yt−1, wt) and (˜xt, ˜yt) = fπ(˜xt−1, ˜yt−1, ˜wt). Then, for any policy π, we have +(i) ∥xt − ˜x0∥2 ≤ tαdY + ∥x0 − ˜x0∥2, +(ii) ∥xt − ˜xt∥2 ≤ 2tαdY + ∥x0 − ˜x0∥2, +(iii) ∥yt − ˜yt∥2 ≤ 2tdY + ∥y0 − ˜y0∥2. +Proof. Lemma A.1 is a consequence of Assumption 1. +The following two lemmas are about properties of the Bellman operators H and ˜H (recall that +˜H is the frozen-state version). +Lemma A.2. For any state (x, y) and any two value functions V, V ′ : X × Y → R, we have +|( ˜HtV )(x, y) − ( ˜HtV ′)(x, y)| ≤ γt max +y∈Ytx +��V (x, y) − V ′(x, y) +��, +where Yt +s is the set of fast states reachable from s = (x, y) after t transitions of fY(x, ·, ·, ·). +Proof. The result follows by the contraction property of the Bellman operators. +Lemma A.3 (Discrepancy between H and ˜H). Consider a value function V : X ×Y → R. Suppose +there exists LV > 0 such that for any states (x, y) and (˜x, ˜y), it holds that |V (x, y) − V (˜x, ˜y)| ≤ +LV ∥(x, y) − (˜x, ˜y)∥2. Then, +��(HtV )(x, y) − ( ˜HtV )(˜x, ˜y) +�� +≤ +��(x, y) − (˜x, ˜y) +�� +2 +� +Lr +t−1 +� +i=0 +(γLf)i + LV (γLf)t +� ++ αdY +� +Lr +t−1 +� +i=1 +γi +i−1 +� +j=0 +Lj +f + LV γt +t−1 +� +j=0 +Lj +f +� +. +49 + +Proof. We need to show that for each t ≥ 1, +��(HtV )(x, y) − ( ˜HtV )(˜x, ˜y) +�� ≤ φt,1 +��(x, y) − (˜x, ˜y) +�� +2 + φt,2 (αdY), +(26) +for coefficients φt,1 and φt,2 defined as +φt,1 = +� +Lr +t−1 +� +i=0 +(γLf)i + LV (γLf)t +� +and +φt,2 = +� +Lr +t−1 +� +i=1 +γi +i−1 +� +j=0 +Lj +f + LV γt +t−1 +� +j=0 +Lj +f +� +. +Let (x′, y′) = (fX (x, y, a, w), fY(x, y, a, w)) and (˜x′, ˜y′) = (fX (˜x, ˜y, a, w), fY(˜x, ˜y, a, w)) be one-step +transitions starting from (x, y) and (˜x, ˜y), according to the true system dynamics. For t = 1: +��(HV )(x, y) − ( ˜HV )(˜x, ˜y) +�� += +���max +a +� +r(x, y, a) + γ E[V (x′, y′)] +� +− max +˜a +� +r(˜x, ˜y, ˜a) + γ E[V (˜x, ˜y′)] +���� +≤ Lr +��(x, y) − (˜x, ˜y) +�� +2 + γ max +a +E +��V (x′, y′) − V (˜x, ˜y′) +�� +≤ Lr +��(x, y) − (˜x, ˜y) +�� +2 + LV γ max +a +E ∥(x′, y′) − (˜x, ˜y′)∥2 += Lr +��(x, y) − (˜x, ˜y) +�� +2 + LV γ max +a +E +� +∥(x′, y′) − (˜x′, ˜y′)∥2 + ∥(˜x′, ˜y′) − (˜x, ˜y′)∥2 +� +≤ Lr +��(x, y) − (˜x, ˜y) +�� +2 + LV γ +� +Lf∥(x, y) − (˜x, ˜y)∥2 + αdY +� += φ1,1 +��(x, y) − (˜x, ˜y) +�� +2 + φ1,2 (αdY), +which verifies the base case. Let us now assume that (26) holds for t − 1. +��(HtV )(x, y) − ( ˜HtV )(˜x, ˜y) +�� += +���max +a +� +r(x, y, a) + γ E[(HV )t−1(x′, y′)] +� +− max +˜a +� +r(˜x, ˜y, ˜a) + γ E[( ˜HV )t−1(˜x, ˜y′)] +���� +≤ Lr +��(x, y) − (˜x, ˜y) +�� +2 + γ max +a +E +��(HV )t−1(x′, y′) − ( ˜HV )t−1(˜x, ˜y′) +�� +≤ Lr +��(x, y) − (˜x, ˜y) +�� +2 + γ +� +φt−1,1 +��(x′, y′) − (˜x, ˜y′) +�� +2 + φt−1,2 (αdY) +� +≤ Lr +��(x, y) − (˜x, ˜y) +�� +2 + γ +� +φt−1,1 +� +Lf +��(x, y) − (˜x, ˜y) +�� +2 + αdY +� ++ φt−1,2 (αdY) +� +≤ +� +Lr + γLfφt−1,1 +���(x, y) − (˜x, ˜y) +�� +2 + +� +γφt−1,1 + γφt−1,2 +� +(αdY), +where the second inequality follows by the induction hypothesis and the third inequality follows by +50 + +the same steps as in the case of t = 1. It is straightforward to verify that +φt,1 = Lr + γLfφt−1,1 +and +φt,2 = γφt−1,1 + γφt−1,2, +which completes the induction step and the proof. +A.2 +Proof of Proposition 3.1 +We consider an MDP ⟨S, A, W, f, r, γ⟩ and note that U ∗ is the unique optimal solution of the base +model (5), and there exists a stationary optimal policy ν∗(x, y) = arg max U ∗(x, y) that attains this +optimal value (Bertsekas and Shreve, 2004, Proposition 4.3). Fix a state s0 ∈ S and for t > 0 and +a sequence of policies π0, . . . , πt−1, define the notation: +s1(π0) = fπ0(s0, w1) +and +st′+1(π0, . . . , πt′) = fπt′(st′(π0, . . . , πt′−1), wt′+1) +for t′ ≥ 1. Therefore, we have +U ∗(s0) = max +π0 +r(s0, π0) + γ E +� +U ∗(s1(π0)) +� += r(s0, ν∗) + γ E +� +U ∗(s1(ν∗)) +� +. +(27) +By expanding the U ∗(s1(π0)) and U ∗(s1(ν∗)) terms in (27), we have the following: +U ∗(s0) = max +π0,π1 E +� +r(s0, π0) + γ r(s1(π0), π1) + γ2 U ∗(s2(π0, π1)) +� += E +� +r(s0, ν) + γ r(s1(ν∗), ν∗) + γ2 U ∗(s2(ν∗, ν∗)) +� +. +Let π = (π0, π1, . . . , πT−1). Repeating the expansion, we obtain: +U ∗(s0) = max +π +E +�T−1 +� +t=0 +γt r +� +st(π0, . . . , πt−1), πt +� ++ γT U ∗� +sT (π0, . . . , πT−1) +� +� +(28) += E +�T−1 +� +t=0 +γt r +� +st(ν∗, . . . , ν∗), ν∗� ++ γT U ∗� +sT (ν∗, . . . , ν∗) +� +� +. +(29) +Observe that (28) is in same form as the Bellman equation (8) for the hierarchical reformulation +(with T-horizon reward function R and value function ¯U), which has a unique optimal solution ¯U ∗. +Therefore U ∗(s0) = ¯U ∗(s0) and (i) is proved when we recall that s0 was chosen arbitrarily. Part (ii) +51 + +follows because by (29), it is clear that (ν∗, . . . , ν∗) solves (28) and hence also (8). +A.3 +Proof of Proposition 4.1 +Using (17) and (18), the difference between the two reward functions can be expanded as follows: +��E[R(s0, a, π∗)] − E[ ˜R(s0, a, J∗ +1)] +�� += +��r(x0, y0, a) + γ E[(HT−1U ∗)(x1, y1)] − γT E[U ∗(xT , yT )] − r(x0, y0, a) − γ E[( ˜HT−10)(x1, y1)] +�� += γ +��E[(HT−1U ∗)(x1, y1)] − E[( ˜HT−10)(x1, y1)] − γT−1E[U ∗(xT , yT )] +�� +≤ γ E +��(HT−1U ∗)(x1, y1) − ( ˜HT−1U ∗)(x1, y1) +�� +� +�� +� +Term A ++ γ E +��( ˜HT−1U ∗)(x1, y1) − ( ˜HT−10)(x1, y1) − γT−1 U ∗(xT , yT ) +�� +� +�� +� +Term B +, +where (xt, yt) is the state obtained after transitioning from (x0, y0) according to the true dynamics +f = (fX , fY) for t steps. We now work on Terms A and B separately. +Noting that U ∗ has Lipschitz constant LU by Assumption 2, we can apply Lemma A.3 to Term +A to obtain +Term A ≤ αdY +� +Lr +T−2 +� +i=1 +γi +i−1 +� +j=0 +Lj +f + γT−1LU +T−2 +� +j=0 +Lj +f +� +. +(30) +Moving on to Term B, since the reward function r ≥ 0, it follows that U ∗(s) ≥ 0 for all s. Also, +the monotonicity of ˜H implies that ( ˜HT−1U ∗) ≥ ( ˜HT−10). Therefore, applying Lemma A.2, +( ˜HT−1U ∗)(x1, y1) − ( ˜HT−10)(x1, y1) = +��( ˜HT−1U ∗)(x1, y1) − ( ˜HT−10)(x1, y1) +�� +≤ γT−1 maxy∈YT −1 +s1 +��U ∗(x1, y) − 0 +�� += γT−1 maxy∈YT −1 +s1 +U ∗(x1, y), +where YT−1 +s1 +is the set of fast states reachable from s1 = (x1, y1) after T −1 transitions of fY(x1, ·, ·, ·). +Let ˜ys1 = arg maxy∈YT −1 +s1 +U ∗(x1, y) be the fast state that attains the maximum. +Note that ˜ys1 +depends on s1, which is random. Combining with the rest of Term B, we have +Term B ≤ γ E +��γT−1U ∗(x1, ˜ys1) − γT−1 U ∗(xT , yT ) +�� +52 + +≤ max +ω∈Ω γT ��U ∗� +x1(ω), ˜ys1(ω) +� +− U ∗� +xT (ω), yT (ω) +��� +≤ max +ω∈Ω γT LU +� +∥x1(ω) − xT (ω)∥2 + ∥˜ys1(ω) − yT (ω)∥2 +� +(31) +≤ γT LUdY(α + 2)(T − 1), +(32) +where (31) follows by Assumption 2 and (32) comes from Lemma A.1 (we use that xT (ω) is T − 1 +transitions from x1(ω) and both ˜ys1(ω) and yT (ω) are both T − 1 transitions from y1(ω)). Finally, +we have +Terms A + B ≤ αdY +� +Lr +T−2 +� +i=1 +γi +i−1 +� +j=0 +Lj +f + γT−1LU +T−2 +� +j=0 +Lj +f +� ++ γT LUdY(α + 2)(T − 1) += αdY +� +Lr +T−2 +� +i=1 +γi +i−1 +� +j=0 +Lj +f +� ++ γT−1LU +� +αdY +T−2 +� +j=0 +Lj +f + γdY(α + 2)(T − 1) +� +which completes the proof. +B +Proofs for Section 5 +B.1 +Technical Lemmas +Lemma B.1. Consider two MDPs, M1 and M2, who differ in their transition and reward functions: +Mi = ⟨S, A, W, fi, ri, γ⟩. Let U ∗ +i be the optimal value function of Mi. Suppose that +(a) |r1(s, a) − r2(s, a)| ≤ ϵr for all s ∈ S and a ∈ A; +(b) ∥f1(s, a, w) − f2(s, a, w)∥2 ≤ d for all s ∈ S, a ∈ A, and w ∈ W; and +(c) there exists L1 > 0 such that |U ∗ +1 (s) − U ∗ +1 (s′)| ≤ L1∥s − s′∥2 for all s, s′ ∈ S. +Then, the difference in optimal values of the two MDPs can be bounded as follows: +��U ∗ +1 (s) − U ∗ +2 (s) +�� ≤ ϵr + γL1d +1 − γ +for all s ∈ S. +53 + +Proof. Let ˆs = arg maxs∈S |U ∗ +1 (s) − U ∗ +2 (s)|. We will analyze +��U ∗ +1 (ˆs) − U ∗ +2 (ˆs) +��. +��U ∗ +1 (ˆs) − U ∗ +2 (ˆs) +�� = +���max +a1∈A +� +r1(ˆs, a1) + γ E +� +U ∗ +1 +� +f1(ˆs, a1, w) +��� +− max +a2∈A +� +r2(ˆs, a2) + γ E +� +U ∗ +2 (f2(ˆs, a2, w)) +����� +≤ max +a∈A +��r1(ˆs, a) + γ E +� +U ∗ +1 +� +f1(ˆs, a, w) +�� +− r2(ˆs, a) − γ E +� +U ∗ +2 (f2(ˆs, a, w)) +��� +≤ max +a∈A +��r1(ˆs, a) − r2(ˆs, a) +�� + γ max +a∈A +��E +� +U ∗ +1 +� +f1(ˆs, a, w) +�� +− E +� +U ∗ +2 +� +f2(ˆs, a, w) +���� +≤ ϵr + γ max +a∈A +��E +� +U ∗ +1 +� +f1(ˆs, a, w) +� +− U ∗ +1 +� +f2(ˆs, a, w) +���� ++ γ max +a∈A +��E +� +U ∗ +1 +� +f2(ˆs, a, w) +� +− U ∗ +2 +� +f2(ˆs, a, w) +���� +≤ ϵr + γL1 max +a,w ∥f1(ˆs, a, w) − f2(ˆs, a, w)∥2 + γ max +s∈S +��U ∗ +1 (s) − U ∗ +2 (s) +�� +≤ ϵr + γL1d + γ +��U ∗ +1 (ˆs) − U ∗ +2 (ˆs) +��. +Rearranging, we have +��U ∗ +1 (ˆs) − U ∗ +2 (ˆs) +�� ≤ ϵr + γL1d +1 − γ +, +which completes the proof if we recall how ˆs was chosen. +Lemma B.2. Consider two MDPs, M1 and M2, who differ in their transition and reward functions: +Mi = ⟨S, A, W, fi, ri, γ⟩. Let U ∗ +i be the optimal value function of Mi. Suppose that +(a) |r1(s, a) − r2(s, a)| ≤ ϵr for all s ∈ S and a ∈ A; +(b) ∥f1(s, a, w) − f2(s, a, w)∥2 ≤ d for all s ∈ S, a ∈ A and w ∈ W; +(c) there exists L1 > 0 such that |U ∗ +1 (s) − U ∗ +1 (s′)| ≤ L1∥s − s′∥2 for any s, s′ ∈ S; and +(d) |U ∗ +1 (s) − U ∗ +2 (s)| ≤ ϵU for all s ∈ S. +Let ˜π2 be a policy that is an approximation of the optimal policy for M2, in the sense that: +˜π2(s) = arg max +a∈A +� +˜r2(s, a) + γ E +� ˜U2 +� ˜f2(s, a, w) +��� +, +(33) +where |r2(s, a) − ˜r2(s, a)| ≤ ˜ϵr, ∥f2(s, a, w) − ˜f2(s, a, w)∥2 ≤ ˜d, and |U ∗ +2 (s) − ˜U2(s)| ≤ ˜ϵU for all +54 + +s ∈ S, a ∈ A, and w ∈ W. Then, the value of ˜π2 when implemented in M1 has regret bounded by: +��U ∗ +1 − U ˜π2 +1 +�� +∞ ≤ 2(ϵr + ˜ϵr) + 2γ(ϵU + ˜ϵU) + 2γL1(d + ˜d) +1 − γ +. +This lemma is a generalization and extension of Corollary 1 of Singh and Yee (1994). +Proof. Let π∗ +1 be an optimal policy for M1. Using (33), it follows that +˜r2(s, π∗ +1(s)) + γ E +� ˜U2( ˜f2(s, π∗ +1(s), w)) +� +≤ ˜r2(s, ˜π2(s)) + γ E +� ˜U2( ˜f2(s, ˜π2(s), w)) +� +. +(34) +Set ϵU = ϵU + ˜ϵU, ϵr = ϵr + ˜ϵr. +Combining parts (a) and (d) in the statement of the lemma +with the approximation errors of ˜U2 and ˜r2, we know that U ∗ +1 (s) − ϵU ≤ ˜U2(s) ≤ U ∗ +1 (s) + ϵU and +r1(s, a) − ϵr ≤ ˜r2(s, a) ≤ r1(s, a) + ϵr for any s and a. Using these, we can lower bound both +terms on the left-hand-side of (34), upper bound both terms on the right-hand-side of (34), and +then rearrange to obtain +r1(s, π∗ +1(s)) − r1(s, ˜π2(s)) ≤ 2ϵr + 2γϵU + γ E +� +U ∗ +1 ( ˜f2(s, ˜π2(s), w)) − U ∗ +1 ( ˜f2(s, π∗ +1(s), w)) +� +. +(35) +Let state ˆs = arg maxs∈S U ∗ +1 (ˆs) − U ˜π2 +1 (ˆs) be the state that achieves the largest regret (when using +˜π2 in M1). Substituting from (35) gives +U ∗ +1 (ˆs) − U ˜π2 +1 (ˆs) = r1(ˆs, π∗ +1(ˆs)) − r1(ˆs, ˜π2(ˆs)) + γ E +� +U ∗ +1 (f1(ˆs, π∗ +1(ˆs), w)) − U ˜π2 +1 (f1(ˆs, ˜π2(ˆs), w)) +� +≤ 2ϵr + 2γϵU + γ E +� +U ∗ +1 ( ˜f2(ˆs, ˜π2(ˆs), w)) − U ∗ +1 ( ˜f2(ˆs, π∗ +1(ˆs), w)) +� ++ γ E +� +U ∗ +1 (f1(ˆs, π∗ +1(ˆs), w)) − U ˜π2 +1 (f1(ˆs, ˜π2(ˆs), w)) +� += 2ϵr + 2γϵU + γ E +� +U ∗ +1 ( ˜f2(ˆs, ˜π2(ˆs), w)) − U ∗ +1 (f1(ˆs, ˜π2(ˆs), w)) +� ++ γ E +� +U ∗ +1 (f1(ˆs, π∗ +1(ˆs), w)) − U ∗ +1 ( ˜f2(ˆs, π∗ +1(ˆs), w)) +� ++ γ E +� +U ∗ +1 (f1(ˆs, ˜π2(ˆs), w)) − U ˜π2 +1 (f1(ˆs, ˜π2(ˆs), w)) +� +≤ 2ϵr + 2γϵU + 2γL1(d + ˜d) + γ +� +U ∗ +1 (ˆs) − U ˜π2 +1 (ˆs) +� +, +where we have used property (c) and that ∥f1(s, a, w)− ˜f2(s, a, w)∥2 ≤ d+ ˜d. Therefore, we rearrange +55 + +to see that +U ∗ +1 (ˆs) − U ˜π2 +1 (ˆs) ≤ 2ϵr + 2γϵU + 2γL1(d + ˜d) +1 − γ +, +completing the proof. +B.2 +Proof of Lemma 5.1 +To analyze R(µ, π) = ∥ ¯U ∗ − ¯U µ(π)∥∞, we will consider two MDPs that operate on the T-period +timescale, one with optimal value ¯U ∗ and the other with optimal value V ∗(J1, π). +The reason +to study an MDP with optimal value V ∗(J1, π) is because µ can be viewed as an approximation +to the optimal policy for the second MDP, as suggested in (20). Since both MDPs are defined +on the T-period timescale, the transition functions are defined using T-period noise sequences +w = (w1, w2, . . . , wT ). +• For ¯U ∗, let M1 = ⟨S, A, W, f1, r1, γT ⟩ be the MDP associated with the base model refor- +mulation (8), but with the lower-level policy fixed to be π∗. +The reward function r1 is +r1(s, a) = E[R(s, a, π∗)]. Given a T-period noise sequence w, an initial state s, and action a, +the “next” state f1(s, a, w) = sT (a, π∗) is the state obtained by first taking action a in state +s and then following policy π∗ for the next T − 1 steps. +• For V ∗(J1, π), let M2 = ⟨S, A, W, f2, r2, γT ⟩ be the MDP associated with the frozen-state +hierarchical approximation (14), where r2 is defined as r2(s, a) = E[ ˜R(s, a, J1)]. The transition +function f2 is defined in the same way as f1 except we replace π∗ by π. +Let ϵr(π∗, J1) = maxs,a |E[R(s, a, π∗)] − E[ ˜R(s, a, J1)]|, so that we have |r1(s, a) − r2(s, a)| ≤ +ϵr(π∗, J1). +Noting that the first steps of f1 and f2 are the same (action a in state s with w1 +revealed), applying parts (ii) and (iii) of Lemma A.1, the maximum discrepancy between f1 and f2 +is: +∥f1(s, a, w) − f2(s, a, w)∥2 ≤ d(α, dY, T) := 2(α + 1)dY(T − 1). +Applying Lemma B.1, we see that +�� ¯U ∗ − V ∗(J1, π) +�� +∞ ≤ +1 +1 − γT +� +ϵr(π∗, J1) + γT LUd(α, dY, T) +� +. +56 + +We also need to account for the fact that µ is greedy with respect to V , an approximation of +the optimal value of M2. More precisely, µ is greedy with respect to r2(s, a) = E +� ˜R(s0, a, J1) +� +, +f2(s, a, w) = sT (a, π), and value function V . We can thus apply Lemma B.2 with ϵr = ϵr(π∗, J1), +d = d(α, dY, T), L1 = LU, ϵU = +�� ¯U ∗ −V ∗(J1, π) +�� +∞, and ˜ϵU = +��V −V ∗(J1, ˜π) +�� +∞. Collecting terms +completes the proof. +C +Proofs for Section 6 +C.1 +Technical Lemmas +Lemma C.1. Consider an MDP ⟨S, A, W, f, r, γ⟩ with reward function r taking values in [0, rmax]. +Suppose the optimal value function is U ∗ and the associated Bellman operator is F. Fix any initial +value function such that 0 ≤ U0(s) ≤ rmax/(1 − γ) for all s and let U k = F k U0 be the result after +iteration k of value iteration. Then, it holds that +∥U k − U ∗∥∞ ≤ γk rmax +1 − γ . +Proof. This is a standard result that follows from the contraction property of F and the fact that +U k = F Uk−1. Therefore, +∥U k − U ∗∥∞ = ∥HU k−1 − HU ∗∥∞ ≤ γ∥U k−1 − U ∗∥∞ ≤ γk∥U 0 − U ∗∥∞ ≤ γk rmax +1 − γ , +where in the last step, we used 0 ≤ U ∗(s) ≤ rmax/(1 − γ) for all s. +Lemma C.2 (Proposition 6.1 of Bertsekas and Tsitsiklis (1996)). Consider an MDP ⟨S, A, W, f, r, γ⟩ +with optimal value function U ∗. Suppose that ν is a policy that is greedy with respect to another +value function U: +ν(s) = arg max +a +� +r(s, a) + E +� +U(f(s, a, w)) +�� +. +If ∥U − U ∗∥∞ ≤ ε, then the performance of ν is bounded as follows: +∥U ν − U ∗∥∞ ≤ 2γε +1 − γ . +57 + +Lemma C.3. Let V k(J∗ +1, π∗) be the value function approximation obtained from running FSVI for +k iterations. Then, the “value iteration error” is given by +��V k(J∗ +1, π∗) − V ∗(J∗ +1, π∗) +�� +∞ ≤ γkT rmax +1 − γ . +Proof. Consider the upper-level MDP. Note that the discount factor is γT and the T-horizon reward +function +˜R(s0, a, J∗ +1) ∈ +� +0, 1 − γT +1 − γ rmax +� +. +The result follows by Lemma C.1. +C.2 +Proof of Proposition 6.1 +Since νk is greedy with respect to Uk, we can apply Lemmas C.1 and C.2 to obtain +∥U νk − U ∗∥∞ ≤ +2γ +1 − γ +γkrmax +1 − γ = 2rmaxγk+1 +(1 − γ)2 . +C.3 +Proof of Theorem 6.1 +We apply Lemma 5.1 with π = ˜π∗, J1 = J∗ +1, and V = V k(J∗ +1, π∗). The result follows by combining +it with the result of Lemma C.3 and noting that by Proposition 4.1, ϵr(π∗, J∗ +1) ≤ ϵr(γ, α, dY, L, T). +D +Proofs for Section 7 +D.1 +Proof of Lemma 7.1 +First, we note that ¯Jt(x, y) nearly satisfies the Bellman equation for the separable MDP, with +the exception of a next state transition that is based on x∗. Let y′(x, y, a) = fY(x, y, a, w) and +∆g(x) = g(x) − g(x∗). We have: +¯Jt(x, y) = +T−t−1 +� +i=0 +γi∆g(x) + max +a +� +g(x∗) + h(y, a) + γ E +� ¯Jt+1(x∗, y′(x∗, y, a)) +�� += +T−t−1 +� +i=0 +γi∆g(x) + max +a +� +g(x∗) + h(y, a) + γ E +� +¯Jt+1(x, y′(x∗, y, a)) − +T−t−2 +� +i=0 +γi∆g(x) +�� +58 + += +T−t−1 +� +i=0 +γi∆g(x) + max +a +� +g(x∗) + h(y, a) + γ E +� +¯Jt+1(x, y′(x∗, y, a)) − +T−t−2 +� +i=0 +γi∆g(x) +�� += ∆g(x) + max +a +� +g(x∗) + h(y, a) + γ E +� ¯Jt+1(x, y′(x∗, y, a)) +�� += max +a +� +g(x) + h(y, a) + γ E +� ¯Jt+1(x, y′(x∗, y, a)) +�� +. +(36) +The main proof is by backward induction. Consider states (x, y) and (˜x, ˜y). When t = T − 1, +the difference between the two values is +�� ¯JT−1(x, y) − J∗ +T−1(˜x, ˜y) +�� = +��g(x) − g(x∗) + max +a +� +g(x∗) + h(y, a) +� +− max +˜a +r(˜x, ˜y, ˜a) +�� +≤ max +a +��g(x) + h(y, a) − r(˜x, ˜y, a) +�� +≤ max +a +��g(x) + h(y, a) − r(x, y, a) +�� + max +a +��r(x, y, a) − r(˜x, ˜y, a) +�� +≤ ζ + Lr +� +∥x − ˜x∥2 + ∥y − ˜y∥2 +� +, +where the last inequality follows from Assumption 3 and (2). Suppose that the desired result holds +for period t, i.e., we have +�� ¯Jt(x, y) − J∗ +t (˜x, ˜y) +�� ≤ +�T−t−1 +� +i=0 +γi +�� +ζ + Lr ∥x − ˜x∥2 +� ++ +�T−t−1 +� +i=0 +γiLi +f +� +Lr∥y − ˜y∥2 ++ +�T−t−1 +� +i=1 +Li +f +T−t−1 +� +j=i +γj +� +Lr ∥x∗ − ˜x∥2. +(37) +Then for period t − 1, the value difference can be expanded as +�� ¯Jt−1(x, y) − J∗ +t−1(˜x, ˜y) +�� += +����max +a +� +g(x) + h(y, a) + γ E +� ¯Jt(x, y′(x∗, y, a)) +�� +− max +˜a +� +r(˜x, ˜y, ˜a) + γ E +� +J∗ +t (˜x, y′(˜x, ˜y, ˜a)) +������ +≤ max +a +��g(x) + h(y, a) − r(˜x, ˜y, a) +�� +� +�� +� +Term A ++ γ max +a +���E +� ¯Jt(x, fY(x∗, y, a, w)) − J∗ +t (˜x, fY(˜x, ˜y, a, w)) +���� +� +�� +� +Term B +, (38) +where we used (36) in the equality above. It is easy to see that by Assumption 3 +Term A ≤ ζ + Lr +� +∥x − ˜x∥2 + ∥y − ˜y∥2 +� +. +59 + +Noting that +��fY(x∗, y, a, w)) − fY(˜x, ˜y, a, w) +�� +2 ≤ Lf +� +∥x∗ − ˜x∥2 + ∥y − ˜y∥2 +� +, +we see from the induction hypothesis (37) that +Term B ≤ γ +��T−t−1 +� +i=0 +γi +�� +ζ + Lr ∥x − ˜x∥2 +� ++ +�T−t−1 +� +i=0 +γiLi +f +� +LrLf +� +∥x∗ − ˜x∥2 + ∥y − ˜y∥2 +� ++ +�T−t−1 +� +i=1 +Li +f +T−t−1 +� +j=i +γj +� +Lr ∥x∗ − ˜x∥2 +� += +�T−t +� +i=1 +γi +�� +ζ + Lr ∥x − ˜x∥2 +� ++ +�T−t +� +i=1 +γiLi +f +� +Lr +� +∥x∗ − ˜x∥2 + ∥y − ˜y∥2 +� ++ +�T−t−1 +� +i=1 +Li +f +T−t +� +j=i+1 +γj +� +Lr ∥x∗ − ˜x∥2 += +�T−t +� +i=1 +γi +�� +ζ + Lr ∥x − ˜x∥2 +� ++ +�T−t +� +i=1 +γiLi +f +� +Lr∥y − ˜y∥2 + +�T−t +� +i=1 +Li +f +T−t +� +j=i +γj +� +Lr ∥x∗ − ˜x∥2, +where the last equality is obtained by from +T−t +� +i=1 +Li +f +T−t +� +j=i +γj = +T−t +� +i=1 +γiLi +f + +T−t−1 +� +i=1 +Li +f +T−t +� +j=i+1 +γj. +Combining Terms A and B completes the proof. +D.2 +Proof of Proposition 7.1 +The difference of the reward functions +��E[ ˜R(s0, a, J∗ +1)]−E[ ˜R(s0, a, ¯J∗ +1)] +�� can be expanded as follows, +��E[ ˜R(s0, a, J∗ +1)] − E[ ˜R(s0, a, ¯J1)] +�� = γ +��E +� +J∗ +1 +� +f(s0, a, w) +�� +− E +� ¯J1(f(s0, a, w)) +��� +≤ +T−1 +� +i=1 +γiζ + +�T−2 +� +i=1 +Li +f +T−1 +� +j=i+1 +γj +� +Lr max +x +∥x∗ − x∥2, +where the inequality is by Lemma 7.1. +60 + +E +Proofs for Section 8 +E.1 +Technical Lemmas +Lemma E.1. For any vectors J ∈ RN and J′ ∈ RN, it holds that +��(ΦΦ†)(J) − (ΦΦ†)(J′) +�� +∞ ≤ κ ∥J − J′∥∞ +Proof. For simplicity, let D = Φ +� +Φ†(J) − Φ†(J′) +� +be the term inside the norm on the left hand side. +Then, for any state s, we have |D(s)| = φ⊺(s) +� +Φ†(J)−Φ†(J′) +� +. We select θ1(s), θ1(s), . . . , θM(s) ∈ R +that satisfy Assumption 4, obtaining +|D(s)| = +�����κ +� M +� +m=1 +θm(s)φ⊺(sm) +� +� +Φ†(J) − Φ†(J′) +� +����� +≤ κ max +m +��φ⊺(sm) +� +Φ†(J)) − Φ†(J′) +��� += κ max +m +|D(sm)| += κ max +m +|J(sm) − J′(sm)| +≤ κ ∥J − J′∥∞ +where the third equality uses the fact that sm is a pre-selected state. +Lemma E.2. Given a lower-level value function ˆJ(ωt+1), recall that one approximate Bellman step +in the lower level of FSAVI yields ˆJ(ωt) = ΦΦ† ¯H ˆJ(ωt+1) in the value function space. If ωT = 0, +�� ˆJ(ω1) +�� +∞ ≤ κrmax +T−1 +� +i=0 +(κγ)i = (κγ)T − 1 +κγ − 1 +κrmax. +Moreover, the upper-level reward function can be bounded as follows: +��E +� ˜R(s0, a, ˆJ1(ω1)) +��� ≤ (κγ)T+1 − 1 +κγ − 1 +κrmax. +61 + +Proof. The proof follows by Assumption 4 and some manipulation: +�� ˆJ(ωt)(s) +�� = +�����κ +� M +� +m=1 +θm(s)φ⊺(sm) +� +� +Φ† ¯H ˆJ(ωt+1) +� +����� +≤ κ max +m +��φ⊺(sm) +� +Φ† ¯H ˆJ(ωt+1) +��� += κ max +m +��� ¯H ˆJ(ωt) +� +(sm) +�� += κ rmax + κγ +�� ˆJ(ωt+1) +�� +∞. +Applying the above inequality T − 1 times yields the first result. Next, we see that +��E +� ˜R(s0, a, ˆJ1(ω1)) +��� ≤ rmax + γ +�� ˆJ(ω1) +�� +∞ +≤ κrmax + κγ +�� ˆJ(ω1) +�� +∞ +≤ κrmax +T +� +i=0 +(κγ)i, +where we used the fact that κ ≥ 1 in the second inequality. +Lemma E.3. Suppose (ω∗ +1, ω∗ +2, . . . , ω∗ +T ) satisfies ω∗ +t = ¯H′ω∗ +t+1 for all t. Then, we have +��J∗ +t − ˆJt(ω∗ +t ) +�� +∞ ≤ +� +1 + γ + 1 +γ +T−t +� +i=1 +(κγ)i +� +εlow = +� 1 + κ +1 − κγ − (κγ)T (1 + γ) +γ − κγ2 +� +εlow +a bound on the error of the value function approximation. +Proof. Let ε′ = εlow + δ for some δ > 0. For each t, choose a parameter vector ¯ωt ∈ RM such that +∥J∗ +t − ˆJt(¯ωt)∥∞ < ε′, which is possible by the definition of εlow. Then, it holds that +�� ˆJt(¯ωt) − Φ ¯H′(¯ωt+1) +�� +∞ = +��Φ¯ωt − ΦΦ† ¯H ˆJt+1(¯ωt+1) +�� +∞ += +��ΦΦ†Φ¯ωt − ΦΦ† ¯H ˆJt+1(¯ωt+1) +�� +∞ +≤ κ +��Φ¯ωt − ¯H ˆJt+1(¯ωt+1) +�� +∞ +(39) += κ +�� ˆJt(¯ωt) − ¯H ˆJt+1(¯ωt+1) +�� +∞ +≤ κ +��� ˆJt(¯ωt) − J∗ +t +�� +∞ + +��J∗ +t − ¯H ˆJt+1(¯ωt+1) +�� +∞ +� +< κ +� +ε′ + ∥ ¯HJ∗ +t+1 − ¯H ˆJt+1(¯ωt+1)∥∞ +� +62 + +≤ κ +� +ε′ + γ +��J∗ +t+1 − ˆJt+1(¯ωt+1) +�� +∞ +� +(40) +< κ(γ + 1) ε′. +where (39) is by Lemma E.1 and (40) follows by the contraction property of ¯H. The next step is +the quantify the difference between ˆJt(¯ωt) and ˆJt(ω∗ +t ). Let ε′′ = κ(γ + 1) ε′. +�� ˆJt(¯ωt) − ˆJt(ω∗ +t ) +�� +∞ ≤ +�� ˆJt(¯ωt) − Φ ¯H′(¯ωt+1) +�� +∞ + +��Φ ¯H′(¯ωt+1) − ˆJt(ω∗ +t ) +�� +∞ +≤ ε′′ + +��ΦΦ† ¯H ˆJt+1(¯ωt+1) − ΦΦ† ¯H ˆJt+1(ω∗ +t+1) +�� +∞ +≤ ε′′ + γ′ +γ +�� ¯H ˆJt+1(¯ωt+1) − ¯H ˆJt+1(ω∗ +t+1) +�� +∞ +(41) +≤ ε′′ + γ′�� ˆJt+1(¯ωt+1) − ˆJt+1(ω∗ +t+1) +�� +∞ +(42) +≤ ε′′ + γ′� +ϵ′′ + γ′�� ˆJt+2(¯ωt+2) − ˆJt+2(ω∗ +t+2) +�� +∞ +� +≤ · · · +≤ ε′′ +T−t−1 +� +i=0 +(γ′)i + (γ′)T−t �� ˆJT (¯ωT ) − ˆJT (ω∗ +T ) +�� +∞ += γ + 1 +γ +ε′ +T−t +� +i=1 +(γ′)i, +(43) +where (41) is by Lemma E.1, (42) is by the contraction property of ¯H, and (43) because ω∗ +T = ¯ωT = 0 +(since JT (s) = 0 for all s). Therefore, +��J∗ +t − ˆJt(ω∗ +t ) +�� +∞ ≤ +��J∗ +t − ˆJt(¯ωt) +�� +∞ + +�� ˆJt(¯ωt) − ˆJt(ω∗ +t ) +�� +∞ +≤ ε′ +� +1 + γ + 1 +γ +T−t +� +i=1 +(γ′)i +� += ε′ +� +1 + γ + 1 +γ +T−t +� +i=1 +(κγ)i +� += ε′ +� +1 + γ + 1 +γ +κγ − (κγ)T +1 − κγ +� += ε′ +� 1 + κ +1 − κγ − (κγ)T (1 + γ) +γ − κγ2 +� +. +Since δ can be arbitrarily small, the proof is complete. +63 + +Lemma E.4. Given an upper-level value function ˆV (βi), recall that one approximate Bellman step +in the upper level of FSAVI yields ˆV (βi+1) = ΦΦ†Fω∗ ˆV (βi) in the value function space. We have +�� ˆV (β∗) +�� ≤ +κ2 − κ2(κγ)T+1 +(1 − κγT )(1 − κγ) rmax, +where β∗ is a fixed point of F ′ +ω∗. +Proof. Again, the proof follows by Assumption 4 and some manipulation: +�� ˆV (βi+1)(s) +�� = +�����κ +� M +� +m=1 +θm(s)φ⊺(sm) +� +� +Φ†Fω ˆV (βi) +� +����� +≤ κ max +m +��φ⊺(sm) +� +Φ†Fω ˆV (βi) +��� += κ max +m +��� +Fω ˆV (βi) +� +(sm) +�� +≤ κ Rmax + κγT �� ˆV (βi) +�� +∞, +where Rmax is an upper bound on +��E +� ˜R(s0, a, ˆJ1(ω1)) +��� from Lemma E.2. Starting with βi = 0 and +applying the inequality repeatedly, we see that +�� ˆV (β∗) +�� +∞ ≤ κRmax +∞ +� +j=0 +(κγT )j ≤ κRmax +1 − κγT , +which completes the proof. +Lemma E.5. For any ω, the parameter space Bellman operator for the upper-level problem F ′ +ω = +Φ† ◦ Fω ◦ Φ is a γ′-contraction with respect to a norm ∥ · ∥Φ on RM defined by ∥β∥Φ = ∥Φβ∥∞, i.e., +��F ′ +ω(β) − F ′ +ω(β′) +�� +Φ ≤ κγT ��β − β′�� +Φ, +where β, β′ ∈ RM. Therefore, there exists a fixed point β∗ of F ′ +ω. +Proof. The proof follows Theorem 3a of Tsitsiklis and Van Roy (1996). We include the steps here +in our notation for completeness: +��F ′ +ω(β) − F ′ +ω(β′) +�� +Φ = +��(Φ† ◦ Fω ◦ Φ)(β) − (Φ† ◦ Fω ◦ Φ)(β′) +�� +Φ +64 + += +��Φ(Φ† ◦ Fω ◦ Φ)(β) − Φ(Φ† ◦ Fω ◦ Φ)(β′) +�� +∞ +≤ κ +��Fω(Φβ) − Fω(Φβ′) +�� +∞ +≤ κγT ��Φβ − Φβ′�� +∞ += κγT ��β − β′�� +Φ, +where first inequality follows by Lemma E.1 and the second inequality follows by the γT -contraction +property of Fω. +Lemma E.6. Let ω∗ be the solution of the lower level of FSAVI and let β∗ be the fixed point of +F ′ +ω∗. The approximate value iteration of the upper level of FSAVI, which produces βk, has a “value +iteration” error of: +�� ˆV (βk) − ˆV (β∗) +�� +∞ ≤ (κγT )k κ2 − κ2(κγ)T+1 +(1 − κγT )(1 − κγ) rmax. +Proof. We have: +�� ˆV (βk) − ˆV (β∗) +�� +∞ = +��ΦF ′ +ω∗βk−1 − ΦF ′ +ω∗β∗�� +∞ += +��F ′ +ω∗βk−1 − F ′ +ω∗β∗�� +Φ +≤ κγT ��Φβk−1 − Φβ∗�� +∞ +≤ κγT �� ˆV (βk−1) − ˆV (β∗) +�� +∞ +≤ (κγT )k�� ˆV (β0) − ˆV (β∗) +�� +∞ +≤ (κγT )k κ2 − κ2(κγ)T+1 +(1 − κγT )(1 − κγ) rmax. +The first inequality is by Lemma E.5 and the last inequality follows from β0 = 0 and Lemma +E.4. +Lemma E.7. Consider any ω. If β∗ is the fixed point of F ′ +ω, i.e., β∗ = F ′ +ω β∗, which exists by +Lemma E.5, then it holds that +��V ∗ +ω − ˆV (β∗) +�� +∞ ≤ +� 1 + κ +1 − κγT +� +εup +65 + +where V ∗ +ω is the fixed point of Fω. +Proof. Let ε′ = εup + δ for some δ > 0. Choose ¯β ∈ RM such that +��V ∗ +ω − ˆV ( ¯β) +�� +∞ < ε′. Then, +�� ˆV ( ¯β) − ΦF ′ +ω( ¯β) +�� +∞ = +��Φ ¯β − ΦΦ†Fω ˆV ( ¯β) +�� +∞ += +��ΦΦ†Φ ¯β − ΦΦ†Fω ˆV ( ¯β) +�� +∞ +< κ +��Φ ¯β − Fω ˆV ( ¯β) +�� +∞ +(44) += κ +�� ˆV ( ¯β) − Fω ˆV ( ¯β) +�� +∞ +≤ κ +��� ˆV ( ¯β) − V ∗ +ω +�� +∞ + +��V ∗ +ω − Fω ˆV ( ¯β) +�� +∞ +� +< κ +� +ε′ + +��FωV ∗ +ω − Fω ˆV ( ¯β) +�� +∞ +� +< κ +� +ε′ + γT ϵ′� += κ +� +1 + γT � +ε′, +(45) +where (44) is by Lemma E.1 and (45) is by the γT -contraction property of Fω. +Now, we let +ε′′ = κ(1 + γT ) ε′ and see that +�� ˆV ( ¯β) − ˆV (β∗) +�� +∞ ≤ +�� ˆV ( ¯β) − ΦF ′ +ω( ¯β) +�� +∞ + +��ΦF ′ +ω( ¯β) − ˆV (β∗) +�� +∞ +< ϵ′′ + +��ΦΦ†Fω ˆV ( ¯β) − ΦΦ†Fω ˆV (β∗) +�� +∞ +< ϵ′′ + κ +��Fω ˆV ( ¯β) − Fω ˆV (β∗) +�� +∞ +≤ ϵ′′ + κγT �� ˆV ( ¯β) − ˆV (β∗) +�� +∞. +It thus follows that +�� ˆV ( ¯β) − ˆV (β∗) +�� +∞ ≤ κ + κγT +1 − κγT ε′. +Putting the pieces together, we have +��V ∗ +ω − ˆV (β∗) +�� +∞ ≤ +��V ∗ +ω − ˆV ( ¯β) +�� +∞ + ∥ ˆV ( ¯β) − ˆV (β∗) +�� +∞ +≤ ε′ + κ + κγT +1 − κγT ε′ +≤ +1 + κ +1 − κγT ε′. +Since δ can be arbitrarily small, the proof is complete. +66 + +E.2 +Proof of Theorem 8.1 +We apply Lemma 5.1 with π = ˆπω∗, J1 = ˆJ1(ω∗), and V = ˆV (βk). First, to compute the reward +error ϵr(π∗, ˆJ1(ω∗ +1)), we have +ϵr(π∗, ˆJ1(ω∗ +1)) = max +s,a +��E[R(s, a, π∗)] − E[ ˜R(s, a, ˆJ1(ω∗ +1))] +�� +≤ ϵr(γ, α, dY, L, T) + max +s,a +��E[ ˜R(s, a, J∗ +1)] − E[ ˜R(s, a, ˆJ1(ω∗ +1))] +�� +≤ ϵr(γ, α, dY, L, T) + γ +��J∗ +1 − ˆJ1(ω∗ +1) +�� +∞ +≤ ϵr(γ, α, dY, L, T) + +� +γ + (γ + 1) +T−1 +� +i=1 +(γ′)i +� +εlow, +where the last inequality follows from Lemma E.3. The other term to analyze is +��V ∗ +ω∗ − ˆV (βk) +�� +∞, +where we remind the reader of our usage of the shorthand notation V ∗ +ω∗ = V ∗( ˆJ1(ω∗), ˆπω∗): +��V ∗ +ω∗ − ˆV (β∗) +�� +∞ ≤ +��V ∗ +ω∗ − ˆV (β∗) +�� +∞ + +�� ˆV (β∗) − ˆV (βk) +�� +∞ +≤ +� 1 + κ +1 − κγT +� +εup + (κγT )k +� κ2 − κ2(κγ)T+1 +(1 − κγT )(1 − κγ) +� +rmax, +which follows by Lemmas E.4 and E.6. This completes the proof. +F +Bounds on LU in Terms of Lr and Lf +We start with an assumption that, if true, leads to a simple bound on the Lipschitz constant LU. +The main result is in Proposition F.1. +Assumption 5. Suppose that γLf < 1, where the constant Lf, as defined in (3), is the sensitivity +of the transition function to small changes in (s, a). +Lemma F.1. Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩ and let U : S → R be a value +function such that there exists LU > 0 where for any states s and ˜s, +|U(s) − U(˜s)| ≤ LU ∥s − ˜s∥2. +(46) +Define the state-action value function Q(s, a) = r(s, a) + γ E +� +U(f(s, a, w)) +� +. Then, for any state- +67 + +action pairs (s, a) and (˜s, ˜a), the state-action value function Q satisfies +��Q(s, a) − Q(˜s, ˜a) +�� ≤ (Lr + γLULf) +� +∥s − ˜s∥2 + ∥a − ˜a∥2 +� +. +Proof. For any state-action pairs (s, a), (˜s, ˜a) ∈ S × A, we have +��Q(s, a) − Q(˜s, ˜a) +�� ≤ |r(s, a) − r(˜s, ˜a)| + γ +��E +� +U(f(s, a, w)) − U(f(˜s, ˜a, w)) +��� +≤ Lr +� +∥s − ˜s∥2 + ∥a − ˜a∥2 +� ++ γLU max +w +��f(s, a, w) − f(˜s, ˜a, w)) +�� +2 +(47) +≤ Lr +� +∥s − ˜s∥2 + ∥a − ˜a∥2 +� ++ γLULf +� +∥s − ˜s∥2 + ∥a − ˜a∥2 +� +(48) +≤ (Lr + γLULf) +� +∥s − ˜s∥2 + ∥a − ˜a∥2 +� +, +where (47) follows by (46) and (48) follows by the definition of Lf in (3). +Lemma F.2. Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩. +Let Q : S × A → R be a +state-action value function and assume there exists LQ > 0 where for any states (s, a) and (˜s, ˜a), +��Q(s, a) − Q(˜s, ˜a) +�� ≤ LQ +� +∥s − ˜s∥2 + ∥a − ˜a∥2 +� +. +(49) +Define U(s) = maxa Q(s, a). Then, for any states s and ˜s, the value function U satisfies +��U(s) − U(˜s) +�� ≤ LQ ∥s − ˜s∥2. +Proof. Note that: +��U(s) − U(˜s) +�� = +��max +a +Q(s, a) − max +˜a +Q(˜s, ˜a) +�� +≤ max +a +��Q(s, a) − Q(˜s, a) +�� +≤ LQ∥s − ˜s∥2, +where the last inequality is by (49). +Lemma F.3. Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩. Starting with U0 = 0, recur- +68 + +sively define Qk+1 and Uk+1 as follows: +Qk+1(s, a) = r(s, a) + γ E +� +Uk(f(s, a, w) +� +and +Uk+1(s) = max +a +Qk+1(s, a). +Then Uk is Lipschitz continuous and its Lipschitz constant LUk satisfies +LUk = Lr + γLfLUk−1. +(50) +Proof. The proof is by induction. +For k = 1, note that Q1(s, a) = r(s, a) and therefore has +Lipschitz constant Lr by (2). By Lemma F.2, it follows that U1 also has Lipschitz constant Lr. +Since LU0 = 0, we see that LU1 = Lr satisfies (50). Now, assume that LUk satisfies (50) for k ≥ 1. +Then, by Lemma F.1, Qk+1 is (Lr + γLfLUk)-Lipschitz continuous and by Lemma F.2, Uk+1 is +(Lr + γLfLUk)-Lipschitz continuous. +Proposition F.1. Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩ and suppose Assumption 5 +holds. Then, the optimal value U ∗, as defined in (5), satisfies: +��U ∗(s) − U ∗(˜s) +�� ≤ +Lr +1 − γLf +��s − ˜s +�� +2 +for any states s, ˜s ∈ S. +Proof. According to Proposition 7.3.1 of Bertsekas (2012), the value Uk in Lemma F.3 converges to +the optimal value U ∗ (value iteration). The recursion (50) can be written as: +LUk = Lr + γLfLr + · · · + (γLf)k−1Lr = +k−1 +� +i=0 +(γLf)iLr, +a convergent sequence since Assumption 5 is satisfied. Letting k → ∞, we see that U ∗ has Lipschitz +constant +lim +k→∞ LUk = +∞ +� +i=0 +(γLf)iLr = +Lr +1 − γLf +, +completing the proof. +69 + diff --git a/n9AzT4oBgHgl3EQfAPpu/content/tmp_files/load_file.txt b/n9AzT4oBgHgl3EQfAPpu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7dd5c08a8a52b6f82478c710cacd06b20d9bc79 --- /dev/null +++ b/n9AzT4oBgHgl3EQfAPpu/content/tmp_files/load_file.txt @@ -0,0 +1,2076 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf,len=2075 +page_content='Faster Approximate Dynamic Programming by Freezing Slow States Yijia Wang and Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Jiang University of Pittsburgh January 4, 2023 Abstract We consider infinite horizon Markov decision processes (MDPs) with fast-slow structure, meaning that certain parts of the state space move “fast” (and in a sense, are more influential) while other parts transition more “slowly.” Such structure is common in real-world problems where sequential decisions need to be made at high frequencies, yet information that varies at a slower timescale also influences the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Examples include: (1) service allocation for a multi-class queue with (slowly varying) stochastic costs, (2) a restless multi-armed bandit with an environmental state, and (3) energy demand response, where both day-ahead and real-time prices play a role in the firm’s revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Models that fully capture these problems often result in MDPs with large state spaces and large effective time horizons (due to frequent decisions), rendering them computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We propose an approximate dynamic programming algorithmic framework based on the idea of “freezing” the slow states, solving a set of simpler finite-horizon MDPs (the lower-level MDPs), and applying value iteration (VI) to an auxiliary MDP that transitions on a slower timescale (the upper-level MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We also extend the technique to a function approximation setting, where a feature-based linear architecture is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' On the theoretical side, we analyze the regret incurred by each variant of our frozen-state approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Finally, we give empirical evidence that the frozen-state approach generates effective policies using just a fraction of the computational cost, while illustrating that simply omitting slow states from the decision modeling is often not a viable heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='00922v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='AI] 3 Jan 2023 1 Introduction We consider sequential decision problems, modeled as Markov decision processes (MDPs), that are endowed with a new “fast-slow” structure: a fast-slow MDP has a state that can be divided into two parts, a slow state and a fast state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' At each time step, the transition of the slow state results in a change that is relatively small compared to that of the fast state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' An alternative view from the perspective of the reward function (rather than the transition function) is that the reward is less sensitive to changes in the slow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Fast-slow structure is common in important real-world problems where sequential decisions need to be made at high frequencies, yet information that varies at a slower timescale also influences the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The following examples illustrate this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Service allocation in multi-class queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The first example is a dynamic service allocation problem for a multi-class queue (Ansell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Brown and Haugh, 2017), with the addition of stochastic holding costs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', the cost of leaving items in the queue) that vary slowly and can be viewed as the slow state (Lee and Vojnovic, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' One prominent motivation is the case of energy-aware job scheduling in data centers, where variations of electricity prices over time can influence the holding cost (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Restless multi-armed bandit with an environmental state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our second example ap- plication is the restless multi-armed bandit (Whittle, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Weber and Weiss, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Killian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Zhang and Frazier, 2021) with an environmental state, a model that is applicable to a problems ranging from machine maintenance (Smallwood and Sondik, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Ruiz-Hernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2020) to dynamic assortment planning (Brown and Smith, 2020) to public health and preventative healthcare (Mate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The restless bandit model involves making intervention decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', whether to per- form maintenance) on “arms” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', machines), each of which is associated with an evolving internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Here, the environmental state can be viewed as the slow state, because it often transitions slowly relative to the arms’ internal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Energy demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We can also apply the fast-slow framework in sequential deci- sion problems from the realm of demand response in the electricity market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Specifically, we consider the problem faced an energy aggregator who observes a day-ahead price and then 2 simultaneously bids a reduction quantity into the demand response market and sets the com- pensation for demand reduction from consumers (Albadi and El-Saadany, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Eid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Khezeli and Bitar, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Khezeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Essentially, the aggre- gator hopes to generate profit from the difference between the contracted price for delivery of demand reduction to the market and the price that offers customers for that reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' However, the aggregator has to consider the demand elasticity of its customers, along with the stochasticity of day-ahead prices and real-time prices (which determine the “penalty” for mistmatch between the promised and realized quantities of demand reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since real- time prices are much more volatile compared to the day-ahead prices, it is reasonable to view day-ahead prices as the slow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Attempts to optimally solve a model that incorporates the full state space along with the true decision-making frequency often encounter computational issues, due to the challenge of solving an MDP with a large state space over a large number of periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Anecdotal evidence suggests that to improve tractability, both practitioners and academic researchers may elect to design simplified decision models that ignore the effect of the slow state on components of their problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In other words, these states might be intentionally left out of the state variable by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', fixing them to constant values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Although such an approach results in policies that can be obtained in a compu- tationally tractable manner, we see in Section 9 that they can incur significant regret compared to the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Main Contributions In this paper, we propose somewhat of a compromise between the solving the full MDP and com- pletely ignoring slow states, by designing a framework around periodically “freezing” and “releasing” slow states, and re-using policies that are computed based on a frozen slow-state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Specifically, we make the following contributions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We first consider a fast-slow MDP and provide an (exact) reformulation into an MDP with hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The upper level is a slow-timescale infinite horizon MDP and the lower level is a fast-timescale finite horizon MDP with T periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' One period of the upper- level problem is composed of a complete lower-level problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We propose a frozen-state 3 approximation to the reformulated MDP, along with an associated frozen-state value iteration (FSVI) algorithm, where the slow state is frozen in the lower-level problem, while each period in the upper-level problem “releases” the slow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Computational benefits arise in several ways: (1) re-use of the lower-level policy (which is computed once) when applying value iteration in the upper level, (2) frozen states simplify the dynamics of the lower-level MDP (dramatically fewer successor states), and (3) the lower-level MDP thus becomes separable into independent MDPs, opening the door to speedups via parallel computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Solving the frozen-state approximation gives a policy that switches between the one action from upper- level policy and T − 1 actions from the lower-level policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We give a theoretical analysis that upper bounds the expected regret from applying this policy compared to the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We then discuss an additional step of approximation that further reduces computational re- quirements, called the nominal-state approximation, which takes advantage of a factored re- ward function assumption and approximates the lower-level MDP using a fixed set of “nominal” slow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The consequence is that instead of solving the lower-level MDP for all slow states, this approximation allows us to solve it only for the set of nominal slow states, which are then used to approximate the lower-level value for other slow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We also provide an upper bounds on the expected regret of the policy obtained from the nominal state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Next, we show how the fast-slow framework can also be exploited in an approximate dynamic programming (ADP) setting (Bertsekas and Tsitsiklis, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Powell, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Specifically, we design a frozen-state approximate value iteration (FSAVI) algorithm that mimics FSVI but uses a linear architecture to approximate the value function in both the lower and upper levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The linear architecture combines estimated values from a set of pre-selected states to form approximations of the value function at other states, based on the technique introduced in Tsitsiklis and Van Roy (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We provide an analysis of the expected regret for policies generated by FSAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lastly, we perform a systematic empirical study on three problem settings (service alloca- tion in multi-class queues, restless bandit with an environmental state, and energy demand response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We show that the proposed algorithms based on the frozen-state approximation quickly converge to good policies using significantly less computation compared to standard 4 methods (value iteration, approximate value iteration, Q-learning, deep Q-networks, and a baseline that ignores slow states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Notably, our results show that ignoring the slow state leads particularly poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We also give qualitative evidence that policies generated by the frozen-state approach have structural features resembling those of the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 Related Work In this section, we provide a brief review of related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' First, there exists a stream of literature focused on sequential decision making problems with exact hierarchical, multi-timescale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2003) study multi-timescale MDPs, which are composed of M different decisions that are made on M different discrete timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The authors consider the impact of upper-level states and actions on the transition of the lower levels, an idea is also present in our fast-slow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Multi-timescale MDPs have often been applied in supply chain problems, including production plan- ning in semiconductor fabrication (Panigrahi and Bhatnagar, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Bhatnagar and Panigrahi, 2006), hydropower portfolio management (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2006), and strategic network growth for reverse supply chains (Wongthatsanekorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2018) propose a row-generation-based algo- rithm to solve a linear programming formulation of the multi-timescale MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Jacobson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (1999) consider “piecewise stationary” MDPs, where the transition and reward functions are “renewed” ev- ery N + 1 periods, motivated by problems where routine decisions are periodically interrupted by higher-level decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the case of large renewal periods, they propose a policy called the “initially stationary policy” which uses a fixed decision rule for some number of initial periods in each renewal cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our fast-slow model focuses on a novel fast-slow structure present in many MDPs and unlike the above work, does not assume any natural/exact hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Instead, we focus on how a particular type of (frozen-state) hierarchical structure can be used as an approximation to the true MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' However, we note that many MDPs with natural two-timescale structure can also fit into our framework, and therefore, given that perspective, our model can be viewed as a generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our proposed frozen-state algorithms are also related to literature on hierarchical reinforce- ment learning, which are methods that artificially decompose a complex problem into smaller sub- problems (Barto and Mahadevan, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Approaches include the options framework (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 1999), the hierarchies of abstract machines (HAMs) approach (Parr and Russell, 1998), and MAXQ 5 value function decomposition (Dietterich, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Out of these three approaches, the options framework is most closely related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A Markov option (also called a macro-action or temporally extended action) is composed of a policy, a termination condition, and an initiation set (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Precup, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' One of the biggest challenges is to automatically construct options that can effectively speed up reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A large portion of work in this direction is based on subgoals, states that might be beneficial to reach (Digney, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' McGovern and Barto, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Jonsson and Barto, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Ciosek and Silver, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The subgoals are identified by utilizing the learned model of the environment (Menache et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Mannor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Şimşek and Barto, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Şimşek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2005), or through trajectories without learning a model (McGovern and Barto, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Stolle and Precup, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The options (and subgoals) framework is largely motivated by robotics and navigation-related tasks, while we are particularly interested in solving problems that arise in the operations research and operations management domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The problems that we study do not decompose naturally into “subgoals” — leading us to identify and focus on the fast-slow structure, which does indeed arise naturally for many problems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Another work that is related to the options framework is Song and Xu (2020), who divide finite-horizon MDPs into two sub-problems along the time horizon, and concatenate their optimal solutions to generate an overall solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our paper also, in a sense, divides MDPs along the time horizon, but our work is quite different from Song and Xu (2020) in that we work on infinite horizon problems and convert them into auxiliary problems that operate on a slower timescale, which takes advantage of reusable lower-level policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' More importantly, the various methods we propose all build on the idea of freezing certain states to reduce computational cost, which is unique to our approach and to our knowledge, this is a novel direction that has not been proposed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 3 Fast-Slow MDPs In this section, we introduce the base model, the original MDP to be solved and formally introduce the notion of a fast-slow MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We then provide a hierarchical reformulation of the base model using fixed-horizon policies, and show the equivalence (in optimal value) between the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Base Model Consider a discrete-time MDP ⟨S, A, W, f, r, γ⟩, where S is the finite state space, A is the finite action space, W is the space of possible realizations of an exogenous, independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=') noise process {wt} defined on a discrete probability space (Ω, F, P), f : S × A × W → S is the transition function, r : S × A → [0, rmax] is the bounded reward function, and γ ∈ [0, 1) is the discount factor for future rewards (Puterman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The objective is U ∗(s) = max {νt} E � ∞ � t=0 γt r � st, νt(st) � ��� s0 = s � , (1) where states transition according to st+1 = f(st, at, wt+1) and we optimize over sequences of policies νt : S → A, which are deterministic mappings from states to actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The expectation is taken over exogenous noise process {wt}∞ t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We assume throughout that S, A, X, Y, S × A, and X × Y are equipped with the Euclidean metric,1 which is naturally the case for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Assumption 1 (Separability and the Fast-Slow Property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose the following hold: (i) The state space S is separable and can be written as S = X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We call X the “slow state space” and Y the “fast state space.” (ii) Let st = (xt, yt) ∈ S, where xt ∈ X is the slow state and yt ∈ Y the fast state, at ∈ A, and wt+1 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The transition dynamics st+1 = f(st, at, wt+1) ∈ S can be written with the notation: xt+1 = fX (xt, yt, at, wt+1) ∈ X and yt+1 = fY(xt, yt, at, wt+1) ∈ Y, for some fX : S × A × W → X and fY : S × A × W → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (iii) For any state (x, y) ∈ S, action a ∈ A, and exogenous noise w ∈ W, suppose the one-step transitions of x and y satisfy: ��y − fY(x, y, a, w) �� 2 ≤ dY and ��x − fX (x, y, a, w) �� 2 ≤ αdY, 1However, as long as the relevant spaces are metric spaces, the framework continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We choose Euclidean metrics as they are natural for our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 7 for some dY < ∞ and α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that one particularly instructive example is the case of exogenous slow states, where xt+1 = fX (xt, wt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Here, the transition does not depend on the action at, nor does it depend on the fast state yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Such a model is common in practice: examples of exogenous slow states include prices, weather conditions, and other environmental variables that are not influenced by the decision maker’s actions or the states of the primary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', Yu and Mannor (2009), who study a related model called the “arbitrarily modulated MDP.” Assumption 2 (Lipschitz Properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that the reward function r, transition function f, and optimal value function U ∗ are Lipschitz with respect to ∥ · ∥2: |r(s, a) − r(s′, a′)| ≤ Lr ��(s, a) − (s′, a′) �� 2, (2) ��f(s, a, w) − f(s′, a′, w) �� 2 ≤ Lf ��(s, a) − (s′, a′) �� 2, (3) ��U ∗(s) − U ∗(s′) �� 2 ≤ LU ��s − s′�� 2, (4) for some Lipschitz constants Lr, Lf, and LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lipschitz assumptions are common in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' see, for example, Ok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2018), Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2021), Sinclair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2020), and Sinclair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In Appendix F, we give bounds on LU in terms of Lr and Lf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' While we could have used those results directly and omitted the assumption on LU, we opt to include (4) to increase the clarity of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Definition 1 (Fast-Slow MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' An MDP ⟨S, A, W, f, r, γ⟩ is called a (α, dY)-fast-slow MDP if Assumptions 1 and 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given any state s = (x, y), noise w, and policy ν, we use the notation fν(s, w) = f(s, ν(s), w), fν X (x, y, w) = fX � x, y, ν(x, y), w � , fν Y(x, y, w) = fY � x, y, ν(x, y), w � , and r(x, y, ν) = r(x, y, ν(x, y)) throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The value of a stationary policy2 ν at state (x, y) is the expected cumulative reward starting from state (x, y) following policy ν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', U ν(x, y) = E � ∞ � t=0 γtr � xt, yt, ν � ��� (x0, y0) = (x, y) � = r � x, y, ν � + γ E � U ν(x′, y′) � , 2It is well-known that there exists an optimal policy to (1) that is both stationary and deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Puterman (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8 where (x′, y′) = fν(x, y, w) and (xt+1, yt+1) = fν(xt, yt, wt) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The optimal value function at state U ∗(x, y), as defined in (1), satisfies the Bellman equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', U ∗(x, y) = max a r(x, y, a) + γ E � U ∗(x′, y′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (5) Denote by H the Bellman operator of the base model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' for any state (x, y) and value function U, (HU)(x, y) = max a r(x, y, a) + γ E � U(f(x, y, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (6) A policy that is greedy with respect to the optimal value function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', ν∗(x, y) = arg max a r(x, y, a) + γ E � U ∗(x′, y′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' is an optimal policy, and the optimal value U ∗ and the value of the optimal policy U ν∗ are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Hierarchical Reformulation using Fixed-Horizon Policies In this section, we derive an exact hierarchical reformulation with the original timescale broken up into groups of T periods each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The reformulation holds for a general MDP ⟨S, A, W, f, r, γ⟩, but the concepts that we introduce in this section will serve as the basis for developing our frozen-state computational approach for fast-slow MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Denote (µ, π) a T-horizon policy, which is a sequence of T policies (µ, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πT−1), µ : S → A, πt : S → A and π = (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πT−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Following (µ, π) means that we take the first action according to µ and then next T − 1 actions according to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given any state s0, the T-period reward function (of the base model) associated with (µ, π) is written as: R(s0, µ(s0), π) = r(s0, µ) + T−1 � t=1 γt r(st, πt), (7) where s1 = fµ(s0, w1) and st+1 = fπt(st, wt+1) for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A T-periodic policy (µ, π) refers to the infinite sequence that repeatedly applies the T-horizon policy (µ, π), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', (µ, π, µ, π, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that despite it not being a stationary policy, the T-periodic policy (µ, π) can be implemented in the infinite horizon problem defined in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The value of the 9 T-periodic policy (µ, π) at state s0 is ¯U µ(s0, π) = E � ∞ � k=0 γkT R(sk, µ(sk), π) ��� s0 = s � = E � R(s0, µ(s0), π) + γT ¯U µ(sT , π) � , where, again, s1 = fµ(s0, w1) and st+1 = fπt(st, wt+1) for t > 0 within each cycle of T periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Figure 1 compares stationary policy ν and a T-periodic policy (µ, π) for the case of T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In the figure, we also illustrate how rewards can be written in an “aggregated” fashion over the T periods using (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ν ν ν ν ν ν ν ⋯ ν π1 π2 π3 ⋯ μ π1 π2 π3 μ R(s0, μ(s0), π) R(s4, μ(s4), π) Figure 1: Illustration of a stationary policy µ (upper timeline) and a T-periodic policy (µ, π) (lower timeline) for T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The periods covered by the T-period reward associated with (µ, π) is shown in the lower timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The optimal value function satisfies the following Bellman equation: ¯U ∗(s0) = max (µ,π) E � R(s0, µ(s0), π) + γT ¯U ∗(sT ) � , (8) where the “action” now involves selecting the π as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Denote (µ∗, π∗) an optimal T-periodic policy, which solves (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, we prove that the base model (5) and the hierarchical reformulation (8) are equivalent in a certain sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given an MDP ⟨S, A, W, f, r, γ⟩, the following hold: (i) The optimal value of the base model (5) is equal to the optimal value of the hierarchical refor- mulation (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', U ∗ = ¯U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (ii) An optimal stationary policy ν∗ with respect to the base model (5) is also an optimal policy for the hierarchical reformulation (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', ¯U ∗ = ¯U ν∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 10 Part (i) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 is most relevant to our situation in the sense that the optimal T- periodic policy (µ∗, π∗) is no better than the stationary optimal policy ν∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, solving the hierarchical reformulation (8) allows us to achieve the same value as the ν∗, the optimal policy to the original base model (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that, at this point, we have simply reformulated the problem, but (8) is no easier to solve that (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Despite the more favorable discount factor γT in (8), its action space is now effectively the space of T-horizon policies, rather than a single action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In the next section, we propose an approximation that allows us to fix a lower-level policy π and only optimize µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This allows us to enjoy the γT discount factor while maintaining the same action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4 The Frozen-State Approximation We propose a frozen-state approximation, where we make two simplifications to the T-period finite- horizon problem with terminal value U ∗ that is embedded in each T-period “time step” of (8), termed the lower-level problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' First, motivated by the slow transitions of x given in Assumption 1, we “freeze” slow states for all T periods of the lower-level problem, and second, we decouple the problem from the main MDP by solving an approximation with zero terminal value instead of U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The first simplification reduces the computation needed to solve the finite-horizon MDP, while the second simplification, due to the decoupling from the main problem, allows us to pre-compute an approximation to π∗, which we denote ˜π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' By then fixing ˜π∗, we are able to construct an auxiliary problem that proceeds at a timescale that is a factor of T slower than the MDP of the base model (equivalently, the discount factor becomes γT instead of γ), yet optimizing over the same action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This naturally leads to ADP algorithms with computational benefits (see Sections 6, 7, and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The number of periods T to freeze the state is a parameter to the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Figure 2 for a high-level illustration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' we provide a detailed description of the approach in the next few sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' It is important to note that the freezing of states only occurs “within the algorithm” as a step toward more efficient computation of policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our resulting policies are then implemented in the underlying base model MDP, which proceeds naturally according to its true dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our theoretical and empirical results always attempt to answer the question: how well does a approximate policy, which is computed by pretending certain states are frozen, perform in the true model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 11 π*1 π*2 π*3 x1, y1 x2, y2 x3, y3 U* (a) The lower-level problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', optimizing over π) em- bedded in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π*1 ˜π*2 ˜π*3 x1, y1 x1, y2 x1, y3 J*T ≡ 0 (b) The lower-level problem of the frozen-state approxi- mation, with frozen x1 and J∗ T instead of U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Figure 2: A comparison of the lower-level problem of the hierarchical reformulation vs the lower-level problem of the frozen-state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 The Lower-Level MDP (Frozen Slow States) We view the problem from period 1 to period T as the “lower level” of the frozen-state approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 To form the lower-level problem of the frozen-state approximation, we consider this T − 1 period problem in isolation: J ˜π 1 (x, y) = E �T−1 � t=1 γt−1 r(x1, yt, ˜πt) ��� (x1, y1) = (x, y) � and J∗ 1(x, y) = max ˜π J ˜π t (x, y) (9) where xt+1 = xt = x remains frozen, yt+1 = f ˜πt Y (x, yt, wt+1), and ˜π = (˜π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ˜πT−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The problem (9) can be solved using finite-horizon dynamic programming: accordingly, let the terminal J∗ T ≡ 0 and for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , T − 1, let J∗ t (x, y) = max a r(x, y, a) + γ E � J∗ t+1(x, y′) � , (10) where y′ = fY(x, y, a, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We also have the standard recursion for the performance of a policy: J ˜π t (x, y) = r(x, y, ˜πt(x, y)) + γ E � J ˜π t (x, f ˜πt Y (x, y, wt+1)) � , (11) with J ˜π T ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We denote by ˜H the Bellman operator of the lower-level problem, which is on the same timescale as the base model (hence, the discount factor is γ) and looks similar to the Bellman operator H defined in (6), but the transition of the slow-state x is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any state (x, y) and 3This corresponds to the periods relevant to π from (µ, π) in the hierarchical reformulation (8), whose structure the frozen-state approximation mimics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 12 lower-level value function Jt+1,4 define: � ˜HJt+1 � (x, y) = max a r(x, y, a) + γ E � Jt+1(x, fY(x, y, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (12) Note that (12) can be viewed as an approximation to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Analogously, let ¯H ˜π be the Bellman operator associated with (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Also, let ˜π∗ = (˜π∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ˜π∗ T−1) be the finite-horizon policy that is greedy with respect to J∗ t : ˜π∗ t (x, y) = arg max a r(x, y, a) + γ E � J∗ t+1(x, y′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' It may not immediately be clear why freezing slow states is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' There are two main computa- tional benefits to solving (10) instead of an analogous version of (10) without freezing x: In algorithms like value iteration (Puterman, 2014), each update requires computing expecta- tions over successor states, and therefore the number of successor states impacts the number of operations for each step of value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' When x is frozen, the number of successor states is much smaller since we only have successor fast states (y′): in other words, we only need to compute E � J∗ t+1(x, y′) � instead of E � J∗ t+1(x′, y′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5 Second, (10) can effectively be viewed as |X| independent MDPs, one for each x ∈ X, allowing for the possibility of computing the policy with additional parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In the nominal-state approximation discussed Section 7, we analyze the error of an approach that solves only a small number out of the |X| independent MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 The Upper-Level MDP (True State Dynamics) Let us now consider the upper-level problem of the frozen-state approximation, which is an infinite horizon problem with groups of T periods aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Denote the stationary upper-level policy by µ : S → A, which is the policy that we are attempting to optimize in the upper-level problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The upper-level problem takes two “inputs” related to the lower-level problem: (1) J1, an approximation 4We include time indexing on the value function to emphasize that this Bellman operator is used in a finite-horizon (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', non-stationary) setting, but the definition of ˜H itself does not depend on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 5Even in the case that the expectation is approximated via sampling, the former requires sampling from a lower- dimensional successor state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 13 of the optimal lower-level value J∗ 1, (2) π, a lower-level finite-horizon policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Fixing these inputs, the value at state s0 = (x0, y0) by executing policy µ is V µ(s0, J1, π) = E � ˜R(s0, µ(s0), J1) + γT V µ(sT (µ, π), J1, π) � , where sT (µ, π) is the state reached according to the true system dynamics by following (µ, π), starting from s0 and ˜R(s0, a, J1) = r(s0, a) + γ J1 � f(s0, a, w) � (13) is a one-step approximation to the T-period reward function R, defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Figure 3 helps to visualize the upper-level MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π1 π2 π3 ⋯ μ π1 π2 π3 μ x1 x1 x1 x0 x4 x5 x5 x5 γT J1 J1 Figure 3: Illustration of the upper-level problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Notably, the discount factor is γT and the reward function, from the point of view of µ, depends on the lower-level value function J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This value function is computed by freezing states, as visualized by the grey box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The optimal value (for this approximation) at state s0 can be written as V ∗(s0, J1, π) = max a E � ˜R(s0, a, J1) + γT V ∗(sT (a, π), J1, π) � , (14) where sT (a, π) is the state reached according to the true system dynamics by first taking action a and then following π, starting from s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Throughout the paper, we use the notation V µ(J1, π) and V ∗(J1, π) to refer to the value function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', S → R) obtained when the MDP is evaluated or solved for a fixed J1 and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We also define the Bellman operator associated with (14): � FJ1,πV � (s0) = max a E � ˜R(s0, a, J1) + γT V (sT (a, π)) � , (15) 14 which will become useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Recall that the optimal lower-level policy (that solves the frozen-state model) is denoted ˜π∗ and its optimal value is J∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ˜µ∗ be the optimal upper-level policy corresponding to these inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', the policy greedy with respect to V ∗(s0, J∗ 1, ˜π∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Thus, (˜µ∗, ˜π∗) is the resulting T-periodic policy from the frozen-state hierarchical approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' we refer to it as the T-periodic frozen-state policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 Characterizing the Exact and Frozen-State Reward Functions Recall that (µ∗, π∗) is an optimal T-periodic policy of the base model’s hierarchical reformulation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose π∗ is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, the Bellman equation of the base model reformulation is U ∗(x0, y0) = ¯U ∗(x0, y0) = max a E � R(x0, y0, a, π∗) + γT ¯U ∗(xT , yT ) � = max a E � r(x0, y0, a) + T−1 � t=1 γt r(xt, yt, π∗ t ) + γT U ∗(xT , yT ) � = max a E � r(x0, y0, a) + γ � HT−1U ∗� (x1, y1) � , (16) where the notation Hk is shorthand for k applications of the operator H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', HkU = H(Hk−1U) and H1U = HU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, the expected T-horizon reward can be written as E � R(x0, y0, a, π∗) � = E � r(x0, y0, a) + γ � HT−1U ∗� (x1, y1) − γT U ∗(xT , yT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (17) Given the optimal value J∗ 1 of the lower level (10), the T-horizon reward of the upper level (14) can be written as E � ˜R(x0, y0, a, J∗ 1) � = r(x0, y0, a) + γ E � J∗ 1(x1, y1) � = r(x0, y0, a) + γ � ˜HT−1J∗ T � (x1, y1), = r(x0, y0, a) + γ � ˜HT−1 0 � (x1, y1), (18) where 0 is the all-zero value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The difference between (17) and (18) can be interpreted as follows: in the former, we follow a lower-level policy that is aware of a terminal value U ∗ (but 15 exclude that value when defining the T-horizon reward), while in the latter, we follow a lower-level policy that sees zero terminal reward at the end of the T − 1 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The first step to understanding the performance of the frozen-state policy is to analyze the reward approximation E[ ˜R(s0, a, J∗ 1)] compared to the true reward E[R(s0, a, π∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 shows how the difference between two reward functions is dependent on the number of frozen periods T, along with the problem parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 (Reward Approximation Error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ⟨S, A, W, f, r, γ⟩ be a (α, dY)-fast-slow MDP satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let π∗ be the optimal lower-level policy for the base model reformulation (8) and J∗ 1 be the optimal (first-stage) value of the lower-level problem in the frozen-state approximation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any state s0 = (x0, y0) and action a, the approximation error between the T-horizon reward of hierarchical reformulation and the frozen-state approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', the discrepancy between (17) and (18), can be bounded as: ��E � R(s0, a, π∗) � − E � ˜R(s0, a, J∗ 1) ��� ≤ αdY � Lr T−2 � i=1 γi i−1 � j=0 Lj f � + γT−1LU � αdY T−2 � j=0 Lj f + γdY(α + 2)(T − 1) � , (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The detailed proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For more convenient notation, we define ϵr(γ, α, dY, Lr, Lf, T) to be the right-hand-side of (19): ϵr(γ, α, dY, L, T) = αdY � Lr T−2 � i=1 γi i−1 � j=0 Lj f � + γT−1LU � αdY T−2 � j=0 Lj f + γdY(α + 2)(T − 1) � , where L = (Lr, Lf, LU) emphasizes the dependence on the various Lipschitz constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In subse- quent sections, we use ϵr as an ingredient in analyzing the regret of various frozen-state policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 5 Regret of the Frozen-State Policy (˜µ∗, ˜π∗) In this section, we will show an upper bound on the regret from applying the T-periodic policy (˜µ∗, ˜π∗) instead of the optimal policy ν∗ in the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that this is the policy obtained if we were able to perfectly solve the frozen-state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We start with definitions of the 16 regret of both stationary and T-periodic policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Definition 2 (Regret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider a fast-slow MDP with initial state s0 and optimal policy ν∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The regret of a stationary policy ν is defined as R(s0, ν) = U ν∗(s0) − U ν(s0) and R(ν) = max s0 R(s0, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The regret of the T-periodic policy (µ, π) is defined as: R(s0, µ, π) = U ν∗(s0) − ¯U µ(s0, π) = ¯U ∗(s0) − ¯U µ(s0, π) and R(µ, π) = max s0 R(s0, µ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The second equality in the definition of R(s0, µ, π) uses the value equivalence between the base model and its hierarchical reformulation (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' As a follow-up comment to Remark 2, notice that V ∗(s0, J∗ 1, ˜π∗) does not directly enter the regret definition, as V ∗(s0, J∗ 1, ˜π∗) is just the optimal value of the frozen-state approximation, not the value of its implied greedy policy ˜µ∗ when evaluated in the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' However, the regret of course depends on V ∗(s0, J∗ 1, ˜π∗) indirectly, because ˜µ∗ depends on V ∗(s0, J∗ 1, ˜π∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In this section, we derive a bound on R(˜µ∗, ˜π∗), the regret of applying T-periodic policy (˜µ∗, ˜π∗) to the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' First, we start with a general lemma, that will be used throughout the paper as a tool to analyze variants of FSVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose we have an approximation (π, J1) to the lower-level solution (π∗, U∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Fur- ther, suppose we have an approximation V to the upper-level solution V ∗(J1, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider a T- periodic policy (µ, π), where µ(s0) = arg max a∈A E � ˜R(s0, a, J1) + γT V (sT (a, π)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (20) Then, the regret of (µ, π) can be bounded as follows: R(µ, π) ≤ � 2γT (1 − γT )2 + 2 1 − γT � ϵr(π∗, J1) + � 2γ2T (1 − γT )2 + 2γT 1 − γT � LU d(α, dY, T) + 2γT 1 − γT ��V ∗(J1, π) − V �� ∞, 17 where ϵr(π∗, J1) = maxs,a |E[R(s, a, π∗)] − E[ ˜R(s, a, J1)]| and d(α, dY, T) = 2dY(α + 1)(T − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This result above can be interpreted as the regret being bounded by reward error + end-of-horizon error + V -approximation error, which directly corresponds to the three terms in the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The reward error is due to freezing the slow state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' the end-of-horizon error is due to using zero terminal value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' and the V -approximation error is due to not solving the upper-level problem exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The main result of this section follows directly from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 and is given in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, which shows the expected regret R(˜µ∗, ˜π∗) of applying the policy learned from the frozen-state hierarchical approximation to the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ⟨S, A, W, f, r, γ⟩ be a (α, dY)-fast-slow MDP satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The regret of applying the T-periodic policy (˜µ∗, ˜π∗) in the base model is bounded by R(˜µ∗, ˜π∗) ≤ � 2γT (1 − γT )2 + 2 1 − γT � ϵr(γ, α, dY, L, T) + � 2γ2T (1 − γT )2 + 2γT 1 − γT � LU d(α, dY, T) where d(α, dY, T) = 2dY(α + 1)(T − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 with π = ˜π∗, J1 = J∗ 1, and V = V ∗(J∗ 1, ˜π∗), while noting that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, ϵr(π∗, J1) ≤ ϵr(γ, α, dY, L, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6 Frozen-State Value Iteration In this section, we introduce the our new approach: the frozen-state value iteration (FSVI) algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The main ideas of our approach are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Solve the lower-level MDP with frozen states to obtain a policy ˜π∗ and its value J∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since the lower-level problem is a finite horizon MDP, it can be solved exactly using T − 1 steps of VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Apply value iteration (VI) to the upper-level problem starting with some initial value function V 0, while using J∗ 1 to approximate the T-horizon reward and ˜π∗ for T-step transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note 18 that this is an infinite-horizon MDP that operates at a slower timescale and enjoys a much more favorable discount factor of γT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Before we dive into the details and analysis of FSVI, we start by mentioning that the naive approach to solving the base model MDP (5) is to directly apply standard VI (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', Bertsekas and Tsitsiklis (1996)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For completeness, we provide the full description in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Algorithm 1: Exact VI for the Base Model Input: Initial values U 0, number of iterations k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Output: Approximation to the optimal policy νk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , k do 2 for s in the state space S do 3 U i(s) = maxa r(s, a) + γ E � U i−1(f(s, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4 end 5 end 6 for s in the state space S do 7 νk(s) = arg maxa r(s, a) + γ E � U k(f(s, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8 end Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 is a well-known property that gives the required number of iterations of exact VI on the base model needed for the resulting policy to achieve a desired level of regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let νk be the result of running Algorithm 1 on the base model (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, R(νk) = ∥U νk − U ∗∥∞ ≤ 2rmaxγk+1 (1 − γ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Analysis of FSVI A detailed specification is given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We denote the resulting value function approxima- tion after k iterations of value iteration as V k, from which we obtain a policy ˜µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The T-periodic policy output by FSVI is formed by combining ˜µk with the optimal finite-horizon policy ˜π∗ from the lower-level MDP: (˜µk, ˜π∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 19 Algorithm 2: Frozen-State Value Iteration (FSVI) Input: Initial values J∗ T ≡ 0 and V 0, number of iterations k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Output: Approximation of the T-periodic frozen-state policy (˜µk, ˜π∗) and J∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 for t = T − 1, T − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , 1 do 2 for each slow state x ∈ X do 3 for each fast state y ∈ Y do 4 J∗ t (x, y) = maxa r(x, y, a) + γ E � J∗ t+1(x, fY(x, y, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 5 ˜π∗ t (x, y) = arg maxa r(x, y, a) + γ E � J∗ t+1(x, fY(x, y, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6 end 7 end 8 end 9 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , k do 10 for s0 = (x0, y0) in the state space X × Y do 11 V i(x0, y0, J∗ 1, ˜π∗) = maxa E � ˜R(s0, a, J∗ 1) + γT V i−1(xT , yT , J∗ 1, ˜π∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 12 end 13 end 14 for s0 = (x0, y0) in the state space X × Y do 15 ˜µk(x0, y0) = arg maxa E � ˜R(s0, a, J∗ 1) + γT V k(xT , yT , J∗ 1, ˜π∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 16 end 20 An instance of FSVI is associated with two primary quantities: k, the number of VI iterations, and T, the number of periods the slow state is frozen in the frozen-state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The next theorem makes use of this lemma to provide a bound on the regret of the policy obtained for a particular k and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let (˜µk, ˜π∗) be the resulting T-periodic policy after running FSVI for k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The regret incurred when running (˜µk, ˜π∗) in the base model satisfies R(˜µk, ˜π∗) ≤ � 2γT (1 − γT )2 + 2 1 − γT � ϵr(γ, α, dY, L, T) + � 2γ2T (1 − γT )2 + 2γT 1 − γT � LU d(α, dY, T) + 2rmaxγ(k+1)T (1 − γ)(1 − γT ), where the last term, which depends on k, accounts for the error due to value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Running Time of FSVI It is well-known that each iteration of standard VI has time complexity O � |S|2|A| � , which provides a contraction factor of γ (Littman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The upper level of FSVI, on the other hand, enjoys an improved contraction factor γT with the same per-iteration running time of O � |S|2|A| � , given that we pay a one-time fixed cost of solving the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The O � |S|2|A| � consists of |S||A| due to the number of state-action pairs at which to compute the Bellman update and another factor of |S| due to the number of successor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since freezing slow states restricts the successor states to Y, each iteration of the lower-level VI (Lines 2-7 of Algorithm 2) has running time O � |X||Y|2|A| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' An additional O � |S|2 T � is required to compute the T-step transition probabilities of following ˜π∗, to be used in the upper-level VI, resulting in a one-time fixed cost of O(|X||Y|2|A| T + |S|2 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Particularly when |X| is large, this can be a reasonable fixed cost to pay in order to get the much improved discount factor of γT going forward (as we will show in the numerical results of Section 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In Sections 7 and 8, we propose two extensions to FSVI that further reduce its computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 21 7 An Approximation for Nearly-Factored MDPs One potential drawback of Algorithm 2 is that solving the lower-level problem requires solving an MDP for each slow state x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In this section, we consider the situation where the reward function satisfies a certain nearly-factored assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In such a case, it is possible to design an extension of FSVI that solves the lower-level problem for a nominal slow state (or a small number of slow states) and then leverage the nearly-factored structure to approximate the lower-level values at other slow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Such a scheme would lower the one-time, fixed computational cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', the effort required to solve the lower-level problem) of applying FSVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Assumption 3 (Nearly-Factored Reward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A fast-slow MDP ⟨S, A, W, f, r, γ⟩ has a “nearly-factored” reward if there exists functions g, h, and ζ > 0 such that: |g(x) + h(y, a) − r(x, y, a)| ≤ ζ, for all x ∈ X, y ∈ Y, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In other words, the reward r is a sum of slow and fast components with error at most ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This terminology comes from the notion factored MDPs, a commonly-studied type of weakly- connected structure that notably assumes an additive reward function (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', Boutilier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2000) and Osband and Van Roy (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Although the analysis in this paper is based on additive separability of the reward function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', r(x, y, a) ≈ g(x) + h(y, a)), it is easy to extend the analysis to other types of separable rewards, such as r(x, y, a) ≈ ⟨g(x), h(y, a)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 The Nominal-State Approximation in the Lower Level We now seek to reduce the amount of computation needed to solve the lower-level MDP in the case where Assumption 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The two main ingredients of our approach are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' An approximation the lower-level (frozen-state) MDP by another MDP with reward function exactly equal to g(x) + h(y, a), instead of r(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We call it the separable approximation of the lower-level MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A solution ¯J1(x∗, y) to the separable approximation at a particular nominal state6 x∗ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6In this section, we discuss the case with a single nominal slow state x∗ for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' An extension to multiple 22 For any other x ∈ X, we can use Assumption 3 as a basis to approximate the value at x using only ¯J1(x∗, y), without the need to solve an MDP for slow state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Fix a nominal state x∗ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The Bellman recursion for the separable approximation at x∗ is analogous to (10): let ¯JT ≡ 0 and for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , T − 1, let ¯Jt(x∗, y) = max a g(x∗) + h(y, a) + γ E � ¯Jt+1(x∗, y′) � , (21) where y′ = fY(x∗, y, a, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that x∗ continues to be frozen and we have simply replaced the reward r with g + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ¯π be (T − 1)-period policy associated with (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since ¯Jt is only defined for the slow state x∗, we need to extend it to x ̸= x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ∆g(x) = g(x) − g(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Leveraging the separability of the reward function, we propose ¯Jt(x, y) = T−t−1 � i=0 γi∆g(x) + ¯Jt(x∗, y), (22) where we account for the reward error by applying a correction (but we do not account for error in the transitions from using x∗ instead of x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 for a full analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Such an approximation allows for solving the frozen-state MDP only for x∗ and using the result to approximate the value for other slow states, dramatically reducing the amount of overhead when using FSVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The Nominal FSVI algorithm makes use of this idea and is introduced in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider ¯Jt(x, y), as defined by (21) and (22), which is an approximation to the true frozen-state value J∗ t (x, y), as defined in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Under Assumption 3, it holds that: �� ¯Jt(x, y) − J∗ t (˜x, ˜y) �� ≤ �T−t−1 � i=0 γi �� ζ + Lr ∥x − ˜x∥2 � + �T−t−1 � i=0 γiLi f � Lr∥y − ˜y∥2 + �T−t−1 � i=1 Li f T−t−1 � j=i γj � Lr ∥x∗ − ˜x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The next proposition is a simple consequence of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' nominal slow states is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 23 Algorithm 3: Nominal FSVI Input: A nominal state x∗, initial values ¯JT ≡ 0 and V 0, number of iterations k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Output: Approximation of the T-periodic frozen-state policy (¯µk nom, ¯πnom) and ¯J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 for t = T − 1, T − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , 1 do 2 for each fast state y ∈ Y do 3 ¯Jt(x∗, y) = maxa g(x∗) + h(y, a) + γ E � ¯Jt+1(x∗, fY(x∗, y, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4 end 5 end 6 Define ¯Jt(x, y) using (22) and let ¯πnom be greedy with respect to ¯Jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 7 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , k do 8 for s0 = (x0, y0) in the state space X × Y do 9 V i(x0, y0, ¯J1, ¯πnom) = maxa E � ˜R(s0, a, ¯J1) + γT V i−1(xT , yT , ¯J1, ¯πnom) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 10 end 11 end 12 for s0 = (x0, y0) in the state space X × Y do 13 ¯µk nom(x0, y0) = arg maxa E � ˜R(s0, a, ¯J1) + γT V k(xT , yT , ¯J1, ¯πnom) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 14 end Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Recall from (13) the definition of ˜R(s0, a, J1), the approximation of the T-period reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Under Assumption 3, the error between using a nominal state approximation versus fully optimizing the frozen-state lower level is: ��E � ˜R(s0, a, J∗ 1) � − E � ˜R(s0, a, ¯J1) ��� ≤ T−1 � i=1 γiζ + �T−2 � i=1 Li f T−1 � j=i+1 γj � Lr max x ∥x∗ − x∥2, where J∗ t is computed as in Algorithm 2 and ¯Jt is computed as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let (¯µk nom, ¯πnom) be the result after running Nominal FSVI for k iterations with nominal state x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The regret incurred when running (¯µk nom, ¯πnom) in the base model satisfies R(¯µk nom, ¯πnom) ≤ � 2γT (1 − γT )2 + 2 1 − γT � ϵr,nom(γ, α, dY, L, T, ζ, x∗) + � 2γ2T (1 − γT )2 + 2γT 1 − γT � LU d(α, dY, T) + 2rmaxγ(k+1)T (1 − γ)(1 − γT ), 24 where the reward error is given by ϵr,nom(γ, α,dY, L, T, ζ, x∗) = ϵr(γ, α, dY, L, T) + T−1 � i=1 γiζ + �T−2 � i=1 Li f T−1 � j=i+1 γj � Lr max x ∥x∗ − x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The proof is similar to the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, except we need to compute the ϵr(π∗, ¯J1) term of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Combining Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 with the result in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, we can see that ϵr,nom(γ, α, dY, L, T, ζ, x∗) bounds ϵr(π∗, ¯J1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8 Value Function Approximation with a Linear Architecture So far in this paper, we have proposed a frozen-state approximation that is able to exploit a cer- tain fast-slow problem structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We then proposed another level of approximation using nominal slow states, which is valid when the MDP has a nearly-factored reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Both of these approximations use a tabular value function representation and therefore, each iteration of VI in the upper level requires looping through the entire state space X × Y (although the nominal-state approximation does reduce the computational burden in the lower-level problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In this section, we explore the use of a linear architecture for a more compact representation of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We show how approximate VI (AVI) can be combined with the frozen-state approximation, resulting in an algorithm that can scale to fast-slow MDPs with much larger state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The form of AVI that we use is based on the technique first proposed in Tsitsiklis and Van Roy (1996) and later also used in Zanette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A technical contribution we make here is to prove error bounds when this approximation architecture is used in a hierarchical setting that combines finite-horizon and infinite-horizon components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', our frozen-state VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 The Approximation Architecture Let φ(s) = � φ1(s), φ2(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , φM(s) �⊺ ∈ RM be an M-dimensional feature vector evaluated at state s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' An approximation { ˆJt(ωt)}T t=1 of the lower-level value functions {J∗ t }T t=1 of the frozen- state approximation is given by a sequence of parameter vectors {ωt}T t=1 with ωt ∈ RM, where the 25 component of ˆJt(ωt) associated with s is given by ˆJt(s, ωt) = φ⊺(s) ωt, for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (Note that since J∗ T ≡ 0, we can set ωT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=') The approximation ˆV (β) to the upper-level value function (of the frozen-state model) V ∗(J∗ 1, ˜π∗) is given by a parameter vector β ∈ RM, where ˆV (s, β) = φ⊺(s) β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For simplicity, we have used the same features in both the upper and lower levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' It is well-known that naive specifications of approximate value iteration applied to linear architec- tures can produce divergent behavior (Bertsekas and Tsitsiklis, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To circumvent this potential issue, our algorithmic approach depends on a set of pre-selected states ˜S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , sM}, an idea popularized in Tsitsiklis and Van Roy (1996), who showed that if certain assumptions on these states and the feature vectors are satisfied, then divergence is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A similar algorithm is also described more recently in Zanette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In this section, we need an ordering of the state space, so without loss of generality, we assume that S = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , N} and that the first M are the pre-selected states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', sm = m for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We make the following assumption on the feature vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' this is essentially Assumption 2 of Tsitsiklis and Van Roy (1996), adapted to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ˜S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , sM} be a set of pre-selected anchor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose the fol- lowing conditions on the features φ are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The vectors φ(s1), φ(s2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , φ(sM) are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' There exists some γ′ ∈ [γ, 1) such that for any state s ∈ S, there are coefficients θm(s) ∈ R for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , M satisfying: M � m=1 |θm(s)| ≤ 1 and φ(s) = γ′ γ M � m=1 θm(s) φ(sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The interpretation of this assumption is that the feature space {φ(s) | s ∈ S} lies in the convex hull 26 of the points defined by the pre-selected states: � ±(γ′/γ) φ(sm) �M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To reduce notional clutter, we will define κ = γ′/γ to be the amplification factor induced by the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Following Tsitsiklis and Van Roy (1996), we let Φ ∈ RN×M be a matrix with the s-th row equal to φ⊺(s) and let L ∈ RM×M be a matrix with the m-th row equal to φ⊺(sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' If we let G be the remaining rows of Φ, then we see that Φ = � L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Next, by Assumption 4, the matrix L has a unique matrix inverse L−1 ∈ RM×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We define Φ† ∈ RM×N as follows: for m ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , M}, suppose the m-th column of Φ† is equal to the m-th column of L−1, and let all the other entries of Φ† be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In other words, Φ† = [L−1 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, we see that Φ† is a left inverse of Φ: Φ†Φ = [L−1 0] � �� L G � �� = L−1L = I, where I ∈ RM×M is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Frozen-State Approximate Value Iteration Recall the lower-level Bellman operator ¯H from (12) and the upper-level Bellman operator F ˆJ1,ˆπ de- fined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The high-level idea behind our new approach, frozen-state approximate value iteration (FSAVI) is as follows: Lower-level AVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We first run approximate value iteration (under basis functions Φ) for the lower-level problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Letting ω∗ T = 0, the parameter ω∗ t is estimated by first evalu- ating ¯H ˆJt+1(ω∗ t+1) at the pre-selected states, and then computing ω∗ t so that ˆJt(s, ω∗ t ) = � ¯H ˆJt+1(ω∗ t+1) � (s) for s ∈ ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Upper-level AVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that after solving the lower level, we have parameter vectors ω∗ = (ω∗ 1, ω∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ω∗ T ), implying lower-level value functions ˆJt(ω∗ t ) = Φω∗ t and an associated greedy policy ˆπ(ω∗) = (ˆπ1(ω∗), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ˆπT−1(ω∗)): ˆπt(x, y, ω∗) = arg max a r(x, y, a) + γ E � ˆJt+1(x, y′, ω∗ t+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (23) For the upper level, the parameter βk is updated to βk+1 in iteration k + 1 by first evaluating F ˆJ1(ω∗ 1),ˆπ(ω∗) ˆV (βk) at the pre-selected states, then computing βk+1 so that ˆV (s, βk+1) = 27 � F ˆJ1(ω∗ 1),ˆπ(ω∗) ˆV (βk) � (s) for s ∈ ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that, taking Φ as fixed, the dependence of the upper level on the lower level can be represented succinctly through ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, we will use the simplified notation Fω := F ˆJ1(ω1),ˆπ(ω) going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To start, we define two new Bellman operators for the parameter space: ¯H′ = Φ† ◦ ¯H ◦ Φ and F ′ ω = Φ† ◦ Fω ◦ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To understand the definition of H′, consider the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose we start with a parameter vector ω∗ t+1, representing an approximate value function at time period t+1 given by ˆJt+1(ω∗ t+1) = Φω∗ t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The update to the next parameter vector ω∗ t is obtained by applying ¯H to ˆJt+1(ω∗ t+1), as we would normally do, and then applying Φ† to project back to the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the upper level, a similar logic holds to go from βi to βi+1: we first have the approximate upper-level value function ˆV (βi) = Φβk and then apply the normal Bellman update Fω∗, before lastly obtaining the updated parameter βi+1 using Φ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, we have ω∗ t = ¯H′(ω∗ t+1) and βi+1 = F ′ ω∗(βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We show in the Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5 of the Appendix that F ′ ω∗ is an (κγT )-contraction in the norm ∥ · ∥Φ on RM defined by ∥β∥Φ = ∥Φβ∥∞ and therefore has a fixed point β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We now define two quantities related to the approximation error of the linear architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Define the linear architecture approximation error for the lower level as εlow = maxt∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=',T} infωt∈RM ��J∗ t − ˆJt(ωt) �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (24) Let V ∗ ω be the fixed point of Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the upper level, we define error ϵup as εup = supω infβ∈RM ��V ∗ ω − ˆV (β) �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (25) Both (24) and (25) are related to the approximation errors defined in Tsitsiklis and Van Roy (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' moreover, taking a uniform bound over quantities that need to be approximated resembles the errors defined in Munos and Szepesvári (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 28 Algorithm 4: Frozen-State Approximate Value Iteration (FSAVI) Input: ˜S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , sM}, φ, initial weights ωT = β0 = 0, number of iterations k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Output: Approximation of the T-periodic frozen-state policy � ˆµ(βk, ω∗), ˆπω∗� and ˆJ1(ω∗) 1 for t = T − 1, T − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , 1 do 2 for each pre-selected state s = (x, y) ∈ ˜S do 3 Jt(x, y) = maxa r(x, y, a) + γ E � ˆJt+1(x, fY(x, y, a, w), ωt+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 4 end 5 Set remaining entries of Jt to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Update parameter vector: ω∗ t = Φ†Jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 6 end 7 Let ˆπω∗ be greedy with respect to ˆJt(ω∗ t ) = Φω∗ t , similar to (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , k do 9 for each pre-selected state s0 ∈ ˜S do 10 V i(s0) = maxa E � ˜R(s, a, ˆJ1(ω∗ 1)) + γT ˆV (sT (a, ˜πavi), βi−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 11 Set remaining entries of V i to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Update parameter vector: βi = Φ† V i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 12 end 13 end 14 for s0 in the state space S do 15 ˆµ(βk, ω∗)(s0) = arg maxa E � ˜R(s0, a, ˆJ1(ω∗ 1)) + γT ˆV (sT (a, ˜πω∗), βk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 16 end Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let (ˆµ(βk, ω∗), ˆπω∗) be the result after running FSAVI for k iterations for a given ˜S and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The regret incurred when running (ˆµ(βk, ω∗), ˆπω∗) in the base model satisfies R � ˆµ(βk, ω∗), ˆπω∗� ≤ � 2γT (1 − γT )2 + 2 1 − γT � ϵr,avi(γ, α, dY, L, T, γ′, εlow) + � 2γ2T (1 − γT )2 + 2γT 1 − γT � LU d(α, dY, T) + � 1 + κ 1 − κγT � εup + (κγT )k � κ2 − κ2(κγ)T+1 (1 − κγT )(1 − κγ) � rmax, where the reward error is given by ϵr,avi(γ, α, dY, L, T, γ′, εlow) = ϵr(γ, α, dY, L, T) + � 1 + κ 1 − κγ − (κγ)T (1 + γ) γ − κγ2 � εlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 29 9 Numerical Experiments We now apply our algorithms to three applied examples: (1) dynamic service allocation for a multi- class queue, (2) restless multi-armed bandit for asset maintenance optimization, and (3) energy demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the service allocation and asset maintenance problems, which have relatively smaller state spaces, we compare the VI-based variants: Base VI, Slow-agnostic VI, Q-learning, FSVI, and Nominal FSVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the energy demand response problem, which has a relatively larger state space, we compare the feature-based variants: Base AVI, Slow-agnostic AVI, DQN, FSAVI, and Nominal FSAVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We study the performance of each method with respect to its running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Building off of the discussion in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2, we define the running time of each iteration of a given method to be |Sobs||A||Ssucc|, where Sobs is the set of states at which we compute the Bellman update and Ssucc is the set of successor states evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We now give more details about the methods being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Base VI/AVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We refer to Algorithm 1 as the “Base VI” approach, which is simply standard VI applied to the base model with no changes (with discount factor γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The “Base AVI” base- line uses the VI with the linear architecture described in Section 8 directly on the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' More precisely, it iterates the approximate Bellman operator Φ† ◦ H ◦ Φ, where H is the Bell- man operator for the base model defined in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For succinctness, we have omitted a detailed algorithm specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For Base VI, we use exact transition probabilities when computing the expectation in the Bellman update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' However, because Base AVI is used for problems with larger state spaces, exact evaluation of the expectation is more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Instead, we resort to using a Monte Carlo-based sample average, with a sample of |Ssucc| = 40 successor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Slow-agnostic VI/AVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' As we discussed in the introduction, simplified decision models that ignore the slow state are often used in practice to improve tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To make this precise, we propose the following slow-agnostic Bellman operator to test a particular instantiation of this idea that averages over slow states and ignores it thereafter: � HfastW � (y) = max a∈A |X|−1 � x∈X � r(x, y, a) + γ E � W(fY(x, y, a, w) �� , 30 where W is a value function defined only over y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' “Slow-agnostic VI” iterates the op- erator Hfast, while “Slow-agnostic AVI” iterates Φ† ◦ Hfast ◦ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the AVI version, we use |Ssucc| = 20 successor state samples to approximate the expectation (this is smaller than in Base AVI because we only need to sample over Y, rather than X ×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Upon implementation, the policy ignores the slow state and only uses the value of y to take actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Q-learning/Deep Q-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We also compare our approaches to the well-known rein- forcement learning method, Q-learning (QL) (Watkins, 1989), along with its deep reinforce- ment learning variant, Deep Q-networks (DQN) (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2013, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Ours: FSVI/Nominal FSVI (multiple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Next, we have our VI-based approaches based on the frozen-state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The FSVI and Nominal FSVI methods are described in detail in Algorithms 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' “Nominal FSVI (multiple)” refers to an extension to multiple nominal states (as mentioned in Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The extension uses 9 nominal states for the service allocation problem (3 in each dimension of the 2-dimensional slow state) and 5 nominal states for the other two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The nominal states are equally spaced within the bounds of each slow state dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To apply Line 6 of Algorithm 3 (Nominal FSVI), we select the nearest nominal state by Euclidean distance to (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For both methods, we freeze slow states for T = 10 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Ours: FSAVI/Nominal FSAVI (multiple) Lastly, we have our AVI-based frozen-state methods, which also freeze slow states for T = 10 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' FSAVI is described in Algorithm 4 and Nominal FSAVI (multiple) is the natural combination of Nominal FSVI (multiple) and FSAVI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' the computational benefit here is that since x∗ is fixed, the lower-level linear ap- proximation only needs to be performed over y rather than (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For both AVI methods, we use M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 |S| pre-selected states with Gaussian radial basis functions for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='7 We use |Ssucc| = 250 successor state samples to approximate the expectation, spread over 25 simulated sample paths of the lower-level policy of length T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='8 7The M basis functions operate on normalized state space (with each state variable normalized to [0, 1]), with their centers spaced evenly and width equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 8Note that each iteration of the upper level of FSAVI is more computationally intensive than the upper level of Base AVI due to the need for simulating the lower level policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We account for this accurately in the computational cost calculation to provide a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 31 To evaluate policies, we use a truncated horizon of 100 periods (10T) and each method is evaluated over 10 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In order to create a more fine-grained performance plot, in each Bellman update of the VI-based variants, we allow evaluating the performance of policies “in between” complete iterations of VI, when the Bellman updates have only been executed for a subset of states in the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For each run, the order of state updates is randomly shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The AVI-based variants, however, require all pre-selected states to be observed before the parameter vector can be updated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' hence, we plot the performance of the policy after each full AVI iteration is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 0 1 2 3 4 5 Computational Cost 1e5 60 55 50 45 40 Test Reward QL Base VI Slow-agnostic VI FSVI Nominal FSVI (a) Multi-class service allocation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0 Computational Cost 1e5 8 10 12 14 16 18 Test Reward QL Base VI Slow-agnostic VI FSVI Nominal FSVI (b) Restless two-armed bandit 0 1 2 3 4 5 Computational Cost 1e7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 Test Reward 1e4 DQN Base AVI Slow-agnostic AVI FSAVI Nominal FSAVI (c) Energy demand response Figure 4: Performance of each algorithm on the three example applications: (a) multi-class service allocation and queueing, (b) restless two-armed bandit for asset maintenance optimization, and (c) energy demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The solid lines show median performance and the error bars represent the 10th-90th percentiles across 10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The x-axis shows the computational cost, defined by |Sobs||A||Ssucc|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Multi-Class Service Allocation with Stochastic Holding Costs We study a version of the multi-class service system problem based on the model presented in Ansell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2003) and later Brown and Haugh (2017), extended to the case of stochastic holding 32 costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Many variations of the original problem (without stochastic costs) have been studied in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' to name a few, see Cox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (1961), Harrison (1975), Van Mieghem (1995), and Gittins (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our variant with stochastic holding costs is partially motivated by the model in Lee and Vojnovic (2021), which proposes a learning algorithm for job scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The main idea behind this example is that when the holding cost stochastic process evolves slowly, it becomes a reasonable candidate for the slow state in our frozen-state framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We consider a single server and two classes of customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For class j ∈ {1, 2}, the arrival rate is µj9, service rate is λj, and the queue capacity is Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let yt,j be the number of customers in queue j and let zt be the class of the customer that is currently being served (if zt = 0, this represents when there is no customer and hence, no decision to be made).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Assume λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The holding cost of queue j, applied to each customer in the queue, is represented by an exogenous Markov process {xt,j}t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The dynamics of the system obey the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' With probability µj, a class zt = j customer arrives and yt+1,zt = min(yt,zt + 1, Qzt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' With probability λzt, the server completes serving the current customer and the queue length transitions according to yt+1,zt = yt,zt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' With probability 1 − λzt − � j µt,j, no event happens and yt+1,j = yt,j for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Each time the server completes serving the current customer, an action at ∈ {0, 1, 2} is taken to decide the class of customer to be served next, with at = 0 representing the case where no customer needs to be served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The reward function is simply the negative of the total cost: r(xt,1, xt,2, yt,1, yt,2, at) = − �2 j=1 xt,jyt,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the frozen-state approximation, we let the slow state be (xt,1, xt,2) and the fast state be (yt,1, yt,2, zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the nominal state approximation, we take the following approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We solve the lower- level problem by letting ¯Jt(x1, x2, y1, y2, z) = ¯Jt,1(x1, y1, z) + ¯Jt,2(x2, y2, z), where ¯Jt,j(xj, yj, z) represents the reward of the optimal frozen-state policy associated with queue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The value of ¯Jt,j(xj, yj, z) is computed by setting ¯JT,j(xj, yj, z) = −xjyj and for other t < T, ¯Jt,j(xj, yj, z) = 9We abuse notation here by reusing µ, which was previously used in the paper to denote the upper-level policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 33 rj(xj, yj, a∗) + γ E � ¯Jt+1,j(xj, y′ j, z′) � , where rj(xj, yj, a) = −xjyj, (y′ 1, y′ 2, z′) = fY(s, a, w), and a∗ = arg max a∈A r(x1, x2, y1, y2, a) + γE � ¯Jt+1(x1, x2, y′ 1, y′ 2, z′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We then use a multiplicative decomposition rj(xj, yj, a) = gj(xj)hj(yj) with gj(xj) = −xj and hj(yj) = yj, and apply a multiplicative correction gj(xj)/gj(x∗ j) to get ¯Jt,j(xj, yj, z) from ¯Jt,j(x∗ j, yj, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To make the results more easily interpretable as a function of the holding cost, we consider a case where the two classes have the same arrival rates, service rates, and capacities: µ1 = µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2, λ1 = λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3, and Q1 = Q2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the cost process, let {ai}6 i=1 be six equally spaced values from the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Our cost process transitions on the set {ai}6 i=1 such that if xt,j = ai, then xt+1,j = xt,j with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='9, xt+1,j = a(i−1)∨1 with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='05, and xt+1,j = a(i+1)∧n with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Results and Discussion for Service Allocation Figure 4a shows the performance of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' As a function of the computational effort, Nominal FSVI and FSVI quickly outperform the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Base VI and Q-learning converge more slowly, but eventually they find policies with decent (but not superior) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Not surprisingly, Slow-agnostic VI plateaus quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' These results illustrate that for the multi-class service allocation problem, although the slow state is important enough that it should not be ignored, there are drastic computational benefits of applying the frozen-state idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Figure 5 provides a qualitative comparison of the various policies by visualizing the actions taken in two situations: when the cost of queue 1 is lower and when the cost of queue 2 is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' There are a few main takeaways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' First, the upper level policies, along with the first 8 periods of the lower-level policies, of FSVI and Nominal FSVI resemble the Base VI policy: that is, they tend to serve customers in currently high cost queue, as long as that queue is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Second, we observe deficiencies in the Slow-agnostic VI and Q-learning policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' By ignoring the slow state, Slow-agnostic VI’s policy serves the nonempty queue when the other queue is empty, and tends to serve the shorter queue when both queues are nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Q-learning also finds a suboptimal policy within the computational budget, tending to focus on the longer queue (but not the higher cost queue) in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 34 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (a) Base VI 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (b) Slow-agnostic VI 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (c) Q-learning 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (d) FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' upper 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (e) FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' lower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' t = 8 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (f) FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' lower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' t = 9 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (g) Nominal FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' upper 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (h) Nominal FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' lower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' t = 8 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 < xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 0 1 2 3 Queue 1 0 1 2 3 Queue 2 xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1 > xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 2 (i) Nominal FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' lower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' t = 9 Figure 5: Visualization of the policies learned by all methods for the multi-class service allocation and queueing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In each plot, the x- and y-axes represent the length of the first and the second queues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A red square indicates the policy choosing to serve customers in queue 1 for the majority of runs (replications), while a blue square indicates the policy choosing to serving customers in queue 2 more often than not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The shade of the color represents the frequency of taking that action over 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For each policy, we show two situations: when the holding cost of queue 1 is lower (xt,1 ≤ xt,2, where we expect the optimal policy to primarily serve queue 2), and when the holding cost of queue 2 is lower (xt,1 > xt,2, where we expect the optimal policy to primarily serve queue 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that the first row shows methods that learn stationary policies, while the second and third rows show various snapshots of the non-stationary policies learned by FSVI and Nominal FSVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Third, we see that in the final lower-level period (t = 9), FSVI and Nominal FSVI learn poor policies due to the “end-of-horizon effect” of the zero terminal value approximation in the finite- horizon problem in the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' When both queues are nonempty, regardless of which queue is served, there will be no downstream impact (and the queue will not shorten).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Although the frozen state approximation introduces suboptimal decisions in some periods, we can see that the good actions taken in the upper level and early stages of the lower level outweigh this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 35 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Restless Multi-Armed Bandit with Environmental States We now move on to the case of a restless multi-armed bandit (Whittle, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Weber and Weiss, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Killian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Zhang and Frazier, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This problem arises in a wide variety of settings, including from machine maintenance (Smallwood and Sondik, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Ruiz-Hernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2020), dynamic assortment planning (Brown and Smith, 2020), public health intervention decisions (Bhattacharya, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Mate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2020), and preventative healthcare (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We construct a two-armed instance that features an exogenous environmental context, inspired by maintenance problems, to illustrate our frozen-state algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose there are two arms j ∈ {1, 2} and each arm can either be operational (yt,j = 1) or non-operational (yt,j = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' At the end of each period, the operator chooses whether each arm should receive an intervention (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', maintain or repair): at = (at,1, at,2), with each at,j ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Each intervention incurs a cost of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The state yt+1,j of arm j in the next period depends on three factors: its current state yt,j, whether it is maintained at, and the exogenous environment state that describes the condition of the overall system xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We consider 25 possible values for the environment state: xt ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , 24}), with xt+1 equal to xt + 2, xt + 1, xt, xt − 1 and xt − 2 with probabilities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='15, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='05 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lower values of xt increase the probability of of an arm becoming (and staying) non-operational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' the precise values of the transition probabilities are described in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The immediate reward function is r(xt, yt,1, yt,2, at) = 2 � j yt,j −� j at,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We view the environment state xt as the slow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We use the additive nominal state approximation proposed in Section 7, which trivially applies here because the reward function does not depend on the slow state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Results and Discussion for Restless Bandit Figure 4b shows the performance of the algorithms as a function of the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The trends are similar to what we observed in Figure 4a, except here we see some notable instability of the Slow-agnostic policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A likely explanation is given at the end of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We use Figure 7 to visualize the final policies learned by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The x-axes represent the environment state, while the y-axis represents the operating state of both arms (machines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Darker gray squares on the left (right) panels indicate a high frequency of intervening arm 1 (arm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We 36 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='8 at = 0 at = 1 xt = 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='99 at = 0 at = 1 xt = 24 ⋯ improving environment state Figure 6: The transition probabilities for both arms in the two extreme environment states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' When the environment is in the poorest condition xt = 0, each arm stays in the non-operational state (yt,j = 0 to yt+1,j = 0) with probabilities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='99 (no intervention, at = 0) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5 (with intervention, at = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' They go from operational to non-operational (yt,j = 1 to yt+1,j = 0) with probabilities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='7 (no intervention, at = 0) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 (with intervention, at = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' When the environment is in the best condition xt = 24, the same probabilities are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For other values of the environment xt, the probabilities are equally spaced between the two extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' observe that although the Base VI policy is not particularly stable across the 10 runs, Base VI, FSVI, and Nominal FSVI all learn a policy with similar structure: intervene non-operating arms and always intervene if the environment state is smaller than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given the performance plot in Figure 4b, this type of structure results in high performing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In addition, we observe that the frozen-state variants are significantly more stable across runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The Slow-agnostic policy learns to focus on non-operating arms, but its inability to distinguish between slow states hurts its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To understand the unstable behavior observed in Fig- ure 4b, note that Slow-agnostic VI is only able to produce policies that apply the same action to entire rows of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Considering what a “good” policy looks like (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', the Base VI, FSVI, and Nominal FSVI plots), it becomes clear that different Slow-agnostic VI policies can have dramati- cally different performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the sake of illustration, let us suppose that the FSVI policy for arm 1 is indeed optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' for Slow-agnostic VI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' consider switching from the current policy that intervenes in arm 1 for (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 0) and (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1) to a policy that intervenes for (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' and (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 0) — we suddenly go from having 5 states with suboptimal actions (xt ≤ 4 for yt = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 0)) to 20 states 37 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 1 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 2 (a) Base VI 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 1 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 2 (b) Slow-agnostic VI 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 1 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 2 (c) Q-learning 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 1 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 2 (d) FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' upper and FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' lower t = 5 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 1 0 5 10 15 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='0) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1) Machine 2 (e) Nominal FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' upper and Nominal FSVI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' lower t = 5 Figure 7: Visualization of the resulting policies across methods for the restless bandit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In each subplot, the x-axis is the environment state and the y-axis show the operating state of each arm (machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The left panel shows whether to intervene machine 1, while the right panel shows whether to intervene machine 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Darker gray squares indicate a high frequency of intervention across the 10 runs (replications) of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that in the last two subplots, the policy shown is the same policy for both the upper level and lower level, period t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' with suboptimal actions (xt ≥ 5 for yt = (1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In other words, the lack of flexibility of the policy space can cause widely varying performance across policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Finally, we note that Q-learning does not seem to have learned a policy with any notable structure, except that it more likely to intervene when both arms are in the non-operational state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 Energy Demand Response For our last example application, we consider the problem of an energy aggregator that provides demand response to consumers, while simultaneously selling the demand reduction to the demand response market, inspired by Khezeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' At the beginning of each period t, the aggregator commits to delivering an amount of energy at in the real-time (RT) market using a forward contract, settled at the day-ahead (DA) price xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The DA price process follows a discretized Ornstein- Uhlenbeck process xt+1 − xt = c1(c2 − xt) + ϵda t+1, where c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2237, c2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4095, estimated using data from the California Independent System Operator (CAISO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' To complete the promised delivery, the aggregator uses dynamic pricing and offers payment to elicit demand reduction, or demand response, from its customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Following the model of Khezeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (2017), if the demand reduction (which can be considered equivalent to delivering energy) falls short of the forward contract at, the aggregator purchases the remaining energy at the RT shortage price p− t , and if the demand reduction exceeds at, the aggregator sells the additional energy at the RT overage price p+ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We model p− t and p+ t using multiplicative adjustments to the DA price xt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', p− t = xty− t and p+ t = xty+ t , with yt+ < 1 < y− t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We consider a system with two (large) customers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', a university or company) m ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The demand response is a function of the compensation provided by the aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For convenience, we represent the offered compensation as a fraction of the the DA price, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', qt,m = αt,m xt, where αt,m ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='275, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='45, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='625, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='8} for each m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The vector αt = (αt,1, αt,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , αt,m) represents the pricing decisions made by the aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The demand response follows a linear model dm(xt, αt,m) = bm,1 + bm,2 (αt,mxt) + ϵdr t+1,m, where ϵdr t+1,m is the noise and bm,1, bm,2 are known coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The state of the system is st = (xt, y− t , y+ t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The reward function is r(xt, y+ t ,y− t , at, αt) = xtat − 2 � m=1 qt,m E � dm(xt, αt,m) � + E � xty+ t � 2 � m=1 dm(xt, αt,m) − at �+ − xty− t � at − 2 � m=1 dm(xt, αt,m) �+� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The DA price process xt is rounded/clipped to values in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 increments between 10 and 30, and its noise term ϵda t+1 follows discretized normal distribution with standard deviation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The RT adjustment factors y− t and y+ t are each uniformly distributed over 10 equally spaced discrete values 39 in the ranges [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='8] and [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='05, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='25], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The possible values of the pricing decisions at,m are limited to the set {10, 12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , 20} in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Finally, for the demand response model, we set b1,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='72, b1,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='55, b2,1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='87, b2,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='26, with ϵdr t+1,m follows discretized normal distribution with standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' With this particular set of coefficients, the impact of an additional unit of compensation offered is larger for customer 1 than for customer 2, but maximum expected demand response of customer 2 is larger than that of customer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For the nominal state approximation, we use a multiplicative decomposition, where the reward function is approximated as g(x∗ t ) h(x∗ t , y+ t , y− t , at, αt) with g(x∗ t ) = x∗ t and h(x∗ t ,y+ t , y− t , at, αt) = at − 2 � m=1 αt,m E � dm(x∗ t , αt,m) � + E � y+ t �� m dm(x∗ t , αt,m) − at �+ − y− t � at − 2 � m=1 dm(x∗ t , αt,m) �+� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Instead of the additive correction term used in (22), we apply a multiplicative correction g(x)/g(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 1200 1400 1600 1800 0 50 100 150 (a) Base AVI 1200 1400 1600 1800 0 50 100 150 (b) FSAVI 1200 1400 1600 1800 0 50 100 150 (c) Nominal FSAVI 1200 1400 1600 1800 0 50 100 150 (d) Slow-agnostic AVI 1200 1400 1600 1800 0 50 100 150 (e) DQN Figure 8: Histograms of the cumulative amount bid by the aggregator over 100 periods (x-axis) for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The y-axis are counts over 1000 simulations of the resulting policy, and the dotted vertical line shows the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6 0 50 100 150 Customer 1 Customer 2 (a) Base AVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6 0 50 100 150 Customer 1 Customer 2 (b) FSAVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6 0 50 100 150 Customer 1 Customer 2 (c) Nominal FSAVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6 0 50 100 150 Customer 1 Customer 2 (d) Slow-agnostic AVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6 0 50 100 150 Customer 1 Customer 2 (e) DQN Figure 9: Histograms showing the proportion of the total amount bid that is satisfied by each customer (x-axis) over 1000 simulations of 100 periods each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The y-axis are counts over 1000 simulations, and the dotted vertical lines show the means for the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This is essentially an illustration of the aggregator’s pricing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Results and Discussion for Demand Response The performance comparisons for the demand response problem are shown in Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The results confirm that the trends observed for the VI-based methods in Figures 4a and 4b hold up in the AVI setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The new methods, FSAVI and Nominal FSAVI continue to outperform the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Base AVI and DQN both improve slowly, but continually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Slow-agnostic AVI, however, plateaus and displays a small yet still noticeable oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For a qualitative understanding of the differences between the policies, we show in Figure 8 histograms of the aggregator’s cumulative bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Base AVI, FSAVI and Nominal FSAVI show similar bimodal bidding behavior and mean values, while the bidding behaviors of Slow-agnostic AVI and DQN deviate noticeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The former has a much narrower distribution, while the latter shows more uniform behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Figure 9 shows histograms of the proportions of the overall amount bid that is satisfied by each customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Once again, we observe Base AVI, FSAVI, and Nominal FSAVI consolidating around a particular pricing behavior, with average amount satisfied by customer 1 being slightly higher (DQN exhibits the reverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 41 10 Conclusion In this paper, we studied a new class of MDPs with a type of structure called fast-slow structure, motivated by practical applications where some states move slowly and are relatively less influential than others, but still important enough not to ignore them during modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Based on this structure, we propose a set of new algorithms based on the idea of freezing the slow state for several periods at a time to ease the computational burden of approximate dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For each algorithm, we analyze the regret of the resulting policy using a novel analysis of various Bellman operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Empirically, on three example applications, we show that our new frozen-state methods converge significantly faster to good policies than standard methods, and notably, ignoring the slow state leads to unstable training and low-performing policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, our method can be viewed as a viable compromise between solving the full MDP (often computationally intractable) and completely ignoring states during the modeling process (computationally easy but potentially highly suboptimal in the true model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 42 References M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Albadi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' El-Saadany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A summary of demand response in electricity markets.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Tang, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Desai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Reducing energy costs for ibm blue gene/p via power-aware job scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In Workshop on Job Scheduling Strategies for Parallel Processing, pages 96–115.' metadata={'source': 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via Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' In IECON 2006-32nd Annual Conference on IEEE Industrial Electronics, pages 2883–2888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' IEEE, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 48 A Proofs from Sections 3 and 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Technical Lemmas Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider a (α, dY)-fast-slow MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any states (x0, y0) and (˜x0, ˜y0), let (xt, yt) and (˜xt, ˜yt) be the states reached after t transitions under a policy π = (π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πt−1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', (xt, yt) = fπ(xt−1, yt−1, wt) and (˜xt, ˜yt) = fπ(˜xt−1, ˜yt−1, ˜wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, for any policy π, we have (i) ∥xt − ˜x0∥2 ≤ tαdY + ∥x0 − ˜x0∥2, (ii) ∥xt − ˜xt∥2 ≤ 2tαdY + ∥x0 − ˜x0∥2, (iii) ∥yt − ˜yt∥2 ≤ 2tdY + ∥y0 − ˜y0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 is a consequence of Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The following two lemmas are about properties of the Bellman operators H and ˜H (recall that ˜H is the frozen-state version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any state (x, y) and any two value functions V, V ′ : X × Y → R, we have |( ˜HtV )(x, y) − ( ˜HtV ′)(x, y)| ≤ γt max y∈Ytx ��V (x, y) − V ′(x, y) ��, where Yt s is the set of fast states reachable from s = (x, y) after t transitions of fY(x, ·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The result follows by the contraction property of the Bellman operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 (Discrepancy between H and ˜H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider a value function V : X ×Y → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose there exists LV > 0 such that for any states (x, y) and (˜x, ˜y), it holds that |V (x, y) − V (˜x, ˜y)| ≤ LV ∥(x, y) − (˜x, ˜y)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, ��(HtV )(x, y) − ( ˜HtV )(˜x, ˜y) �� ≤ ��(x, y) − (˜x, ˜y) �� 2 � Lr t−1 � i=0 (γLf)i + LV (γLf)t � + αdY � Lr t−1 � i=1 γi i−1 � j=0 Lj f + LV γt t−1 � j=0 Lj f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 49 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We need to show that for each t ≥ 1, ��(HtV )(x, y) − ( ˜HtV )(˜x, ˜y) �� ≤ φt,1 ��(x, y) − (˜x, ˜y) �� 2 + φt,2 (αdY), (26) for coefficients φt,1 and φt,2 defined as φt,1 = � Lr t−1 � i=0 (γLf)i + LV (γLf)t � and φt,2 = � Lr t−1 � i=1 γi i−1 � j=0 Lj f + LV γt t−1 � j=0 Lj f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let (x′, y′) = (fX (x, y, a, w), fY(x, y, a, w)) and (˜x′, ˜y′) = (fX (˜x, ˜y, a, w), fY(˜x, ˜y, a, w)) be one-step transitions starting from (x, y) and (˜x, ˜y), according to the true system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For t = 1: ��(HV )(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − ( ˜HV )(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� = ���max a � r(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) + γ E[V (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′)] � − max ˜a � r(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜a) + γ E[V (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′)] ���� ≤ Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + γ max a E ��V (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′) − V (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′) �� ≤ Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + LV γ max a E ∥(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′)∥2 = Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + LV γ max a E � ∥(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′) − (˜x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′)∥2 + ∥(˜x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′)∥2 � ≤ Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + LV γ � Lf∥(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y)∥2 + αdY � = φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 (αdY),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' which verifies the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let us now assume that (26) holds for t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ��(HtV )(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − ( ˜HtV )(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� = ���max a � r(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) + γ E[(HV )t−1(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′)] � − max ˜a � r(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜a) + γ E[( ˜HV )t−1(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′)] ���� ≤ Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + γ max a E ��(HV )t−1(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′) − ( ˜HV )t−1(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′) �� ≤ Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + γ � φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 ��(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y′) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y′) �� 2 + φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 (αdY) � ≤ Lr ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + γ � φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 � Lf ��(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + αdY � + φt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 (αdY) � ≤ � Lr + γLfφt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 ���(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y) − (˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y) �� 2 + � γφt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 + γφt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 � (αdY),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' where the second inequality follows by the induction hypothesis and the third inequality follows by 50 the same steps as in the case of t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' It is straightforward to verify that φt,1 = Lr + γLfφt−1,1 and φt,2 = γφt−1,1 + γφt−1,2, which completes the induction step and the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 We consider an MDP ⟨S, A, W, f, r, γ⟩ and note that U ∗ is the unique optimal solution of the base model (5), and there exists a stationary optimal policy ν∗(x, y) = arg max U ∗(x, y) that attains this optimal value (Bertsekas and Shreve, 2004, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Fix a state s0 ∈ S and for t > 0 and a sequence of policies π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πt−1, define the notation: s1(π0) = fπ0(s0, w1) and st′+1(π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πt′) = fπt′(st′(π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πt′−1), wt′+1) for t′ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, we have U ∗(s0) = max π0 r(s0, π0) + γ E � U ∗(s1(π0)) � = r(s0, ν∗) + γ E � U ∗(s1(ν∗)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (27) By expanding the U ∗(s1(π0)) and U ∗(s1(ν∗)) terms in (27), we have the following: U ∗(s0) = max π0,π1 E � r(s0, π0) + γ r(s1(π0), π1) + γ2 U ∗(s2(π0, π1)) � = E � r(s0, ν) + γ r(s1(ν∗), ν∗) + γ2 U ∗(s2(ν∗, ν∗)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let π = (π0, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πT−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Repeating the expansion, we obtain: U ∗(s0) = max π E �T−1 � t=0 γt r � st(π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πt−1), πt � + γT U ∗� sT (π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , πT−1) � � (28) = E �T−1 � t=0 γt r � st(ν∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ν∗), ν∗� + γT U ∗� sT (ν∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ν∗) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (29) Observe that (28) is in same form as the Bellman equation (8) for the hierarchical reformulation (with T-horizon reward function R and value function ¯U), which has a unique optimal solution ¯U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore U ∗(s0) = ¯U ∗(s0) and (i) is proved when we recall that s0 was chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Part (ii) 51 follows because by (29), it is clear that (ν∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ν∗) solves (28) and hence also (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Using (17) and (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' the difference between the two reward functions can be expanded as follows: ��E[R(s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗)] − E[ ˜R(s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' J∗ 1)] �� = ��r(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) + γ E[(HT−1U ∗)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1)] − γT E[U ∗(xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' yT )] − r(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) − γ E[( ˜HT−10)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1)] �� = γ ��E[(HT−1U ∗)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1)] − E[( ˜HT−10)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1)] − γT−1E[U ∗(xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' yT )] �� ≤ γ E ��(HT−1U ∗)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1) − ( ˜HT−1U ∗)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1) �� � �� � Term A + γ E ��( ˜HT−1U ∗)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1) − ( ˜HT−10)(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y1) − γT−1 U ∗(xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' yT ) �� � �� � Term B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' where (xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' yt) is the state obtained after transitioning from (x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y0) according to the true dynamics f = (fX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' fY) for t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We now work on Terms A and B separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Noting that U ∗ has Lipschitz constant LU by Assumption 2, we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 to Term A to obtain Term A ≤ αdY � Lr T−2 � i=1 γi i−1 � j=0 Lj f + γT−1LU T−2 � j=0 Lj f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (30) Moving on to Term B, since the reward function r ≥ 0, it follows that U ∗(s) ≥ 0 for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Also, the monotonicity of ˜H implies that ( ˜HT−1U ∗) ≥ ( ˜HT−10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, applying Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2, ( ˜HT−1U ∗)(x1, y1) − ( ˜HT−10)(x1, y1) = ��( ˜HT−1U ∗)(x1, y1) − ( ˜HT−10)(x1, y1) �� ≤ γT−1 maxy∈YT −1 s1 ��U ∗(x1, y) − 0 �� = γT−1 maxy∈YT −1 s1 U ∗(x1, y), where YT−1 s1 is the set of fast states reachable from s1 = (x1, y1) after T −1 transitions of fY(x1, ·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ˜ys1 = arg maxy∈YT −1 s1 U ∗(x1, y) be the fast state that attains the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that ˜ys1 depends on s1, which is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Combining with the rest of Term B, we have Term B ≤ γ E ��γT−1U ∗(x1, ˜ys1) − γT−1 U ∗(xT , yT ) �� 52 ≤ max ω∈Ω γT ��U ∗� x1(ω), ˜ys1(ω) � − U ∗� xT (ω), yT (ω) ��� ≤ max ω∈Ω γT LU � ∥x1(ω) − xT (ω)∥2 + ∥˜ys1(ω) − yT (ω)∥2 � (31) ≤ γT LUdY(α + 2)(T − 1), (32) where (31) follows by Assumption 2 and (32) comes from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 (we use that xT (ω) is T − 1 transitions from x1(ω) and both ˜ys1(ω) and yT (ω) are both T − 1 transitions from y1(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Finally, we have Terms A + B ≤ αdY � Lr T−2 � i=1 γi i−1 � j=0 Lj f + γT−1LU T−2 � j=0 Lj f � + γT LUdY(α + 2)(T − 1) = αdY � Lr T−2 � i=1 γi i−1 � j=0 Lj f � + γT−1LU � αdY T−2 � j=0 Lj f + γdY(α + 2)(T − 1) � which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' B Proofs for Section 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Technical Lemmas Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider two MDPs, M1 and M2, who differ in their transition and reward functions: Mi = ⟨S, A, W, fi, ri, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let U ∗ i be the optimal value function of Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that (a) |r1(s, a) − r2(s, a)| ≤ ϵr for all s ∈ S and a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (b) ∥f1(s, a, w) − f2(s, a, w)∥2 ≤ d for all s ∈ S, a ∈ A, and w ∈ W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' and (c) there exists L1 > 0 such that |U ∗ 1 (s) − U ∗ 1 (s′)| ≤ L1∥s − s′∥2 for all s, s′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, the difference in optimal values of the two MDPs can be bounded as follows: ��U ∗ 1 (s) − U ∗ 2 (s) �� ≤ ϵr + γL1d 1 − γ for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 53 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ˆs = arg maxs∈S |U ∗ 1 (s) − U ∗ 2 (s)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We will analyze ��U ∗ 1 (ˆs) − U ∗ 2 (ˆs) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ��U ∗ 1 (ˆs) − U ∗ 2 (ˆs) �� = ���max a1∈A � r1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a1) + γ E � U ∗ 1 � f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) ��� − max a2∈A � r2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a2) + γ E � U ∗ 2 (f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) ����� ≤ max a∈A ��r1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) + γ E � U ∗ 1 � f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) �� − r2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) − γ E � U ∗ 2 (f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) ��� ≤ max a∈A ��r1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) − r2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a) �� + γ max a∈A ��E � U ∗ 1 � f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) �� − E � U ∗ 2 � f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) ���� ≤ ϵr + γ max a∈A ��E � U ∗ 1 � f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) � − U ∗ 1 � f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) ���� + γ max a∈A ��E � U ∗ 1 � f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) � − U ∗ 2 � f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) ���� ≤ ϵr + γL1 max a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='w ∥f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) − f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)∥2 + γ max s∈S ��U ∗ 1 (s) − U ∗ 2 (s) �� ≤ ϵr + γL1d + γ ��U ∗ 1 (ˆs) − U ∗ 2 (ˆs) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Rearranging, we have ��U ∗ 1 (ˆs) − U ∗ 2 (ˆs) �� ≤ ϵr + γL1d 1 − γ , which completes the proof if we recall how ˆs was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider two MDPs, M1 and M2, who differ in their transition and reward functions: Mi = ⟨S, A, W, fi, ri, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let U ∗ i be the optimal value function of Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that (a) |r1(s, a) − r2(s, a)| ≤ ϵr for all s ∈ S and a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (b) ∥f1(s, a, w) − f2(s, a, w)∥2 ≤ d for all s ∈ S, a ∈ A and w ∈ W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (c) there exists L1 > 0 such that |U ∗ 1 (s) − U ∗ 1 (s′)| ≤ L1∥s − s′∥2 for any s, s′ ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' and (d) |U ∗ 1 (s) − U ∗ 2 (s)| ≤ ϵU for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ˜π2 be a policy that is an approximation of the optimal policy for M2, in the sense that: ˜π2(s) = arg max a∈A � ˜r2(s, a) + γ E � ˜U2 � ˜f2(s, a, w) ��� , (33) where |r2(s, a) − ˜r2(s, a)| ≤ ˜ϵr, ∥f2(s, a, w) − ˜f2(s, a, w)∥2 ≤ ˜d, and |U ∗ 2 (s) − ˜U2(s)| ≤ ˜ϵU for all 54 s ∈ S, a ∈ A, and w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, the value of ˜π2 when implemented in M1 has regret bounded by: ��U ∗ 1 − U ˜π2 1 �� ∞ ≤ 2(ϵr + ˜ϵr) + 2γ(ϵU + ˜ϵU) + 2γL1(d + ˜d) 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This lemma is a generalization and extension of Corollary 1 of Singh and Yee (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let π∗ 1 be an optimal policy for M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Using (33), it follows that ˜r2(s, π∗ 1(s)) + γ E � ˜U2( ˜f2(s, π∗ 1(s), w)) � ≤ ˜r2(s, ˜π2(s)) + γ E � ˜U2( ˜f2(s, ˜π2(s), w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (34) Set ϵU = ϵU + ˜ϵU, ϵr = ϵr + ˜ϵr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Combining parts (a) and (d) in the statement of the lemma with the approximation errors of ˜U2 and ˜r2, we know that U ∗ 1 (s) − ϵU ≤ ˜U2(s) ≤ U ∗ 1 (s) + ϵU and r1(s, a) − ϵr ≤ ˜r2(s, a) ≤ r1(s, a) + ϵr for any s and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Using these, we can lower bound both terms on the left-hand-side of (34), upper bound both terms on the right-hand-side of (34), and then rearrange to obtain r1(s, π∗ 1(s)) − r1(s, ˜π2(s)) ≤ 2ϵr + 2γϵU + γ E � U ∗ 1 ( ˜f2(s, ˜π2(s), w)) − U ∗ 1 ( ˜f2(s, π∗ 1(s), w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (35) Let state ˆs = arg maxs∈S U ∗ 1 (ˆs) − U ˜π2 1 (ˆs) be the state that achieves the largest regret (when using ˜π2 in M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Substituting from (35) gives U ∗ 1 (ˆs) − U ˜π2 1 (ˆs) = r1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗ 1(ˆs)) − r1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs)) + γ E � U ∗ 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗ 1(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − U ˜π2 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) � ≤ 2ϵr + 2γϵU + γ E � U ∗ 1 ( ˜f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − U ∗ 1 ( ˜f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗ 1(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) � + γ E � U ∗ 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗ 1(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − U ˜π2 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) � = 2ϵr + 2γϵU + γ E � U ∗ 1 ( ˜f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − U ∗ 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) � + γ E � U ∗ 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗ 1(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − U ∗ 1 ( ˜f2(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' π∗ 1(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) � + γ E � U ∗ 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − U ˜π2 1 (f1(ˆs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜π2(ˆs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) � ≤ 2ϵr + 2γϵU + 2γL1(d + ˜d) + γ � U ∗ 1 (ˆs) − U ˜π2 1 (ˆs) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' where we have used property (c) and that ∥f1(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)− ˜f2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)∥2 ≤ d+ ˜d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, we rearrange 55 to see that U ∗ 1 (ˆs) − U ˜π2 1 (ˆs) ≤ 2ϵr + 2γϵU + 2γL1(d + ˜d) 1 − γ , completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 To analyze R(µ, π) = ∥ ¯U ∗ − ¯U µ(π)∥∞, we will consider two MDPs that operate on the T-period timescale, one with optimal value ¯U ∗ and the other with optimal value V ∗(J1, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The reason to study an MDP with optimal value V ∗(J1, π) is because µ can be viewed as an approximation to the optimal policy for the second MDP, as suggested in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since both MDPs are defined on the T-period timescale, the transition functions are defined using T-period noise sequences w = (w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , wT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For ¯U ∗, let M1 = ⟨S, A, W, f1, r1, γT ⟩ be the MDP associated with the base model refor- mulation (8), but with the lower-level policy fixed to be π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The reward function r1 is r1(s, a) = E[R(s, a, π∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given a T-period noise sequence w, an initial state s, and action a, the “next” state f1(s, a, w) = sT (a, π∗) is the state obtained by first taking action a in state s and then following policy π∗ for the next T − 1 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For V ∗(J1, π), let M2 = ⟨S, A, W, f2, r2, γT ⟩ be the MDP associated with the frozen-state hierarchical approximation (14), where r2 is defined as r2(s, a) = E[ ˜R(s, a, J1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The transition function f2 is defined in the same way as f1 except we replace π∗ by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ϵr(π∗, J1) = maxs,a |E[R(s, a, π∗)] − E[ ˜R(s, a, J1)]|, so that we have |r1(s, a) − r2(s, a)| ≤ ϵr(π∗, J1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Noting that the first steps of f1 and f2 are the same (action a in state s with w1 revealed), applying parts (ii) and (iii) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, the maximum discrepancy between f1 and f2 is: ∥f1(s, a, w) − f2(s, a, w)∥2 ≤ d(α, dY, T) := 2(α + 1)dY(T − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Applying Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, we see that �� ¯U ∗ − V ∗(J1, π) �� ∞ ≤ 1 1 − γT � ϵr(π∗, J1) + γT LUd(α, dY, T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 56 We also need to account for the fact that µ is greedy with respect to V , an approximation of the optimal value of M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' More precisely, µ is greedy with respect to r2(s, a) = E � ˜R(s0, a, J1) � , f2(s, a, w) = sT (a, π), and value function V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We can thus apply Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 with ϵr = ϵr(π∗, J1), d = d(α, dY, T), L1 = LU, ϵU = �� ¯U ∗ −V ∗(J1, π) �� ∞, and ˜ϵU = ��V −V ∗(J1, ˜π) �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Collecting terms completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' C Proofs for Section 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Technical Lemmas Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider an MDP ⟨S, A, W, f, r, γ⟩ with reward function r taking values in [0, rmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose the optimal value function is U ∗ and the associated Bellman operator is F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Fix any initial value function such that 0 ≤ U0(s) ≤ rmax/(1 − γ) for all s and let U k = F k U0 be the result after iteration k of value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, it holds that ∥U k − U ∗∥∞ ≤ γk rmax 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This is a standard result that follows from the contraction property of F and the fact that U k = F Uk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, ∥U k − U ∗∥∞ = ∥HU k−1 − HU ∗∥∞ ≤ γ∥U k−1 − U ∗∥∞ ≤ γk∥U 0 − U ∗∥∞ ≤ γk rmax 1 − γ , where in the last step, we used 0 ≤ U ∗(s) ≤ rmax/(1 − γ) for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 of Bertsekas and Tsitsiklis (1996)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider an MDP ⟨S, A, W, f, r, γ⟩ with optimal value function U ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that ν is a policy that is greedy with respect to another value function U: ν(s) = arg max a � r(s, a) + E � U(f(s, a, w)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' If ∥U − U ∗∥∞ ≤ ε, then the performance of ν is bounded as follows: ∥U ν − U ∗∥∞ ≤ 2γε 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 57 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let V k(J∗ 1, π∗) be the value function approximation obtained from running FSVI for k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, the “value iteration error” is given by ��V k(J∗ 1, π∗) − V ∗(J∗ 1, π∗) �� ∞ ≤ γkT rmax 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider the upper-level MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that the discount factor is γT and the T-horizon reward function ˜R(s0, a, J∗ 1) ∈ � 0, 1 − γT 1 − γ rmax � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The result follows by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Since νk is greedy with respect to Uk, we can apply Lemmas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 to obtain ∥U νk − U ∗∥∞ ≤ 2γ 1 − γ γkrmax 1 − γ = 2rmaxγk+1 (1 − γ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 We apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 with π = ˜π∗, J1 = J∗ 1, and V = V k(J∗ 1, π∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The result follows by combining it with the result of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 and noting that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, ϵr(π∗, J∗ 1) ≤ ϵr(γ, α, dY, L, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' D Proofs for Section 7 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 First, we note that ¯Jt(x, y) nearly satisfies the Bellman equation for the separable MDP, with the exception of a next state transition that is based on x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let y′(x, y, a) = fY(x, y, a, w) and ∆g(x) = g(x) − g(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We have: ¯Jt(x, y) = T−t−1 � i=0 γi∆g(x) + max a � g(x∗) + h(y, a) + γ E � ¯Jt+1(x∗, y′(x∗, y, a)) �� = T−t−1 � i=0 γi∆g(x) + max a � g(x∗) + h(y, a) + γ E � ¯Jt+1(x, y′(x∗, y, a)) − T−t−2 � i=0 γi∆g(x) �� 58 = T−t−1 � i=0 γi∆g(x) + max a � g(x∗) + h(y, a) + γ E � ¯Jt+1(x, y′(x∗, y, a)) − T−t−2 � i=0 γi∆g(x) �� = ∆g(x) + max a � g(x∗) + h(y, a) + γ E � ¯Jt+1(x, y′(x∗, y, a)) �� = max a � g(x) + h(y, a) + γ E � ¯Jt+1(x, y′(x∗, y, a)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (36) The main proof is by backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider states (x, y) and (˜x, ˜y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' When t = T − 1, the difference between the two values is �� ¯JT−1(x, y) − J∗ T−1(˜x, ˜y) �� = ��g(x) − g(x∗) + max a � g(x∗) + h(y, a) � − max ˜a r(˜x, ˜y, ˜a) �� ≤ max a ��g(x) + h(y, a) − r(˜x, ˜y, a) �� ≤ max a ��g(x) + h(y, a) − r(x, y, a) �� + max a ��r(x, y, a) − r(˜x, ˜y, a) �� ≤ ζ + Lr � ∥x − ˜x∥2 + ∥y − ˜y∥2 � , where the last inequality follows from Assumption 3 and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that the desired result holds for period t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', we have �� ¯Jt(x, y) − J∗ t (˜x, ˜y) �� ≤ �T−t−1 � i=0 γi �� ζ + Lr ∥x − ˜x∥2 � + �T−t−1 � i=0 γiLi f � Lr∥y − ˜y∥2 + �T−t−1 � i=1 Li f T−t−1 � j=i γj � Lr ∥x∗ − ˜x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (37) Then for period t − 1, the value difference can be expanded as �� ¯Jt−1(x, y) − J∗ t−1(˜x, ˜y) �� = ����max a � g(x) + h(y, a) + γ E � ¯Jt(x, y′(x∗, y, a)) �� − max ˜a � r(˜x, ˜y, ˜a) + γ E � J∗ t (˜x, y′(˜x, ˜y, ˜a)) ������ ≤ max a ��g(x) + h(y, a) − r(˜x, ˜y, a) �� � �� � Term A + γ max a ���E � ¯Jt(x, fY(x∗, y, a, w)) − J∗ t (˜x, fY(˜x, ˜y, a, w)) ���� � �� � Term B , (38) where we used (36) in the equality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' It is easy to see that by Assumption 3 Term A ≤ ζ + Lr � ∥x − ˜x∥2 + ∥y − ˜y∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 59 Noting that ��fY(x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w)) − fY(˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ˜y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' w) �� 2 ≤ Lf � ∥x∗ − ˜x∥2 + ∥y − ˜y∥2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='we see from the induction hypothesis (37) that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Term B ≤ γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='��T−t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='ζ + Lr ∥x − ˜x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γiLi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='LrLf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='∥x∗ − ˜x∥2 + ∥y − ˜y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='T−t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='j=i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Lr ∥x∗ − ˜x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='ζ + Lr ∥x − ˜x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γiLi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Lr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='∥x∗ − ˜x∥2 + ∥y − ˜y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='j=i+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Lr ∥x∗ − ˜x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='ζ + Lr ∥x − ˜x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γiLi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Lr∥y − ˜y∥2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='�T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='T−t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='j=i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='Lr ∥x∗ − ˜x∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' where the last equality is obtained by from T−t � i=1 Li f T−t � j=i γj = T−t � i=1 γiLi f + T−t−1 � i=1 Li f T−t � j=i+1 γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Combining Terms A and B completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 The difference of the reward functions ��E[ ˜R(s0, a, J∗ 1)]−E[ ˜R(s0, a, ¯J∗ 1)] �� can be expanded as follows, ��E[ ˜R(s0, a, J∗ 1)] − E[ ˜R(s0, a, ¯J1)] �� = γ ��E � J∗ 1 � f(s0, a, w) �� − E � ¯J1(f(s0, a, w)) ��� ≤ T−1 � i=1 γiζ + �T−2 � i=1 Li f T−1 � j=i+1 γj � Lr max x ∥x∗ − x∥2, where the inequality is by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 60 E Proofs for Section 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 Technical Lemmas Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any vectors J ∈ RN and J′ ∈ RN, it holds that ��(ΦΦ†)(J) − (ΦΦ†)(J′) �� ∞ ≤ κ ∥J − J′∥∞ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For simplicity, let D = Φ � Φ†(J) − Φ†(J′) � be the term inside the norm on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, for any state s, we have |D(s)| = φ⊺(s) � Φ†(J)−Φ†(J′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We select θ1(s), θ1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , θM(s) ∈ R that satisfy Assumption 4, obtaining |D(s)| = �����κ � M � m=1 θm(s)φ⊺(sm) � � Φ†(J) − Φ†(J′) � ����� ≤ κ max m ��φ⊺(sm) � Φ†(J)) − Φ†(J′) ��� = κ max m |D(sm)| = κ max m |J(sm) − J′(sm)| ≤ κ ∥J − J′∥∞ where the third equality uses the fact that sm is a pre-selected state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given a lower-level value function ˆJ(ωt+1), recall that one approximate Bellman step in the lower level of FSAVI yields ˆJ(ωt) = ΦΦ† ¯H ˆJ(ωt+1) in the value function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' If ωT = 0, �� ˆJ(ω1) �� ∞ ≤ κrmax T−1 � i=0 (κγ)i = (κγ)T − 1 κγ − 1 κrmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Moreover, the upper-level reward function can be bounded as follows: ��E � ˜R(s0, a, ˆJ1(ω1)) ��� ≤ (κγ)T+1 − 1 κγ − 1 κrmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 61 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The proof follows by Assumption 4 and some manipulation: �� ˆJ(ωt)(s) �� = �����κ � M � m=1 θm(s)φ⊺(sm) � � Φ† ¯H ˆJ(ωt+1) � ����� ≤ κ max m ��φ⊺(sm) � Φ† ¯H ˆJ(ωt+1) ��� = κ max m ��� ¯H ˆJ(ωt) � (sm) �� = κ rmax + κγ �� ˆJ(ωt+1) �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Applying the above inequality T − 1 times yields the first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Next, we see that ��E � ˜R(s0, a, ˆJ1(ω1)) ��� ≤ rmax + γ �� ˆJ(ω1) �� ∞ ≤ κrmax + κγ �� ˆJ(ω1) �� ∞ ≤ κrmax T � i=0 (κγ)i, where we used the fact that κ ≥ 1 in the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose (ω∗ 1, ω∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' , ω∗ T ) satisfies ω∗ t = ¯H′ω∗ t+1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, we have ��J∗ t − ˆJt(ω∗ t ) �� ∞ ≤ � 1 + γ + 1 γ T−t � i=1 (κγ)i � εlow = � 1 + κ 1 − κγ − (κγ)T (1 + γ) γ − κγ2 � εlow a bound on the error of the value function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ε′ = εlow + δ for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For each t, choose a parameter vector ¯ωt ∈ RM such that ∥J∗ t − ˆJt(¯ωt)∥∞ < ε′, which is possible by the definition of εlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, it holds that �� ˆJt(¯ωt) − Φ ¯H′(¯ωt+1) �� ∞ = ��Φ¯ωt − ΦΦ† ¯H ˆJt+1(¯ωt+1) �� ∞ = ��ΦΦ†Φ¯ωt − ΦΦ† ¯H ˆJt+1(¯ωt+1) �� ∞ ≤ κ ��Φ¯ωt − ¯H ˆJt+1(¯ωt+1) �� ∞ (39) = κ �� ˆJt(¯ωt) − ¯H ˆJt+1(¯ωt+1) �� ∞ ≤ κ ��� ˆJt(¯ωt) − J∗ t �� ∞ + ��J∗ t − ¯H ˆJt+1(¯ωt+1) �� ∞ � < κ � ε′ + ∥ ¯HJ∗ t+1 − ¯H ˆJt+1(¯ωt+1)∥∞ � 62 ≤ κ � ε′ + γ ��J∗ t+1 − ˆJt+1(¯ωt+1) �� ∞ � (40) < κ(γ + 1) ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' where (39) is by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 and (40) follows by the contraction property of ¯H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The next step is the quantify the difference between ˆJt(¯ωt) and ˆJt(ω∗ t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ε′′ = κ(γ + 1) ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' �� ˆJt(¯ωt) − ˆJt(ω∗ t ) �� ∞ ≤ �� ˆJt(¯ωt) − Φ ¯H′(¯ωt+1) �� ∞ + ��Φ ¯H′(¯ωt+1) − ˆJt(ω∗ t ) �� ∞ ≤ ε′′ + ��ΦΦ† ¯H ˆJt+1(¯ωt+1) − ΦΦ† ¯H ˆJt+1(ω∗ t+1) �� ∞ ≤ ε′′ + γ′ γ �� ¯H ˆJt+1(¯ωt+1) − ¯H ˆJt+1(ω∗ t+1) �� ∞ (41) ≤ ε′′ + γ′�� ˆJt+1(¯ωt+1) − ˆJt+1(ω∗ t+1) �� ∞ (42) ≤ ε′′ + γ′� ϵ′′ + γ′�� ˆJt+2(¯ωt+2) − ˆJt+2(ω∗ t+2) �� ∞ � ≤ · · · ≤ ε′′ T−t−1 � i=0 (γ′)i + (γ′)T−t �� ˆJT (¯ωT ) − ˆJT (ω∗ T ) �� ∞ = γ + 1 γ ε′ T−t � i=1 (γ′)i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (43) where (41) is by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, (42) is by the contraction property of ¯H, and (43) because ω∗ T = ¯ωT = 0 (since JT (s) = 0 for all s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, ��J∗ t − ˆJt(ω∗ t ) �� ∞ ≤ ��J∗ t − ˆJt(¯ωt) �� ∞ + �� ˆJt(¯ωt) − ˆJt(ω∗ t ) �� ∞ ≤ ε′ � 1 + γ + 1 γ T−t � i=1 (γ′)i � = ε′ � 1 + γ + 1 γ T−t � i=1 (κγ)i � = ε′ � 1 + γ + 1 γ κγ − (κγ)T 1 − κγ � = ε′ � 1 + κ 1 − κγ − (κγ)T (1 + γ) γ − κγ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since δ can be arbitrarily small, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 63 Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Given an upper-level value function ˆV (βi), recall that one approximate Bellman step in the upper level of FSAVI yields ˆV (βi+1) = ΦΦ†Fω∗ ˆV (βi) in the value function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We have �� ˆV (β∗) �� ≤ κ2 − κ2(κγ)T+1 (1 − κγT )(1 − κγ) rmax, where β∗ is a fixed point of F ′ ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Again, the proof follows by Assumption 4 and some manipulation: �� ˆV (βi+1)(s) �� = �����κ � M � m=1 θm(s)φ⊺(sm) � � Φ†Fω ˆV (βi) � ����� ≤ κ max m ��φ⊺(sm) � Φ†Fω ˆV (βi) ��� = κ max m ��� Fω ˆV (βi) � (sm) �� ≤ κ Rmax + κγT �� ˆV (βi) �� ∞, where Rmax is an upper bound on ��E � ˜R(s0, a, ˆJ1(ω1)) ��� from Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Starting with βi = 0 and applying the inequality repeatedly, we see that �� ˆV (β∗) �� ∞ ≤ κRmax ∞ � j=0 (κγT )j ≤ κRmax 1 − κγT , which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any ω, the parameter space Bellman operator for the upper-level problem F ′ ω = Φ† ◦ Fω ◦ Φ is a γ′-contraction with respect to a norm ∥ · ∥Φ on RM defined by ∥β∥Φ = ∥Φβ∥∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', ��F ′ ω(β) − F ′ ω(β′) �� Φ ≤ κγT ��β − β′�� Φ, where β, β′ ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Therefore, there exists a fixed point β∗ of F ′ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The proof follows Theorem 3a of Tsitsiklis and Van Roy (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We include the steps here in our notation for completeness: ��F ′ ω(β) − F ′ ω(β′) �� Φ = ��(Φ† ◦ Fω ◦ Φ)(β) − (Φ† ◦ Fω ◦ Φ)(β′) �� Φ 64 = ��Φ(Φ† ◦ Fω ◦ Φ)(β) − Φ(Φ† ◦ Fω ◦ Φ)(β′) �� ∞ ≤ κ ��Fω(Φβ) − Fω(Φβ′) �� ∞ ≤ κγT ��Φβ − Φβ′�� ∞ = κγT ��β − β′�� Φ, where first inequality follows by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 and the second inequality follows by the γT -contraction property of Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ω∗ be the solution of the lower level of FSAVI and let β∗ be the fixed point of F ′ ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The approximate value iteration of the upper level of FSAVI, which produces βk, has a “value iteration” error of: �� ˆV (βk) − ˆV (β∗) �� ∞ ≤ (κγT )k κ2 − κ2(κγ)T+1 (1 − κγT )(1 − κγ) rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' We have: �� ˆV (βk) − ˆV (β∗) �� ∞ = ��ΦF ′ ω∗βk−1 − ΦF ′ ω∗β∗�� ∞ = ��F ′ ω∗βk−1 − F ′ ω∗β∗�� Φ ≤ κγT ��Φβk−1 − Φβ∗�� ∞ ≤ κγT �� ˆV (βk−1) − ˆV (β∗) �� ∞ ≤ (κγT )k�� ˆV (β0) − ˆV (β∗) �� ∞ ≤ (κγT )k κ2 − κ2(κγ)T+1 (1 − κγT )(1 − κγ) rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The first inequality is by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5 and the last inequality follows from β0 = 0 and Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider any ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' If β∗ is the fixed point of F ′ ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=', β∗ = F ′ ω β∗, which exists by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='5, then it holds that ��V ∗ ω − ˆV (β∗) �� ∞ ≤ � 1 + κ 1 − κγT � εup 65 where V ∗ ω is the fixed point of Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let ε′ = εup + δ for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Choose ¯β ∈ RM such that ��V ∗ ω − ˆV ( ¯β) �� ∞ < ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, �� ˆV ( ¯β) − ΦF ′ ω( ¯β) �� ∞ = ��Φ ¯β − ΦΦ†Fω ˆV ( ¯β) �� ∞ = ��ΦΦ†Φ ¯β − ΦΦ†Fω ˆV ( ¯β) �� ∞ < κ ��Φ ¯β − Fω ˆV ( ¯β) �� ∞ (44) = κ �� ˆV ( ¯β) − Fω ˆV ( ¯β) �� ∞ ≤ κ ��� ˆV ( ¯β) − V ∗ ω �� ∞ + ��V ∗ ω − Fω ˆV ( ¯β) �� ∞ � < κ � ε′ + ��FωV ∗ ω − Fω ˆV ( ¯β) �� ∞ � < κ � ε′ + γT ϵ′� = κ � 1 + γT � ε′, (45) where (44) is by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 and (45) is by the γT -contraction property of Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Now, we let ε′′ = κ(1 + γT ) ε′ and see that �� ˆV ( ¯β) − ˆV (β∗) �� ∞ ≤ �� ˆV ( ¯β) − ΦF ′ ω( ¯β) �� ∞ + ��ΦF ′ ω( ¯β) − ˆV (β∗) �� ∞ < ϵ′′ + ��ΦΦ†Fω ˆV ( ¯β) − ΦΦ†Fω ˆV (β∗) �� ∞ < ϵ′′ + κ ��Fω ˆV ( ¯β) − Fω ˆV (β∗) �� ∞ ≤ ϵ′′ + κγT �� ˆV ( ¯β) − ˆV (β∗) �� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' It thus follows that �� ˆV ( ¯β) − ˆV (β∗) �� ∞ ≤ κ + κγT 1 − κγT ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Putting the pieces together, we have ��V ∗ ω − ˆV (β∗) �� ∞ ≤ ��V ∗ ω − ˆV ( ¯β) �� ∞ + ∥ ˆV ( ¯β) − ˆV (β∗) �� ∞ ≤ ε′ + κ + κγT 1 − κγT ε′ ≤ 1 + κ 1 − κγT ε′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since δ can be arbitrarily small, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 66 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2 Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 We apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 with π = ˆπω∗, J1 = ˆJ1(ω∗), and V = ˆV (βk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' First, to compute the reward error ϵr(π∗, ˆJ1(ω∗ 1)), we have ϵr(π∗, ˆJ1(ω∗ 1)) = max s,a ��E[R(s, a, π∗)] − E[ ˜R(s, a, ˆJ1(ω∗ 1))] �� ≤ ϵr(γ, α, dY, L, T) + max s,a ��E[ ˜R(s, a, J∗ 1)] − E[ ˜R(s, a, ˆJ1(ω∗ 1))] �� ≤ ϵr(γ, α, dY, L, T) + γ ��J∗ 1 − ˆJ1(ω∗ 1) �� ∞ ≤ ϵr(γ, α, dY, L, T) + � γ + (γ + 1) T−1 � i=1 (γ′)i � εlow, where the last inequality follows from Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The other term to analyze is ��V ∗ ω∗ − ˆV (βk) �� ∞, where we remind the reader of our usage of the shorthand notation V ∗ ω∗ = V ∗( ˆJ1(ω∗), ˆπω∗): ��V ∗ ω∗ − ˆV (β∗) �� ∞ ≤ ��V ∗ ω∗ − ˆV (β∗) �� ∞ + �� ˆV (β∗) − ˆV (βk) �� ∞ ≤ � 1 + κ 1 − κγT � εup + (κγT )k � κ2 − κ2(κγ)T+1 (1 − κγT )(1 − κγ) � rmax, which follows by Lemmas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='4 and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' F Bounds on LU in Terms of Lr and Lf We start with an assumption that, if true, leads to a simple bound on the Lipschitz constant LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The main result is in Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Suppose that γLf < 1, where the constant Lf, as defined in (3), is the sensitivity of the transition function to small changes in (s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩ and let U : S → R be a value function such that there exists LU > 0 where for any states s and ˜s, |U(s) − U(˜s)| ≤ LU ∥s − ˜s∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (46) Define the state-action value function Q(s, a) = r(s, a) + γ E � U(f(s, a, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, for any state- 67 action pairs (s, a) and (˜s, ˜a), the state-action value function Q satisfies ��Q(s, a) − Q(˜s, ˜a) �� ≤ (Lr + γLULf) � ∥s − ˜s∥2 + ∥a − ˜a∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For any state-action pairs (s, a), (˜s, ˜a) ∈ S × A, we have ��Q(s, a) − Q(˜s, ˜a) �� ≤ |r(s, a) − r(˜s, ˜a)| + γ ��E � U(f(s, a, w)) − U(f(˜s, ˜a, w)) ��� ≤ Lr � ∥s − ˜s∥2 + ∥a − ˜a∥2 � + γLU max w ��f(s, a, w) − f(˜s, ˜a, w)) �� 2 (47) ≤ Lr � ∥s − ˜s∥2 + ∥a − ˜a∥2 � + γLULf � ∥s − ˜s∥2 + ∥a − ˜a∥2 � (48) ≤ (Lr + γLULf) � ∥s − ˜s∥2 + ∥a − ˜a∥2 � , where (47) follows by (46) and (48) follows by the definition of Lf in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Let Q : S × A → R be a state-action value function and assume there exists LQ > 0 where for any states (s, a) and (˜s, ˜a), ��Q(s, a) − Q(˜s, ˜a) �� ≤ LQ � ∥s − ˜s∥2 + ∥a − ˜a∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (49) Define U(s) = maxa Q(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, for any states s and ˜s, the value function U satisfies ��U(s) − U(˜s) �� ≤ LQ ∥s − ˜s∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Note that: ��U(s) − U(˜s) �� = ��max a Q(s, a) − max ˜a Q(˜s, ˜a) �� ≤ max a ��Q(s, a) − Q(˜s, a) �� ≤ LQ∥s − ˜s∥2, where the last inequality is by (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Starting with U0 = 0, recur- 68 sively define Qk+1 and Uk+1 as follows: Qk+1(s, a) = r(s, a) + γ E � Uk(f(s, a, w) � and Uk+1(s) = max a Qk+1(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then Uk is Lipschitz continuous and its Lipschitz constant LUk satisfies LUk = Lr + γLfLUk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' (50) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The proof is by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' For k = 1, note that Q1(s, a) = r(s, a) and therefore has Lipschitz constant Lr by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' By Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2, it follows that U1 also has Lipschitz constant Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Since LU0 = 0, we see that LU1 = Lr satisfies (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Now, assume that LUk satisfies (50) for k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1, Qk+1 is (Lr + γLfLUk)-Lipschitz continuous and by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='2, Uk+1 is (Lr + γLfLUk)-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Consider an (α, dY)-fast-slow MDP ⟨S, A, W, f, r, γ⟩ and suppose Assumption 5 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Then, the optimal value U ∗, as defined in (5), satisfies: ��U ∗(s) − U ∗(˜s) �� ≤ Lr 1 − γLf ��s − ˜s �� 2 for any states s, ˜s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' According to Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='1 of Bertsekas (2012), the value Uk in Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content='3 converges to the optimal value U ∗ (value iteration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' The recursion (50) can be written as: LUk = Lr + γLfLr + · · · + (γLf)k−1Lr = k−1 � i=0 (γLf)iLr, a convergent sequence since Assumption 5 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' Letting k → ∞, we see that U ∗ has Lipschitz constant lim k→∞ LUk = ∞ � i=0 (γLf)iLr = Lr 1 − γLf , completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} +page_content=' 69' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9AzT4oBgHgl3EQfAPpu/content/2301.00922v1.pdf'} diff --git a/nNE2T4oBgHgl3EQfegfv/content/tmp_files/2301.03918v1.pdf.txt b/nNE2T4oBgHgl3EQfegfv/content/tmp_files/2301.03918v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..324b894ce735ac82d0fcdc4a2b5bd92e4f9ed436 --- /dev/null +++ b/nNE2T4oBgHgl3EQfegfv/content/tmp_files/2301.03918v1.pdf.txt @@ -0,0 +1,743 @@ +arXiv:2301.03918v1 [physics.plasm-ph] 10 Jan 2023 +Feasibility of the EDICAM camera for runaway electron +detection in JT-60SA disruptions +Soma Olasz1,2, Mathias Hoppe3, Tam´as Szepesi2, Kensaku Kamiya4, Gergo +I. Pokol1,2 +1 Institute of Nuclear Techniques, Budapest University of Technology and Economics, +M˝uegyetem rkp. 3, Budapest 1111, Hungary +2 Fusion Plasma Physics Department, Centre For Energy Research, Budapest, Hungary +3 Ecole Polytechnique F´ed´erale de Lausanne, Swiss Plasma Center, CH-1015 Lausanne, +Switzerland +4 National Institutes for Quantum and Radiological Science and Technology, Naka, Japan +Abstract +The visible camera system EDICAM, recently installed on JT-60SA, is sim- +ulated to assess whether it can be used for measuring synchrotron radia- +tion from relativistic runaway electrons. In this simulation, the SOFT syn- +thetic synchrotron diagnostic framework is used to compute the synthetic +synchrotron camera images from a JT-60SA-like disruption modelled with +the DREAM disruption simulation code. In the studied scenario, a large +amount of argon is added to the plasma and a disruption is simulated by +starting a prescribed exponential temperature drop and finishing with fur- +ther cooling provided by the argon in a self-consistent simulation of the cur- +rent quench. The background plasma evolution is calculated by DREAM +self-consistently with the fast electron population, which is modelled kinet- +ically. The resulting runaway electron distribution function along with the +parameters of the EDICAM visible camera system are used as an input to +the SOFT synthetic synchrotron diagnostic framework to assess the feasibil- +ity of the camera for runaway electron detection. We find that the runaway +electron beam formed in the disruption can produce synchrotron radiation +observable by the EDICAM system, thus enabling the use of the EDICAM +for the characterization of runaway electron beams. +Keywords: +runaway electrons, synchrotron radiation, visible diagnostics, disruption +Preprint submitted to Fusion Engineering and Design +January 11, 2023 + +1. Introduction +Runaway electron generation has been extensively studied in recent years [1, +2, 3, 4, 5] due to the major threat they pose for future large-scale experi- +ments [6, 7]. The generated runaway current is exponentially dependent on +the pre-disruption plasma current due to the avalanching effect [8, 9], so in +future large current devices the problem of runaway electrons is expected to +be more severe than in present day tokamaks. Extensive study of runaway +electrons is hence important. +Runaway electrons can emit various forms of radiation and this is used for +their detection in current devices [10, 11]. Bremsstrahlung radiation is pro- +duced due to interaction of runaway electrons and other plasma particles and +is typically measured in the order of 1 MeV photon energy [10]. It is routinely +used for detection of runaway electron presence in a tokamak. Synchrotron +radiation can be detected by visible and infrared camera systems [12, 13] and +it can be used to gather information on the electron distribution function – +both in space and momentum space [14, 15, 16, 17]. +Synchrotron radiation images on various devices and modelling of run- +away electron synchrotron radiation were used in synthetic diagnostic tools +to get information on the runaway electron distribution function. One such +study was done with the KORC code, which has been used to study the +relation between the characteristics of the runaway electron population and +the background plasma parameters [17]. These simulations include full orbit +effects of the electrons. The SOFT synthetic diagnostic framework, which +works in guiding centre coordinates has been used to model synchrotron ra- +diation on several tokamaks [14, 15, 16]. +In this paper, we assess the feasibility of the recently installed EDICAM +visible camera system on the JT-60SA tokamak [18] for runaway electron +detection and characterization. +The EDICAM parameters were given to +the SOFT synthetic diagnostic framework, which was used to simulate a +synthetic synchrotron radiation image from a runaway electron distribution +generated in a JT-60SA-like disruption. The disruption was simulated using +the DREAM disruption runaway electron modelling tool [19]. +The characteristics of the EDICAM system are introduced in section 2. A +description of the modelling tools used in the simulations is given in section 3, +while the simulation settings, results and their interpretation are presented +2 + +EDICAM +QE=20% +QE=15% +QE=10% +Figure 1: The spectral sensitivity of the EDICAM camera system. The quantum efficiency +(QE) shows the fraction of the photon energy absorbed by the sensors. The lines shown +correspond to 10, 15 and 20% efficiency. This already includes the effect of non-sensitive +areas of the pixels. The EDICAM system sensors have a quantum efficiency of about 15% +between 500 and 700 nm, but it can detect photons up to 1000 nm. +in section 4. Finally, we conclude in section 5. +2. The EDICAM system +The parameters of the EDICAM diagnostic are discussed in this section, +with a focus on the camera characteristics, such as location, field of view, +sensibility range, required for the simulation. +An EDICAM visible camera system is installed in the P18 sector of the +JT-60SA tokamak. It features non-destructive read-out, meaning it can read +out data from the sensors without affecting the exposure process, which +enables simultaneous observation of multiple different regions in the camera +field of view, specific regions-of-interest (ROI), as well as the full field of view +(FOV), with different sampling rates. This makes it ideal for the observation +of various fast phenomena, including some phases of disruptions [18]. +The details of the camera characteristics can be found in [20], and the +3 + +0.12 +0.1 +0.08 +Response (A/W) +QE-10% +0.06 +QE-15% +QE-20% +0.04 +LUPA-1300 +0.02 +0 +400 +500 +600 +700 +800 +900 +1000 +Waveleng th (nm)Ip +Figure 2: Schematic top-view of the EDICAM camera (black filled circle) viewing direction +and the direction of the toroidal plasma current (Ip). +JT-60SA-specific installation parameters in [18]. +The optical parameters +relevant to the modelling are listed in table 1. These show that EDICAM +represents a typical general purpose visible camera with a wide field of view +in a tangential direction into the tokamak from a position close to the mid- +plane. The spectral sensitivity of the camera system is shown in figure 1. +It can be seen that the EDICAM is most effective in the 520 to 720 nm +spectral range. A sketch of the camera position and viewing direction relative +to the torus is shown in figure 2 along with the direction of the plasma +current. As can be seen, the camera viewing direction and corresponding +plasma current direction does not prohibit the detection of the synchrotron +radiation emitted by runaway electrons. The simulated camera image from +the EDICAM location is shown in figure 3. The above mid-plane position +of the camera necessitates a more detailed study on the actual feasibility of +synchrotron detection, which motivates the rest of the paper. +3. Modelling tools +This section describes the modelling tools used in this study: first the +DREAM [19] code, used for the disruption simulation, then the SOFT code, +used for the modelling of the expected synchrotron radiation image. +To assess the feasibility of the EDICAM system for runaway radiation +detection, a JT-60SA-like disruption was simulated with the DREAM code. +DREAM is a runaway electron modelling tool developed for the study of +4 + +R +p +XTable 1: Optical parameters of the EDICAM system as installed on JT-60SA. +Parameter +Value +Position (m) +(-4.5304, -1.7552, 0.2312) +Viewing direction (m) +(0.60915, 1.01379, 0) +Field of view (FOV) (degrees) +80 +Spectral range (nm) +520–720 +Entrance pupil (mm) +5 +Figure 3: The simulated camera view of the vacuum vessel from the EDICAM location. +The gray circle is the 80 degrees field of view seen by the camera system. +5 + +disruption runaway electrons. It calculates the evolution of the electron pop- +ulation self-consistently along with the background plasma. The electrons +can be handled with different levels of sophistication ranging from treating +all electrons using a fluid model to resolving all electrons kinetically. +The background plasma evolution includes the temperature evolution due +to ohmic heating, radiation losses, and collisional energy exchange. +The +density evolution of the plasma constituent elements is calculated including +both the main ions and the impurities and their charge state evolution . +The electric field and the current density evolution is calculated by solving +Amp`ere’s and Faraday’s laws. [19]. +In the current work, DREAM was used in a kinetic mode, meaning that +the entire electron population was evolved by the bounce averaged Fokker- +Planck equation with a relativistic test particle collision operator [19]. This +calculates the electron distribution function on separate 3D phase-space grid +momentum grid for the thermal and suprathermal electron population and +a grid for the runaway electron population. The Dreicer and hot-tail gen- +eration of runaway electrons are calculated from the collisional effects and +the Rosenbluth-Putvinski source term is used for the avalanche generation +mechanism. +The SOFT synthetic diagnostic framework [21] was used to calculate the +synchrotron radiation expected from the electron distribution function as +seen by the EDICAM camera system. The camera parameters listed in ta- +ble 1 were added to the model, along with the runaway distribution function +calculated by DREAM. The synchrotron radiation was calculated using the +cone approximation [21]. In this approach the synchrotron radiation is as- +sumed to be along the guiding centre trajectory of the runaway particles in a +hollow cone shape with an infinitely thin side and an opening angle θp, that +is the particle pitch angle. In the presented simulations, SOFT modelled the +tokamak geometry as a circular torus from the major and minor radii, and a +linear q-profile was assumed for simplicity. +4. Modelling results +The disruption simulation and the resulting radiation image are presented +next. The main plasma parameters, the runaway electron distribution func- +tion and the results of the SOFT simulation are shown, along with a discus- +sion on the outstanding features. The aim of the disruption simulation was +to produce a high energy but realistic runaway electron beam in JT-60SA +6 + +geometry that can be used as an input for the synchrotron image simula- +tion. Accordingly, the disruption simulation input parameters, such as the +input argon amount for example, were chosen to produce a significant and +energetic runaway electron population, and the scenario is not optimized for +disruption mitigation purposes. +4.1. Disruption simulation +The DREAM code was used to simulate an argon MMI-induced disrup- +tion. The initial current density profile was taken from the simulation of the +5.5 MA Scenario 2 of the JT-60SA research plan [22]. This was assumed to +be fully ohmic current. The density and temperature profiles used by the +EFIT magnetic equilibrium calculation were used as initial conditions in the +DREAM modelling. The magnetic field strength, major and minor radii and +the total plasma current were also chosen based on Scenario 2 of the JT- +60SA research plan. The EFIT data was interpolated to the radial grid of +DREAM, which was set to 20 radial grid points. +The simulation was done in five separate phases. First the initial electric +field was calculated to give the desired initial plasma current based on the +given density and temperature profiles. Then 1020 m−3 argon gas density +was introduced to the plasma. The transient ionization dynamics were re- +solved using a very short time step, lasting 1 µs after which the prescribed +exponential temperature decay started. This temperature decay was further +simulated on a longer timescale to reach below 100 eV temperatures. It was +artificially prescribed at every radial point until the temperature reached +100 eV at the innermost radial point. After the prescribed phase ended, the +temperature evolution was calculated self-consistently, mainly governed by +the radiation of the injected argon gas and the ohmic heating of the plasma +current. The simulation finished with the calculation of the runaway plateau +phase during which the electron population can experience significant pitch +angle scattering [23]. The total simulated time was about 8.6 ms. +The temperature as a function of radius at different times throughout the +simulation is shown in figure 4a. The initial temperature at the magnetic axis +is 13.4 keV. It can be seen that until the temperature reaches 100 eV at the +centre, the temperature drops exponentially at every radial point, keeping +the initial profile shape. Below 100 eV the temperature is calculated using an +energy-balance model and it equilibrates along the radius. The plasma cools +slower at the outer regions, and simultaneously the current density, shown in +figure 4c, relaxes at a slower pace, as seen around 0.7 normalized radius. The +7 + +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized radius [-] +100 +101 +102 +103 +104 +Temperature [eV] +Temperature evolution +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized radius [-] +0 +50 +100 +150 +Electric field [V/m] +Electric field evolution evolution +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized radius [-] +0.0 +0.5 +1.0 +1.5 +Current density [MA/m2] +Current density evolution +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized radius [-] +0 +20 +40 +60 +80 +Normalized electric field [-] +Electric field evolution evolution +Time 0.10 ms +Time 0.15 ms +Time 0.25 ms +Time 0.35 ms +Time 0.65 ms +Time 0.95 ms +Time 1.25 ms +Time 1.55 ms +Time 1.85 ms +Time 3.35 ms +Time 8.35 ms +Time 0.10 ms +Time 0.15 ms +Time 0.25 ms +Time 0.35 ms +Time 0.65 ms +Time 0.95 ms +Time 1.25 ms +Time 1.55 ms +Time 1.85 ms +Time 3.35 ms +Time 8.35 ms +Figure 4: The time evolution of the main plasma parameters. Plot a) shows the temper- +ature, and plot b) shows the electric field evolution. The change in the current density +is plotted in graph c), while the electric field normalized to the effective critical electric +field is plotted in graph d). Note the logarithmic vertical axis on the temperature plot +compared to the other plots. +final current density is completely provided by runaway electrons, generated +by the electric field induced during the disruption. The accelerating electric +field can be seen in figure 4b, while the electric field normalized to the effective +critical field [24] is shown in figure 4d. The overall shape of the two electric +field plots is similar since the effective critical field is mainly dependent on +the density and weakly on the temperature. As the density profile is mostly +flat during the simulation, the critical field will be similar at every radial +point and the normalization will preserve the shape of the electric field. In +figure 4 a very high electric field can be seen around 0.65 ms, when the +temperature dropped to below 10 eV. This peak is off-axis, located at about +0.75 normalized radius due to the decay of the current peak seen in figure b. +This electric field accelerates the electrons further and yields a more energetic +runaway population at this location. +The plasma current evolution is plotted in figure 5, with the starting times +of the different simulation phases indicated with vertical lines. The start of +the ionization phase and the start of the exponential temperature decay phase +overlap, as the ionization of the injected argon, was simulated in 1 µs. The +thermal quench is simulated between the yellow and the blue vertical lines. +The current quench is finished by 1 ms with a current of slightly less than +8 + +Figure 5: The time evolution of the total plasma current. +The starting times of the +simulation phases were indicated with vertical lines. The ionization phase was 1 µs, so +the line indicating the start of this phase overlaps with the start of the exponential decay +phase. +3 MA. In the runaway plateau phase the plasma current seems to increase +until the end of the simulation. +This is due to the finite wall resistance +used as a boundary condition for the poloidal flux. This allows for current +to flow in the wall, decaying with a characteristic time, which results in an +increase in the plasma current. The runaway electrons are not significantly +accelerated to higher energies during this phase as the accelerating electric +field is proportional to the current decay time. The current drops much faster +in the current quench phase, hence the acceleration in the runaway plateau +phase should be negligible. At the end of the simulation a sufficient runaway +electron population was achieved, so the plasma evolution was not calculated +further. +The angle-averaged distribution function at the end of the simulation is +shown in figure 6 for all the simulated radial locations. The hot-tail seed +formed during the thermal quench can be seen as accelerated to higher ener- +gies, as seen by the local peaks of the distribution function at various points +at different radial coordinates. The maximum runaway electron momentum +reached is about 70mec or about 35 MeV at about 1 m away from the mag- +netic axis, corresponding to about 0.75 normalized radius. +The runaway +distribution in the centre only reached about 10-20mec, as the electric field +9 + +Plasma current evolution +5.5 +Start of the ionization phase +Start of exponential temperature decay +Start of the self consistent temperature phase +5.0 +current [MA] +Start of the runaway plateau phase +4.5 +4.0 +lasma +3.5 +P +3.0 +1 +2 +3 +4 +5 +6 +7 +8 +Time [ms]Figure 6: The runaway electron distribution function plotted for each radial grid point +at the last time step as a function of normalized radius p. The highest energy runaway +electron population is located at about 1 m minor radius. +did not penetrate into the plasma deep enough to accelerate the popula- +tion at the magnetic axis, as seen in figure 4. The synchrotron radiation is +highly dependent on the runaway energy and pitch angle, so the radiation is +expected to be mainly off-axis. +4.2. Simulation of the radiation image +The runaway electron distribution was given to the SOFT code, which +calculated the synchrotron radiation as seen by the EDICAM camera. The +simulated camera image is shown in figure 7. Significant radiation can be +seen on the high field side in a crescent-like shape while the radiation is +negligible at the magnetic axis. The simulated image resembles observations +of synchrotron radiation emitted by runaway electrons in other tokamaks [2]. +The origin of the radiation in momentum space is illustrated by figure 8, +where the Green’s function weighted with the DREAM distribution function +is plotted as a function of normalized parallel and perpendicular momenta. +The Green’s function contains information on the tokamak geometry and +the momentum dependence of the radiation. It can be seen that most of +the radiation originates between p = 50-65mec. This location corresponds +to the highest energies of the runaway electron distribution function seen in +figure 6. The radiation is also sensitive to the pitch angle of the particles, +10 + +Runaway distribution function +r=0.034m +1014 +r=0.103m +r=0.171m +r=0.240m +r=0.308m +r= 0.377m +r= 0.445m +1011 +r=0.514m +(fRE) +r=0.582m +r=0.651m +r=0.719m +r = 0.788m +r=0.856m +108 +r=0.925m +r=0.993m +r=1.062m +r= 1.130m +r= 1.199m +r = 1.267m +105 +r= 1.336m +20 +40 +60 +80 +pFigure 7: The synchrotron radiation of the runaway beam during a JT-60SA-like disrup- +tion as seen by the EDICAM visible camera system. +11 + +Figure 8: The Green’s function from the SOFT simulation weighted with the distribution +function from DREAM. Most of the synchrotron radiation comes from the region between +p∥ = 50mec to p∥ = 65mec with p⊥ ≈ 12. +which explains the shape of the maximum in figure 8. +There is a ridge +extending to 30 mec which is most likely a result of the runaway electron +population at the inner radii. The dip between the two maxima is probably an +artefact of the interpolation between the DREAM and the SOFT grids. The +dominant radiation spot corresponds to the high energy runaway population +at about 0.75 normalized radius. The hollow radiation shape can be explained +by the enhancing effect of the high magnetic field strength on the synchrotron +radiation as well as the high runaway electron energies at larger radii. Since +more and more energetic runaway electrons are generated at the edge, the +main source of radiation is coming from the edge at the high field side of the +tokamak. +5. Conclusions +The DREAM and SOFT codes were used to simulate the synchrotron ra- +diation originating from a runaway population generated in a JT-60SA-like +disruption as seen by the EDICAM visible camera system. DREAM calcu- +lated the runaway electron population during an argon-induced disruption. +12 + +Origin of the radiation +0.9474 +30 +0.8421 +0.7368 +0.6316 +mc +20 +0.5263 +/Td +0.4211 +0.3158 +0.2105 +10 +0.1053 +0.0000 +-60 +-50 +-40 +-30 +-20 +pμ/mcThe resulting runaway distribution was given to SOFT to calculate the syn- +chrotron radiation to assess the feasibility of the EDICAM camera system +for synchrotron radiation detection. It was found that the synchrotron radi- +ation from runaway electrons is emitted in the right direction to be observed +by the installed EDICAM system. Moreover, the EDICAM characteristics +make it ideal to observe fast phenomena in tokamaks, such as disruptions. +Our main conclusion is that the EDICAM camera system is capable of de- +tecting synchrotron radiation from high-energy runaway electrons generated +in major disruptions. +In this feasibility study, the argon injection simulated by a simple model +of uniform argon density introduced simultaneously to the plasma at every +radial point. The simulation could be improved by considering a more real- +istic injection of the argon gas, but it would most likely change the radiation +shape, not the radiation detection capability of the camera system. +The +radiation image could also be improved by adding a realistic camera view, +such as the exact shape of the tokamak, to the SOFT code or another model, +but it is beyond the scope of this study. Present results might motivate a +more detailed study of the range of applicability of the EDICAM system, and +its applicability for early detection of runaway electron beams in mitigated +disruptions and other operation regimes. +Acknowledgments +The authors are grateful to N. Hajnal for the simulated camera image of +the EDICAM camera system and O. Asztalos for the EFIT data provided for +the input to DREAM. This work has been carried out within the framework +of the EUROfusion Consortium, funded by the European Union via the Eu- +ratom Research and Training Programme (Grant Agreement No 101052200 +— EUROfusion). Views and opinions expressed are however those of the +author(s) only and do not necessarily reflect those of the European Union +or the European Commission. Neither the European Union nor the Euro- +pean Commission can be held responsible for them. G. I. Pokol and S. Olasz +acknowledge the support of the National Research, Development and Inno- +vation Office (NKFIH) Grant FK132134. This work was supported in part +by the Swiss National Science Foundation. +13 + +References +[1] C. Paz-Soldan, C. Reux, K. Aleynikova, P. Aleynikov, V. Bandaru, M. +Beidler, N. Eidietis, Y.Q. Liu, C. Liu, A. Lvovskiy, S. Silburn, L. Bar- +doczi, L. Baylor, I. Bykov, D. Carnevale, D. Del-Castillo Negrete, X. +Du, O. Ficker, S. Gerasimov, M. Hoelzl, E. Hollmann, S. Jachmich, S. +Jardin, E. Joffrin, C. Lasnier, M. Lehnen, E. Macusova, A. Manzanares, +G. Papp, G. Pautasso, Z. Popovic, F. Rimini, D. Shiraki, C. Sommariva, +D. Spong, S. Sridhar, G. Szepesi, C. Zhao, the DIII-D Team, JET Con- +tributors, A novel path to runaway electron mitigation via deuterium +injection and current-driven MHD instability, Nuclear Fusion, 61, 11, +(2021), 116058, https://doi.org/10.1088/1741-4326/ac2a69 +[2] C. Reux, C. Paz-Soldan, P. Aleynikov, V. Bandaru, O. Ficker, S. Silburn, +M. 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F¨ul¨op, Effect of +partially ionized impurities and radiation on the effective critical electric +field for runaway generation, Plasma Physics and Controlled Fusion, 60, +7, (2018), 074010, https://dx.doi.org/10.1088/1361-6587/aac33e +17 + diff --git a/nNE2T4oBgHgl3EQfegfv/content/tmp_files/load_file.txt b/nNE2T4oBgHgl3EQfegfv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..374a0b1dbf6087cfe7fd06e5fca16fce2d5f713a --- /dev/null +++ b/nNE2T4oBgHgl3EQfegfv/content/tmp_files/load_file.txt @@ -0,0 +1,702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf,len=701 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='03918v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='plasm-ph] 10 Jan 2023 Feasibility of the EDICAM camera for runaway electron detection in JT-60SA disruptions Soma Olasz1,2, Mathias Hoppe3, Tam´as Szepesi2, Kensaku Kamiya4, Gergo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Pokol1,2 1 Institute of Nuclear Techniques, Budapest University of Technology and Economics, M˝uegyetem rkp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 3, Budapest 1111, Hungary 2 Fusion Plasma Physics Department, Centre For Energy Research, Budapest, Hungary 3 Ecole Polytechnique F´ed´erale de Lausanne, Swiss Plasma Center, CH-1015 Lausanne, Switzerland 4 National Institutes for Quantum and Radiological Science and Technology, Naka, Japan Abstract The visible camera system EDICAM, recently installed on JT-60SA, is sim- ulated to assess whether it can be used for measuring synchrotron radia- tion from relativistic runaway electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In this simulation, the SOFT syn- thetic synchrotron diagnostic framework is used to compute the synthetic synchrotron camera images from a JT-60SA-like disruption modelled with the DREAM disruption simulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In the studied scenario, a large amount of argon is added to the plasma and a disruption is simulated by starting a prescribed exponential temperature drop and finishing with fur- ther cooling provided by the argon in a self-consistent simulation of the cur- rent quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The background plasma evolution is calculated by DREAM self-consistently with the fast electron population, which is modelled kinet- ically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The resulting runaway electron distribution function along with the parameters of the EDICAM visible camera system are used as an input to the SOFT synthetic synchrotron diagnostic framework to assess the feasibil- ity of the camera for runaway electron detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' We find that the runaway electron beam formed in the disruption can produce synchrotron radiation observable by the EDICAM system, thus enabling the use of the EDICAM for the characterization of runaway electron beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Keywords: runaway electrons, synchrotron radiation, visible diagnostics, disruption Preprint submitted to Fusion Engineering and Design January 11, 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Introduction Runaway electron generation has been extensively studied in recent years [1, 2, 3, 4, 5] due to the major threat they pose for future large-scale experi- ments [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The generated runaway current is exponentially dependent on the pre-disruption plasma current due to the avalanching effect [8, 9], so in future large current devices the problem of runaway electrons is expected to be more severe than in present day tokamaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Extensive study of runaway electrons is hence important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Runaway electrons can emit various forms of radiation and this is used for their detection in current devices [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Bremsstrahlung radiation is pro- duced due to interaction of runaway electrons and other plasma particles and is typically measured in the order of 1 MeV photon energy [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It is routinely used for detection of runaway electron presence in a tokamak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Synchrotron radiation can be detected by visible and infrared camera systems [12, 13] and it can be used to gather information on the electron distribution function – both in space and momentum space [14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Synchrotron radiation images on various devices and modelling of run- away electron synchrotron radiation were used in synthetic diagnostic tools to get information on the runaway electron distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' One such study was done with the KORC code, which has been used to study the relation between the characteristics of the runaway electron population and the background plasma parameters [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' These simulations include full orbit effects of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The SOFT synthetic diagnostic framework, which works in guiding centre coordinates has been used to model synchrotron ra- diation on several tokamaks [14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In this paper, we assess the feasibility of the recently installed EDICAM visible camera system on the JT-60SA tokamak [18] for runaway electron detection and characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The EDICAM parameters were given to the SOFT synthetic diagnostic framework, which was used to simulate a synthetic synchrotron radiation image from a runaway electron distribution generated in a JT-60SA-like disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The disruption was simulated using the DREAM disruption runaway electron modelling tool [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The characteristics of the EDICAM system are introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' A description of the modelling tools used in the simulations is given in section 3, while the simulation settings, results and their interpretation are presented 2 EDICAM QE=20% QE=15% QE=10% Figure 1: The spectral sensitivity of the EDICAM camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The quantum efficiency (QE) shows the fraction of the photon energy absorbed by the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The lines shown correspond to 10, 15 and 20% efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This already includes the effect of non-sensitive areas of the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The EDICAM system sensors have a quantum efficiency of about 15% between 500 and 700 nm, but it can detect photons up to 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Finally, we conclude in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The EDICAM system The parameters of the EDICAM diagnostic are discussed in this section, with a focus on the camera characteristics, such as location, field of view, sensibility range, required for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' An EDICAM visible camera system is installed in the P18 sector of the JT-60SA tokamak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It features non-destructive read-out, meaning it can read out data from the sensors without affecting the exposure process, which enables simultaneous observation of multiple different regions in the camera field of view, specific regions-of-interest (ROI), as well as the full field of view (FOV), with different sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This makes it ideal for the observation of various fast phenomena, including some phases of disruptions [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The details of the camera characteristics can be found in [20], and the 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='08 Response (A/W) QE-10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='06 QE-15% QE-20% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='04 LUPA-1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='02 0 400 500 600 700 800 900 1000 Waveleng th (nm)Ip Figure 2: Schematic top-view of the EDICAM camera (black filled circle) viewing direction and the direction of the toroidal plasma current (Ip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' JT-60SA-specific installation parameters in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The optical parameters relevant to the modelling are listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' These show that EDICAM represents a typical general purpose visible camera with a wide field of view in a tangential direction into the tokamak from a position close to the mid- plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The spectral sensitivity of the camera system is shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It can be seen that the EDICAM is most effective in the 520 to 720 nm spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' A sketch of the camera position and viewing direction relative to the torus is shown in figure 2 along with the direction of the plasma current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' As can be seen, the camera viewing direction and corresponding plasma current direction does not prohibit the detection of the synchrotron radiation emitted by runaway electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The simulated camera image from the EDICAM location is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The above mid-plane position of the camera necessitates a more detailed study on the actual feasibility of synchrotron detection, which motivates the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Modelling tools This section describes the modelling tools used in this study: first the DREAM [19] code, used for the disruption simulation, then the SOFT code, used for the modelling of the expected synchrotron radiation image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' To assess the feasibility of the EDICAM system for runaway radiation detection, a JT-60SA-like disruption was simulated with the DREAM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' DREAM is a runaway electron modelling tool developed for the study of 4 R p XTable 1: Optical parameters of the EDICAM system as installed on JT-60SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Parameter Value Position (m) (-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5304, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='7552, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2312) Viewing direction (m) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='60915, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='01379, 0) Field of view (FOV) (degrees) 80 Spectral range (nm) 520–720 Entrance pupil (mm) 5 Figure 3: The simulated camera view of the vacuum vessel from the EDICAM location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The gray circle is the 80 degrees field of view seen by the camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 5 disruption runaway electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It calculates the evolution of the electron pop- ulation self-consistently along with the background plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The electrons can be handled with different levels of sophistication ranging from treating all electrons using a fluid model to resolving all electrons kinetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The background plasma evolution includes the temperature evolution due to ohmic heating, radiation losses, and collisional energy exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The density evolution of the plasma constituent elements is calculated including both the main ions and the impurities and their charge state evolution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The electric field and the current density evolution is calculated by solving Amp`ere’s and Faraday’s laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In the current work, DREAM was used in a kinetic mode, meaning that the entire electron population was evolved by the bounce averaged Fokker- Planck equation with a relativistic test particle collision operator [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This calculates the electron distribution function on separate 3D phase-space grid momentum grid for the thermal and suprathermal electron population and a grid for the runaway electron population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The Dreicer and hot-tail gen- eration of runaway electrons are calculated from the collisional effects and the Rosenbluth-Putvinski source term is used for the avalanche generation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The SOFT synthetic diagnostic framework [21] was used to calculate the synchrotron radiation expected from the electron distribution function as seen by the EDICAM camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The camera parameters listed in ta- ble 1 were added to the model, along with the runaway distribution function calculated by DREAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The synchrotron radiation was calculated using the cone approximation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In this approach the synchrotron radiation is as- sumed to be along the guiding centre trajectory of the runaway particles in a hollow cone shape with an infinitely thin side and an opening angle θp, that is the particle pitch angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In the presented simulations, SOFT modelled the tokamak geometry as a circular torus from the major and minor radii, and a linear q-profile was assumed for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Modelling results The disruption simulation and the resulting radiation image are presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The main plasma parameters, the runaway electron distribution func- tion and the results of the SOFT simulation are shown, along with a discus- sion on the outstanding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The aim of the disruption simulation was to produce a high energy but realistic runaway electron beam in JT-60SA 6 geometry that can be used as an input for the synchrotron image simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Accordingly, the disruption simulation input parameters, such as the input argon amount for example, were chosen to produce a significant and energetic runaway electron population, and the scenario is not optimized for disruption mitigation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Disruption simulation The DREAM code was used to simulate an argon MMI-induced disrup- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The initial current density profile was taken from the simulation of the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5 MA Scenario 2 of the JT-60SA research plan [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This was assumed to be fully ohmic current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The density and temperature profiles used by the EFIT magnetic equilibrium calculation were used as initial conditions in the DREAM modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The magnetic field strength, major and minor radii and the total plasma current were also chosen based on Scenario 2 of the JT- 60SA research plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The EFIT data was interpolated to the radial grid of DREAM, which was set to 20 radial grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The simulation was done in five separate phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' First the initial electric field was calculated to give the desired initial plasma current based on the given density and temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Then 1020 m−3 argon gas density was introduced to the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The transient ionization dynamics were re- solved using a very short time step, lasting 1 µs after which the prescribed exponential temperature decay started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This temperature decay was further simulated on a longer timescale to reach below 100 eV temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It was artificially prescribed at every radial point until the temperature reached 100 eV at the innermost radial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' After the prescribed phase ended, the temperature evolution was calculated self-consistently, mainly governed by the radiation of the injected argon gas and the ohmic heating of the plasma current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The simulation finished with the calculation of the runaway plateau phase during which the electron population can experience significant pitch angle scattering [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The total simulated time was about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='6 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The temperature as a function of radius at different times throughout the simulation is shown in figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The initial temperature at the magnetic axis is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It can be seen that until the temperature reaches 100 eV at the centre, the temperature drops exponentially at every radial point, keeping the initial profile shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Below 100 eV the temperature is calculated using an energy-balance model and it equilibrates along the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The plasma cools slower at the outer regions, and simultaneously the current density, shown in figure 4c, relaxes at a slower pace, as seen around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='7 normalized radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 Normalized radius [-] 100 101 102 103 104 Temperature [eV] Temperature evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 Normalized radius [-] 0 50 100 150 Electric field [V/m] Electric field evolution evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 Normalized radius [-] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5 Current density [MA/m2] Current density evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 Normalized radius [-] 0 20 40 60 80 Normalized electric field [-] Electric field evolution evolution Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='10 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='15 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='25 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='35 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='65 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='95 ms Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='25 ms Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='55 ms Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='85 ms Time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='35 ms Time 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='35 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='10 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='15 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='25 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='35 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='65 ms Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='95 ms Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='25 ms Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='55 ms Time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='85 ms Time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='35 ms Time 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='35 ms Figure 4: The time evolution of the main plasma parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Plot a) shows the temper- ature, and plot b) shows the electric field evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The change in the current density is plotted in graph c), while the electric field normalized to the effective critical electric field is plotted in graph d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Note the logarithmic vertical axis on the temperature plot compared to the other plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' final current density is completely provided by runaway electrons, generated by the electric field induced during the disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The accelerating electric field can be seen in figure 4b, while the electric field normalized to the effective critical field [24] is shown in figure 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The overall shape of the two electric field plots is similar since the effective critical field is mainly dependent on the density and weakly on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' As the density profile is mostly flat during the simulation, the critical field will be similar at every radial point and the normalization will preserve the shape of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In figure 4 a very high electric field can be seen around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='65 ms, when the temperature dropped to below 10 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This peak is off-axis, located at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='75 normalized radius due to the decay of the current peak seen in figure b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This electric field accelerates the electrons further and yields a more energetic runaway population at this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The plasma current evolution is plotted in figure 5, with the starting times of the different simulation phases indicated with vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The start of the ionization phase and the start of the exponential temperature decay phase overlap, as the ionization of the injected argon, was simulated in 1 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The thermal quench is simulated between the yellow and the blue vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The current quench is finished by 1 ms with a current of slightly less than 8 Figure 5: The time evolution of the total plasma current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The starting times of the simulation phases were indicated with vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The ionization phase was 1 µs, so the line indicating the start of this phase overlaps with the start of the exponential decay phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 3 MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In the runaway plateau phase the plasma current seems to increase until the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This is due to the finite wall resistance used as a boundary condition for the poloidal flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This allows for current to flow in the wall, decaying with a characteristic time, which results in an increase in the plasma current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The runaway electrons are not significantly accelerated to higher energies during this phase as the accelerating electric field is proportional to the current decay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The current drops much faster in the current quench phase, hence the acceleration in the runaway plateau phase should be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' At the end of the simulation a sufficient runaway electron population was achieved, so the plasma evolution was not calculated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The angle-averaged distribution function at the end of the simulation is shown in figure 6 for all the simulated radial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The hot-tail seed formed during the thermal quench can be seen as accelerated to higher ener- gies, as seen by the local peaks of the distribution function at various points at different radial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The maximum runaway electron momentum reached is about 70mec or about 35 MeV at about 1 m away from the mag- netic axis, corresponding to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='75 normalized radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The runaway distribution in the centre only reached about 10-20mec, as the electric field 9 Plasma current evolution 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5 Start of the ionization phase Start of exponential temperature decay Start of the self consistent temperature phase 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 current [MA] Start of the runaway plateau phase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 lasma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5 P 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0 1 2 3 4 5 6 7 8 Time [ms]Figure 6: The runaway electron distribution function plotted for each radial grid point at the last time step as a function of normalized radius p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The highest energy runaway electron population is located at about 1 m minor radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' did not penetrate into the plasma deep enough to accelerate the popula- tion at the magnetic axis, as seen in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The synchrotron radiation is highly dependent on the runaway energy and pitch angle, so the radiation is expected to be mainly off-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Simulation of the radiation image The runaway electron distribution was given to the SOFT code, which calculated the synchrotron radiation as seen by the EDICAM camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The simulated camera image is shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Significant radiation can be seen on the high field side in a crescent-like shape while the radiation is negligible at the magnetic axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The simulated image resembles observations of synchrotron radiation emitted by runaway electrons in other tokamaks [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The origin of the radiation in momentum space is illustrated by figure 8, where the Green’s function weighted with the DREAM distribution function is plotted as a function of normalized parallel and perpendicular momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The Green’s function contains information on the tokamak geometry and the momentum dependence of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It can be seen that most of the radiation originates between p = 50-65mec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This location corresponds to the highest energies of the runaway electron distribution function seen in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The radiation is also sensitive to the pitch angle of the particles, 10 Runaway distribution function r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='034m 1014 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='103m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='171m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='240m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='308m r= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='377m r= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='445m 1011 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='514m (fRE) r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='582m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='651m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='719m r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='788m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='856m 108 r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='925m r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='993m r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='062m r= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='130m r= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='199m r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='267m 105 r= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='336m 20 40 60 80 pFigure 7: The synchrotron radiation of the runaway beam during a JT-60SA-like disrup- tion as seen by the EDICAM visible camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 11 Figure 8: The Green’s function from the SOFT simulation weighted with the distribution function from DREAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Most of the synchrotron radiation comes from the region between p∥ = 50mec to p∥ = 65mec with p⊥ ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' which explains the shape of the maximum in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' There is a ridge extending to 30 mec which is most likely a result of the runaway electron population at the inner radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The dip between the two maxima is probably an artefact of the interpolation between the DREAM and the SOFT grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The dominant radiation spot corresponds to the high energy runaway population at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='75 normalized radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The hollow radiation shape can be explained by the enhancing effect of the high magnetic field strength on the synchrotron radiation as well as the high runaway electron energies at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Since more and more energetic runaway electrons are generated at the edge, the main source of radiation is coming from the edge at the high field side of the tokamak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Conclusions The DREAM and SOFT codes were used to simulate the synchrotron ra- diation originating from a runaway population generated in a JT-60SA-like disruption as seen by the EDICAM visible camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' DREAM calcu- lated the runaway electron population during an argon-induced disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' 12 Origin of the radiation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='9474 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='8421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='7368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='6316 mc 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='5263 /Td 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='4211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='3158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='2105 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='1053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='0000 60 50 40 30 20 pμ/mcThe resulting runaway distribution was given to SOFT to calculate the syn- chrotron radiation to assess the feasibility of the EDICAM camera system for synchrotron radiation detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' It was found that the synchrotron radi- ation from runaway electrons is emitted in the right direction to be observed by the installed EDICAM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Moreover, the EDICAM characteristics make it ideal to observe fast phenomena in tokamaks, such as disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Our main conclusion is that the EDICAM camera system is capable of de- tecting synchrotron radiation from high-energy runaway electrons generated in major disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' In this feasibility study, the argon injection simulated by a simple model of uniform argon density introduced simultaneously to the plasma at every radial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The simulation could be improved by considering a more real- istic injection of the argon gas, but it would most likely change the radiation shape, not the radiation detection capability of the camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' The radiation image could also be improved by adding a realistic camera view, such as the exact shape of the tokamak, to the SOFT code or another model, but it is beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Present results might motivate a more detailed study of the range of applicability of the EDICAM system, and its applicability for early detection of runaway electron beams in mitigated disruptions and other operation regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Acknowledgments The authors are grateful to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Hajnal for the simulated camera image of the EDICAM camera system and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Asztalos for the EFIT data provided for the input to DREAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Eu- ratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Neither the European Union nor the Euro- pean Commission can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Pokol and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content=' Olasz acknowledge the support of the National Research, Development and Inno- vation Office (NKFIH) Grant FK132134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} 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7, (2018), 074010, https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} +page_content='1088/1361-6587/aac33e 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfegfv/content/2301.03918v1.pdf'} diff --git a/ndE0T4oBgHgl3EQf8gJ9/content/tmp_files/2301.02789v1.pdf.txt b/ndE0T4oBgHgl3EQf8gJ9/content/tmp_files/2301.02789v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..56c81895023b9884dcf3d520f24ca28155012089 --- /dev/null +++ b/ndE0T4oBgHgl3EQf8gJ9/content/tmp_files/2301.02789v1.pdf.txt @@ -0,0 +1,1289 @@ +CGI-Stereo: Accurate and Real-Time Stereo Matching via Context and +Geometry Interaction +Gangwei Xu +Huan Zhou +Xin Yang† +School of EIC, Huazhong University of Science & Technology +{gwxu, huanzhou, xinyang2014}@hust.edu.cn +Abstract +In this paper, we propose CGI-Stereo, a novel neural net- +work architecture that can concurrently achieve real-time +performance, state-of-the-art accuracy, and strong gener- +alization ability. The core of our CGI-Stereo is a Context +and Geometry Fusion (CGF) block which adaptively fuses +context and geometry information for more accurate and +efficient cost aggregation and meanwhile provides feedback +to feature learning to guide more effective contextual fea- +ture extraction. The proposed CGF can be easily embedded +into many existing stereo matching networks, such as PSM- +Net, GwcNet and ACVNet. The resulting networks are im- +proved in accuracy by a large margin. Specially, the model +which integrates our CGF with ACVNet could rank 1st on +the KITTI 2012 leaderboard among all the published meth- +ods. We further propose an informative and concise cost +volume, named Attention Feature Volume (AFV), which ex- +ploits a correlation volume as attention weights to filter a +feature volume. Based on CGF and AFV, the proposed CGI- +Stereo outperforms all other published real-time methods +on KITTI benchmarks and shows better generalization abil- +ity than other real-time methods. The code is available at +https://github.com/gangweiX/CGI-Stereo. +1. Introduction +Stereo matching, which aims to estimate depth (or dis- +parity) from a pair of rectified stereo images, is a funda- +mental task for many robotics and computational photog- +raphy applications, such as 3D reconstruction, robot navi- +gation and autonomous driving. Despite a plethora of re- +search works in the literature, state-of-the-art methods still +have difficulties in handling repetitive structures, texture- +less/transparent objects and occlusions. Meanwhile, con- +currently achieving a high accuracy and real-time perfor- +mance is critical for practical applications yet remains chal- +lenging. +†Corresponding author. +Recently, stereo matching methods [2,9,10,19,31] based +on convolutional neural networks (CNNs) have achieved +impressive results. +State-of-the-art stereo models extract +CNN features based on which a cost volume is constructed. +The initial cost volume encodes sufficient local matching +costs yet lacks non-local knowledge, thus it is often am- +biguous in occluded regions or large textureless/reflective +regions. +To address this problem, several cost aggrega- +tion networks have been proposed to aggregate contextual +matching costs. For instance, GC-Net [10] proposes a 3D +encoder-decoder structure to learn semantic context from +the cost volume and incorporate it over the volume to rea- +son about global geometry of the scene. Following GC-Net +[10], PSMNet [2] and GwcNet [9] design new stacked 3D +encoder-decoder structures to regularize the cost volume. +These methods demand very deep stacked 3D encoder- +decoder layers to aggregate or regularize the cost volume. +Although they have shown impressive improvements in ac- +curacy, their computational and memory consumptions are +quite high due to enormous 3D convolutions. To reduce +the memory and computational costs, GANet [38] designs +two guided aggregation layers to replace 3D convolutions. +AANet [33] proposes intra-scale and inter-scale cost aggre- +gation layers. CoEx [1] improves cost aggregation by uti- +lizing extracted image features, which excites the cost vol- +ume channels with weights computed from the reference +image features. Recently developed iterative methods, rep- +resented by RAFT-Stereo [16], replace 3D CNN-based ag- +gregation using a recurrent GRU-based updater and repeat- +edly indexes from the original all-pairs correlation volume +to update disparity maps. However, without aggregation +the original cost volume is very noisy. As a result, RAFT- +Stereo [16] demands many GRUs [5] iterations and a long +inference time to obtain a satisfactory disparity map. +Cost aggregation is critical to accuracy and efficiency in +stereo matching yet existing methods require a compromise +between accuracy and speed. This paper asks the question, +can we design a lightweight 3D encoder-decoder aggrega- +tion net to concurrently obtain accurate high-resolution ge- +ometry and semantic context? +arXiv:2301.02789v1 [cs.CV] 7 Jan 2023 + +(a) Left image +(b) CGF-ACV +(c) CFNet +(d) ACVNet +Figure 1. Visual comparisons with other methods [24, 31] on KITTI 2012 [8] and 2015 [20] test set. Our CGF-ACV performs well in +occluded regions and reflective surfaces, and preserves more details. +Motivated by CoEx [1] that context features from refer- +ence image can guide more accurate understanding of stereo +geometry than solely on geometry features, this paper iden- +tifies the importance of interaction between contextual fea- +tures obtained from feature extraction and geometry fea- +tures encoded in the cost volume and proposes a novel ag- +gregation layer called Context and Geometry Fusion (CGF). +On the one hand, CGF efficiently produces spatial attention +weights to adaptively fuse multi-scale context features with +geometry features in the cost volume, thus the subsequent +3D decoding layers can decode accurate and high-resolution +geometry information with the guidance of context informa- +tion. On the other hand, the CGF establishes a direct con- +nection between feature extraction and the decoding layers +of cost aggregation. By the connection, the context features +can be better learned by directly back-propagating the gra- +dients from aggregation and regression to feature extraction. +We experimentally show that when truncating the gradient +back-propagation flow, i.e. context features only serve as a +guidance for geometry feature aggregation, the performance +degrades significantly, as shown in Tab. 4. We further pro- +pose an informative and concise cost volume, named At- +tention Feature Volume (AFV), which exploits a correla- +tion volume as attention weights to filter a feature volume. +Compared to concatenation volume [2, 10], combined vol- +ume [9] and attention concatenation volume [31], our AFV +can greatly improve the efficiency while maintaining a com- +parable accuracy (see Tab. 2). +Based on the proposed CGF and AFV, we design an ac- +curate and real-time stereo matching network called CGI- +Stereo. At the time of writing, our CGI-Stereo ranks 1st +on the popular KITTI 2012 [8] and 2015 [20] benchmarks +among all the published real-time methods. When trained +only on synthetic Scene Flow [19] dataset, our CGI-Stereo +can generalize very well to real datasets such as KITTI 2012 +and 2015, ETH3D [23], and Middlebury [22], outperform- +ing all other real-time methods. Our CGI-Stereo has also +better generalization ability than DSMNet [39] on KITTI +and Middlebury while is 55× faster than it. +In summary, our main contributions are: +• We propose Context and Geometry Fusion (CGF) +which enables effective interaction between context +features and geometry features to improve accuracy +and efficiency. +• We propose a new cost volume, name Attention Fea- +ture Volume (AFV) to encode both matching and con- +tent information, which is more efficient than com- +bined volume and attention concatenation volume. +• Based on CGF and AFV, we design an accurate and +real-time stereo matching network CGI-Stereo, which +outperforms all other published real-time methods on +KITTI benchmarks and shows better generalization +ability than other real-time methods. +• The proposed CGF can be embedded into many ex- +isting stereo matching networks, such as PSMNet [2], +GwcNet [9] and ACVNet [31]. The resulting networks +outperform the original models by a large margin. Spe- +cially, the model CGF-ACV which integrates our CGF +with ACVNet ranks 1st on the KITTI 2012 leader- +board among all the published methods. +2. Related Work +Recently, CNNs have shown great potential for stereo +matching tasks. To handle complex real world scenes, es- +pecially texture-less regions and occlusions, popular stereo +methods [2, 9, 15, 21, 24, 31, 35, 41] usually use 2D convo- +lutions to extract robust features based on which a cost vol- +ume is constructed. Then they use a large number of 3D +convolutions to regularize the cost volume. GC-Net [10] +constructs a 4D cost volume by concatenating the left and + +1 +32 +1 +16 +1 +8 +1 +4 +CGF +CGF +CGF +Expand +Add +Conv +Sigmoid +Context and Geometry Fusion (CGF) +Add +Conv +Expand +Correlation +volume +Attention +feature volume +Context feature +Geometry feature +Fused feature +Spatial attention +Multi-scale context features +Hadamard product +Figure 2. Overview of our proposed CGI-Stereo. To improve the effectiveness and efficiency of aggregation, we propose Context and +Geometry Fusion (CGF) which produces spatial attention weights to adaptively fuse context information and geometry information. We +also construct an informative and concise cost volume, named Attention Feature Volume (AFV), which exploits a correlation volume as +attention weights to filter a feature volume. +right CNN features and then utilizes a 3D encoder-decoder +network to aggregate and regularize the cost volume. Fol- +lowing GC-Net [10], PSMNet [2] and GwcNet [31] ex- +ploit the stacked 3D encoder-decoder structure to regular- +ize cost volume, achieving great improvement in accuracy. +However, the massively stacked 3D convolution layers are +computationally expensive and memory-consuming. To im- +prove efficiency of cost aggregation, GANet [38] proposes +two guided aggregation layers to replace abundant 3D con- +volutions, and AANet [33] proposes intra-scale and inter- +scale aggregation layers. Without 3D convolutions, RAFT- +Stereo [16] exploits GRUs to iteratively update disparity +maps from all-pairs correlation costs. However, these meth- +ods still have difficulties achieving high accuracy and real- +time performance concurrently. +Several studies [7,26,27,30,32,37] focus on lightweight +stereo networks to achieve real-time performance and +meanwhile maintain satisfactory accuracy. +They usually +construct a low-resolution cost volume or a sparse cost vol- +ume to greatly reduce the amount of computation. For ex- +ample, StereoNet [11] and BGNet [30] construct and ag- +gregate a low-resolution cost volume, and then use an edge- +preserving refinement module and an up-sampling module +based on the learned bilateral grid respectively to improve +accuracy. DeepPruner [7] prunes the disparity search range +to build a sparse cost volume. +DecNet [37] runs dense +matching at a very low resolution and uses sparse match- +ing at high resolution. Fast-ACV [32] proposes to exploit a +low-resolution correlation volume to generate disparity hy- +potheses with high likelihood and the corresponding atten- +tion weights, and then construct sparse attention concate- +nation volume. Although these methods achieve real-time +performance, they lead to great degradation in accuracy. +Different from existing methods, our proposed CGF can +adaptively fuse context features into geometry features, +which helps to decode accurate high-resolution geometry +features. In addition, we construct an informative and con- +cise cost volume to reduce the amount of parameters and +computation for aggregation. +Our proposed CGI-Stereo +achieves both real-time performance and high accuracy, +even surpassing some top-performing methods [2,9]. +3. Method +As illustrated in Fig. 2, our CGI-Stereo consists of multi- +scale feature extraction, attention feature volume construc- +tion, cost aggregation and disparity prediction. The cost +aggregation includes a 3D encoder and a decoder based on +context and geometry fusion (CGF). We exploit CGF to de- +code accurate and high-resolution geometry features from a +low-resolution cost volume. In this section, we first describe +the design of our CGF (Sec. 3.1). Then in Sec. 3.2, we +present details of the network architecture of CGI-Stereo. +Finally, in Sec. 3.3, we describe the loss functions used to +train our CGI-Stereo. +3.1. Context and Geometry Fusion +To decode accurate and high-resolution geometry infor- +mation from low-resolution geometry with assistance of +context information, we propose context and geometry fu- +sion (CGF) for efficient and flexible cost aggregation. +Given context features C ∈ RB×C0×H0×W0 (obtained +by feature extraction module denoted by light blue squares +in Fig. 2) and geometry features G ∈ RB×C0×D0×H0×W0 +(obtained by applying 3D convolutions to AFV denoted +using gray cubes in Fig. 2), we expand the size of C to +B × C0 × D0 × H0 × W0 (B: batch, C0: channel, D0: + +disparity, H0: height, W0: width), denoted as Cexpand. +Rather than adding Cexpand and G, we use an attention +mechanism to fuse features. Geometry features tend to have +a unimodal distribution in the disparity dimension. Thus +directly adding expanded context features to geometry fea- +tures, that is, context features are shared across the disparity +channels, completely ignores the differences in disparity di- +mension and in turn yields an ineffective fusion. Inspired +by [28], we generate spatial attention weights by exploring +the spatial relationship of context and geometry features. +The spatial attention weights can adaptively select ”impor- +tant” regions and incorporate context features for enhance- +ment. To compute the spatial attention weights, we first sum +Cexpand and G and then apply a convolution operation as, +As = σ(f 5×5(G + Cexpand)), +(1) +where σ represents the sigmoid function and f 5×5 denotes +a convolution operation with the filter size of 1 × 5 × 5. +The spatial attention weights As ∈ RB×C0×D0×H0×W0 en- +codes where to emphasize or suppress. Accordingly, we +fuse Cexpand and G as, +Gfused = f 5×5(G + As ⊙ Cexpand), +(2) +where ⊙ denotes the Hadamard Product. +3.2. Network Architecture of CGI-Stereo +3.2.1 +Multi-scale Feature Extraction +Given an input stereo image pair whose size is H × W × 3, +we use a pretrained MobileNetV2 on ImageNet [6] as our +backbone to obtain four scales of feature maps whose res- +olutions are 1/4, 1/8, 1/16, and 1/32 of the original resolu- +tion respectively. Following CoEx [1], we repeatedly per- +form up-sampling on the feature map, until its size reaches +H/4 × W/4. In more details, each up-sampling block ap- +plies a transpose convolution with kernel 4 × 4 and stride +2 to up-sample feature map of coarser resolution. Features +are concatenated with skip-connection, and then a 3 × 3 +convolution is applied to merge the skipped and upsampled +features for the current resolution. Finally, we obtain multi- +scale context features which are then used for correlation +volume construction, AFV construction and CGF as shown +in Fig. 2. +3.2.2 +Attention Feature Volume Construction +We construct an informative and concise cost volume, +named Attention Feature Volume (AFV), which exploits +correlation volume as attention weights to filter a feature +volume. Firstly, we exploit the feature maps at 1/4 resolu- +tion to construct a correlation volume Vcorr as, +Vcorr(:, d, x, y) = +< fl(:, x, y), fr(:, x − d, y) > +||fl(:, x, y)||2 · ||fr(:, x − d, y)||2 +, (3) +where d is disparity index, and (x, y) represents the pixel +coordinate. fl and fr are the left and right feature maps. The +calculated correlation volume by cosine similarity has only +one channel, thus we perform a 3 × 3 convolution followed +by a BatchNorm and a leaky ReLU to increase its channel +to 8, denoted as Acorr ∈ RB×C×D/4×H/4×W/4 (C = 8). +Then we expand the size of the left feature maps at 1/4 res- +olution to B × C × D/4 × H/4 × W/4, denoted as Fl. +Finally, the attention feature volume VAF is computed as, +VAF = Acorr ⊙ Fl. +(4) +The attention feature volume encodes both matching and +context information. Compared with combined volume pro- +posed by [9] and attention concatenation volume proposed +by [31], our attention feature volume is more efficient. As +both combined volume and attention concatenation volume +require additional cost to construct the concatenation vol- +ume. +3.2.3 +Cost Aggregation +To decode accurate and high-resolution geometry features, +we propose CGF to fuse multi-scale context features with +geometry features (see Fig. 2). First, we exploit three down- +sampling modules to aggregate matching features and in- +crease features’ receptive field while reducing computation. +Each down-sampling module consists of a 3 × 3 × 3 3D +convolution with stride 2 and a 3 × 3 × 3 3D convolution +with stride 1. After down-sampling, the size of geometry +features is B × 6C × D/32 × H/32 × W/32. Then we +alternately use CGF and an up-sampling module to decode +high-resolution geometry features. Each up-sampling mod- +ule consists of a 4 × 4 × 4 3D transposed convolution with +stride 2 and two 3 × 3 × 3 3D convolutions with stride 1. +3.2.4 +Disparity Prediction +For the aggregated cost volume, we pick out the top 2 values +at every pixel following CoEx [1], and perform softmax on +these values to compute the expected disparity. The com- +puted disparity map d0 has the size of B ×1×H/4×W/4. +Then we exploit ”superpixel” weights surrounding each +pixel as [34] to up-sample disparity map d0 to the original +resolution d1 ∈ RB×1×H×W . +3.3. Loss Function +The whole network is trained in a supervised end-to-end +manner. The final loss function is given by, +L = λ0SmoothL1(d0 − dgt) + λ1SmoothL1(d1 − dgt) +(5) +where d0 is disparity map of 1/4 resolution, d1 is the fi- +nal disparity map at full resolution. The dgt denotes the +ground-truth disparity map. + +Method +EPE (px) +D1 (%) +>1px (%) +>2px (%) +>3px (%) +Time (ms) +Baseline +0.82 +3.08 +9.67 +5.20 +3.76 +27 +AFV +0.74 +2.67 +8.31 +4.51 +3.28 +28 +CGF +0.66 +2.34 +7.47 +4.03 +2.92 +28 +AFV+CGF (CGI-Stereo) +0.64 +2.24 +7.17 +3.86 +2.80 +29 +Table 1. Ablation study on Scene Flow [19]. +4. Experiment +4.1. Datasets and Evaluation Metrics +Scene Flow [19] is a collection of synthetic stereo +datasets which provides 35,454 training image pairs and +4,370 testing image pairs with the resolution of 960×540. +This dataset provides dense disparity maps as ground truth. +For Scene Flow, we utilize the widely-used evaluation met- +rics the end point error (EPE) and the percentage of dis- +parity outliers D1 as the evaluation metrics. The outliers +are defined as the pixels whose disparity errors are greater +than max(3px, 0.05dgt), where dgt denotes the ground- +truth disparity. +KITTI includes KITTI 2012 [8] and KITTI 2015 [20], +which are datasets for real-world driving scenes. KITTI +2012 contains 194 training pairs and 195 testing pairs, and +KITTI 2015 contains 200 training pairs and 200 testing +pairs. Both datasets provide sparse ground-truth disparities +obtained with LIDAR. For KITTI 2012, we report the per- +centage of pixels with errors larger than x disparities in both +non-occluded (x-noc) and all regions (x-all), as well as the +overall EPE in both non occluded (EPE-noc) and all the pix- +els (EPE-all). For KITTI 2015, we report the percentage of +pixels with EPE larger than 3 pixels in background regions +(D1-bg), foreground regions (D1-fg), and all (D1-all). We +also use the training sets of KITTI 2012 and KITTI 2015 for +generalization performance evaluation. The >3px (i.e., the +percentage of points with absolute error larger than 3 pixel) +is reported. +Middlebury 2014 [22] is an indoor dataset with 15 train- +ing image pairs and 15 testing image pairs with full, half, +and quarter resolutions. We use the training image pairs +with half resolution to evaluate the cross-domain general- +ization performance. >2px (i.e., the percentage of points +with absolute error larger than 2 pixel) is reported. +ETH3D [23] is a collection of grayscale stereo pairs +from indoor and outdoor scenes. It contains 27 training and +20 testing image pairs with sparse labeled ground-truth. The +training set is used to evaluate cross-domain generalization +performance. We report >1px (i.e., the percentage of points +with absolute error larger than 1 pixel) metric. +Method +EPE +(px) +D1 +(%) +>3px +(%) +Time +(ms) +Combined volume [9] +0.65 +2.29 +2.86 +36 +ACV [31] +0.64 +2.24 +2.81 +35 +AFV (Ours) +0.64 +2.24 +2.80 +29 +Table 2. Cost volume analysis. Compared to combined volume +[9] and ACV [31], our AFV is more efficient while maintaining a +comparable accuracy. +4.2. Implementation Details +We implement our methods with PyTorch and perform +our experiments using NVIDIA RTX 3090 GPUs. For all +the experiments, we use the Adam [12] optimizer, with +β1 = 0.9, β2 = 0.999. The coefficients of two outputs are +set as λ0=0.3, λ1=1.0. On Scene Flow [19], we first train +CGI-Stereo network for 20 epochs, and then fine-tune it for +another 20 epochs. The initial learning rate is set to 0.001 +decayed by a factor of 2 after epoch 10, 14, 16, and 18. For +KITTI, we fine-tune the pre-trained model on Scene Flow +for 600 epochs on the mixed KITTI 2012 [8] and KITTI +2015 [20] training sets. The initial learning rate is 0.001 +and decreases to 0.0001 at the 300th epoch. +4.3. Ablation Study +To validate the effectiveness of AFV and CGF proposed +in this paper, we conduct ablation experiments on the Scene +Flow test set. We integrate AFV and CGF into a baseline +model. The baseline model constructs a correlation volume +and uses a common 3D encoder-decoder structure to reg- +ularize the correlation volume. As shown in Tab. 1, our +proposed AFV and CGF can significantly improve perfor- +mance, especially CGF, which improves EPE metric from +0.82 to 0.66. The best performance is obtained by inte- +grating AFV and CGF simultaneously, which is denoted as +CGI-Stereo. +4.4. Analysis +4.4.1 +Attention Feature Volume +We compare our attention feature volume (AFV) with at- +tention concatenation volume (ACV) proposed by ACVNet + +Position of CGF +EPE +D1 +>3px +Time +Encoder +Decoder +(px) +(%) +(%) +(ms) + + +0.74 +2.67 +3.28 +28 + + +0.70 +2.48 +3.09 +29 + + +0.64 +2.24 +2.80 +29 + + +0.64 +2.25 +2.82 +30 +Table 3. Position of CGF. Significant improvement is obtained +when integrating CGF into the 3D decoder structure. +Method +EPE +(px) +D1 +(%) +>3px +(%) +Time +(ms) +Truncating gradient +0.72 +2.46 +3.10 +29 +No Truncating +0.64 +2.24 +2.80 +29 +Table 4. Interaction analysis of CGF. +[31] and combined volume proposed by GwcNet [9]. For +all comparison models in this study, only the cost volume +construction is different, other components remain the same +with CGI-Stereo. Compared with the combined volume and +ACV which construct a concatenation volume to encode +context information, our feature volume of AFV already en- +codes sufficient context information. Results in Tab. 2 show +that, our AFV is more efficient while maintaining compa- +rable accuracy compared with the top-performing cost vol- +umes. +4.4.2 +Context and Geometry Fusion +Position of CGF. We evaluate the performance of CGF +at different positions in the 3D encoder-decoder structure, +as shown in Tab. 3. When using CGF in the 3D encoder +structure, the improvement is very slight. While significant +improvement is obtained when integrating it into a 3D de- +coder structure. We analyze that the 3D encoder is a down- +sampling process, which does not require context informa- +tion to guide. However, the 3D decoder is difficult to re- +cover accurate high-resolution geometry information from +low-resolution geometry information. Thus CGF can ex- +ploit context knowledge as an effective guidance. In addi- +tion, through CGF operations, context features can be di- +rectly supervised, which helps to learn better feature repre- +sentations. +Interaction analysis of CGF. The CGF which interacts +context and geometry can facilitate the learning of contex- +tual and geometric features. +As shown in Tab. 4, when +truncating the gradient back-propagation flow of contex- +tual features in CGF, that is, the contextual features only +serve as a guide for geometric features, the performance de- +grades significantly. While by CGF with the gradient back- +propagation flow (without truncating), geometrical features +Method +KITTI 2012 [8] +KITTI 2015 [20] +3-noc +3-all +4-noc +4-all +D1-bg +D1-all +PSMNet [2] +1.49 +1.89 +1.12 +1.42 +1.86 +2.32 +CGF-PSM +1.21 +1.57 +0.91 +1.18 +1.46 +1.80 +GwcNet [9] +1.32 +1.70 +0.99 +1.27 +1.74 +2.11 +CGF-Gwc +1.17 +0.89 +1.12 +1.15 +1.38 +1.71 +ACVNet [31] +1.13 +1.47 +0.86 +1.12 +1.37 +1.65 +CGF-ACV +1.03 +1.34 +0.78 +1.01 +1.31 +1.65 +Table 5. Performance of CGF. Our CGF improves the performance +of the state-of-the-art methods [2,9,31] on KITTI benchmarks by +a large margin. Bold: Best. +Method +EPE (px) +Runtime (ms) +PSMNet [2] +1.09 +410 +GwcNet [9] +0.76 +320 +GANet [38] +0.84 +360 +LEAStereo [4] +0.78 +300 +CFNet [24] +0.97 +180 +StereoNet [11] +1.10 +15 +BGNet [30] +1.17 +28 +DecNet [37] +0.84 +50 +CoEx [1] +0.68 +27 +CGI-Stereo (Ours) +0.64 +29 +Table 6. Comparison with the state-of-the-art on Scene Flow [19]. +can also affect the learning process of contextual features +and improve the effectiveness contextual features. +4.5. Performance of CGF +To demonstrate the superiority of our CGF, we integrate +our CGF into three state-of-the-art models, i.e. PSMNet [2], +GwcNet [9], and ACVNet [31]. We compare the perfor- +mance of the original models with those after using our +CGF. As shown in Tab. 5, our CGF can improve the per- +formance of the original methods on KITTI benchmarks by +a large margin. Specially, our CGF improves PSMNet by +18.8%, GwcNet by 11.4%, and ACVNet by 8.8% on KITTI +2012 [8] for the 3-noc metric. And for D1-all metric on +KITTI 2015 [20], our CGF can improve PSMNet by 22.4%, +GwcNet by 19.0%. +4.6. Comparisons with State-of-the-art +Scene Flow. As shown in Tab. 6, our CGI-Stereo out- +performs all other real-time methods [1, 30, 37] (i.e., the +inference time for a stereo pair is less than 50ms), which +achieves the remarkable EPE of 0.64. In addition, our CGI- +Stereo also outperforms some complex stereo models, in- +cluding PSMNet [2], GwcNet [9], GANet [38], LEASt- +ereo [4] and CFNet [24]. +KITTI 2012 and 2015. As shown in Tab. 7, our CGI- +Stereo ranks 1st on KITTI 2012 and 2015 benchmarks + +(a) Left Image +(b) Ground Truth +(c) CGF-Stereo +(d) CGI-Stereo +Figure 3. Qualitative results of our CGF-Stereo and CGI-Stereo. Our methods are only trained on Scene Flow and tested on KITTI +2012 [8], KITTI 2015 [20], Middlebury 2014 [22] and ETH3D [23] (from top to bottom). +among all the published real-time methods. +Compared +to some real-time methods such as DeepPrunerFast [7], +AANet [33], and DecNet [37], our CGI-Stereo not only +consistently outperforms them by a considerable margin, +but is also faster. +More importantly, our method also +achieves better performance than HITNet [26]. In order to +ensure the fairness of the comparison, the runtime of HIT- +Net is tested on our hardware (RTX 3090) using the open- +source models in PyTorch. Our CGF-ACV, which embeds +the CGF into ACVNet [31], ranks 1st on the KITTI 2012 +leaderboard among all the published methods. Qualitative +comparisons are shown in Fig. 1. + +30 +B34H0L706皖KA·L1644区区区YAMAHAMiddlebury +ALGORITHMSXTarget +Method +KITTI 2012 [8] +KITTI 2015 [20] +3-noc +3-all +4-noc +4-all +EPE +noc +EPE +all +D1-bg +D1-fg +D1-all +Runtime +(ms) +Accuracy +PSMNet [2] +1.49 +1.89 +1.12 +1.42 +0.5 +0.6 +1.86 +4.62 +2.32 +410 +GwcNet [9] +1.32 +1.70 +0.99 +1.27 +0.5 +0.5 +1.74 +3.93 +2.11 +320 +GANet [38] +1.19 +1.60 +0.91 +1.23 +0.4 +0.5 +1.48 +3.46 +1.81 +1800 +LaC+GANet [17] +1.05 +1.42 +0.80 +1.09 +0.4 +0.5 +1.44 +2.83 +1.67 +1800 +CFNet [24] +1.23 +1.58 +0.92 +1.18 +0.4 +0.5 +1.54 +3.56 +1.88 +180 +SegStereo [36] +1.68 +2.03 +1.25 +1.52 +0.5 +0.6 +1.88 +4.07 +2.25 +600 +SSPCVNet [29] +1.47 +1.90 +1.08 +1.41 +0.5 +0.6 +1.75 +3.89 +2.11 +900 +EdgeStereo-V2 [25] +1.46 +1.83 +1.07 +1.34 +0.4 +0.5 +1.84 +3.30 +2.08 +320 +CSPN [3] +1.19 +1.53 +0.93 +1.19 +- +- +1.51 +2.88 +1.74 +1000 +LEAStereo [4] +1.13 +1.45 +0.83 +1.08 +0.5 +0.5 +1.40 +2.91 +1.65 +300 +CREStereo [13] +1.14 +1.46 +0.90 +1.14 +0.4 +0.5 +1.45 +2.86 +1.69 +410 +CGF-ACV (Ours) +1.03 +1.34 +0.78 +1.01 +0.4 +0.5 +1.31 +3.37 +1.65 +260 +Speed +DispNetC [19] +4.11 +4.65 +2.77 +3.20 +0.9 +1.0 +4.32 +4.41 +4.34 +60 +DeepPrunerFast [7] +- +- +- +- +- +- +2.32 +3.91 +2.59 +50∗ +AANet [33] +1.91 +2.42 +1.46 +1.87 +0.5 +0.6 +1.99 +5.39 +2.55 +62 +DecNet [37] +- +- +- +- +- +- +2.07 +3.87 +2.37 +50 +BGNet+ [30] +1.62 +2.03 +1.16 +1.48 +0.5 +0.6 +1.81 +4.09 +2.19 +32 +CoEx [1] +1.55 +1.93 +1.15 +1.42 +0.5 +0.5 +1.79 +3.82 +2.13 +27 +Fast-ACVNet+ [32] +1.45 +1.85 +1.06 +1.36 +0.5 +0.5 +1.70 +3.53 +2.01 +45 +HITNet [26] +1.41 +1.89 +1.14 +1.53 +0.4 +0.5 +1.74 +3.20 +1.98 +54∗ +CGF-Stereo (Ours) +1.47 +1.82 +1.09 +1.34 +0.5 +0.5 +1.72 +3.62 +2.04 +28 +CGI-Stereo (Ours) +1.41 +1.76 +1.05 +1.30 +0.5 +0.5 +1.66 +3.38 +1.94 +29 +Table 7. Comparison with the state-of-the-art on KITTI benchmarks. Bold: Best. ∗ denotes the runtime is tested on our hardware (RTX +3090). +4.7. Generalization Performance +We compare our methods with several other stereo meth- +ods, including the non-real-time methods and real-time +methods. In this evaluation, all the comparison methods are +only trained on the synthetic Scene Flow [19] training set, +and then evaluated on four real-world datasets, i.e. KITTI +2012 [8] and 2015 [20], Middlebury 2014 [22], and ETH3D +[23]. Tab. 8 summarizes the comparisons. Among all real- +time methods, our CGI-Stereo achieves superior general- +ization performance to others. Furthermore, compared to +domain generalized method DSMNet [39], our method not +only has better generalization performance, but also is 55× +faster than it. Qualitative results are shown in Fig. 3. +5. Conclusion +We have proposed CGI-Stereo, a novel neural network +architecture that can concurrently achieves real-time perfor- +mance, state-of-the-art accuracy, and strong generalization +ability. We propose CGF to adaptively interact with context +and geometry information for more accurate and efficient +cost aggregation and meanwhile more effective contextual +feature extraction. We further propose an informative and +Method +KITTI +Middlebury +ETH3D +2012 +2015 +PSMNet [2] +6.0 +6.3 +15.8 +9.8 +GANet [38] +10.1 +11.7 +20.3 +14.1 +DSMNet [39] +6.2 +6.5 +13.8 +6.2 +CFNet [24] +5.1 +6.0 +15.4 +5.3 +STTR [14] +8.7 +6.7 +15.5 +17.2 +RAFT-Stereo [16] +- +5.7 +12.6 +3.3 +FC-PSMNet [40] +5.3 +5.8 +15.1 +9.3 +Graft-PSMNet [18] +4.3 +4.8 +9.7 +7.7 +DeepPrunerFast [7] +7.6 +7.6 +38.7 +36.8 +BGNet [30] +12.5 +11.7 +24.7 +22.6 +CoEx [1] +7.6 +7.2 +14.5 +9.0 +CGF-Stereo (Ours) +6.1 +6.0 +11.6 +6.6 +CGI-Stereo (Ours) +6.0 +5.8 +13.5 +6.3 +Table 8. 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Adaptive unimodal cost +volume filtering for deep stereo matching. +In AAAI, vol- +ume 34, pages 12926–12934, 2020. 2 + diff --git a/ndE0T4oBgHgl3EQf8gJ9/content/tmp_files/load_file.txt b/ndE0T4oBgHgl3EQf8gJ9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43820c8023fcfbd2f1ccbb9f5e4db8904084bc84 --- /dev/null +++ b/ndE0T4oBgHgl3EQf8gJ9/content/tmp_files/load_file.txt @@ -0,0 +1,893 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf,len=892 +page_content='CGI-Stereo: Accurate and Real-Time Stereo Matching via Context and Geometry Interaction Gangwei Xu Huan Zhou Xin Yang† School of EIC, Huazhong University of Science & Technology {gwxu, huanzhou, xinyang2014}@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='cn Abstract In this paper, we propose CGI-Stereo, a novel neural net- work architecture that can concurrently achieve real-time performance, state-of-the-art accuracy, and strong gener- alization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The core of our CGI-Stereo is a Context and Geometry Fusion (CGF) block which adaptively fuses context and geometry information for more accurate and efficient cost aggregation and meanwhile provides feedback to feature learning to guide more effective contextual fea- ture extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The proposed CGF can be easily embedded into many existing stereo matching networks, such as PSM- Net, GwcNet and ACVNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The resulting networks are im- proved in accuracy by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Specially, the model which integrates our CGF with ACVNet could rank 1st on the KITTI 2012 leaderboard among all the published meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We further propose an informative and concise cost volume, named Attention Feature Volume (AFV), which ex- ploits a correlation volume as attention weights to filter a feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Based on CGF and AFV, the proposed CGI- Stereo outperforms all other published real-time methods on KITTI benchmarks and shows better generalization abil- ity than other real-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='com/gangweiX/CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Introduction Stereo matching, which aims to estimate depth (or dis- parity) from a pair of rectified stereo images, is a funda- mental task for many robotics and computational photog- raphy applications, such as 3D reconstruction, robot navi- gation and autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Despite a plethora of re- search works in the literature, state-of-the-art methods still have difficulties in handling repetitive structures, texture- less/transparent objects and occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Meanwhile, con- currently achieving a high accuracy and real-time perfor- mance is critical for practical applications yet remains chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Recently, stereo matching methods [2,9,10,19,31] based on convolutional neural networks (CNNs) have achieved impressive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' State-of-the-art stereo models extract CNN features based on which a cost volume is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The initial cost volume encodes sufficient local matching costs yet lacks non-local knowledge, thus it is often am- biguous in occluded regions or large textureless/reflective regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' To address this problem, several cost aggrega- tion networks have been proposed to aggregate contextual matching costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For instance, GC-Net [10] proposes a 3D encoder-decoder structure to learn semantic context from the cost volume and incorporate it over the volume to rea- son about global geometry of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Following GC-Net [10], PSMNet [2] and GwcNet [9] design new stacked 3D encoder-decoder structures to regularize the cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' These methods demand very deep stacked 3D encoder- decoder layers to aggregate or regularize the cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Although they have shown impressive improvements in ac- curacy, their computational and memory consumptions are quite high due to enormous 3D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' To reduce the memory and computational costs, GANet [38] designs two guided aggregation layers to replace 3D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' AANet [33] proposes intra-scale and inter-scale cost aggre- gation layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' CoEx [1] improves cost aggregation by uti- lizing extracted image features, which excites the cost vol- ume channels with weights computed from the reference image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Recently developed iterative methods, rep- resented by RAFT-Stereo [16], replace 3D CNN-based ag- gregation using a recurrent GRU-based updater and repeat- edly indexes from the original all-pairs correlation volume to update disparity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' However, without aggregation the original cost volume is very noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As a result, RAFT- Stereo [16] demands many GRUs [5] iterations and a long inference time to obtain a satisfactory disparity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Cost aggregation is critical to accuracy and efficiency in stereo matching yet existing methods require a compromise between accuracy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' This paper asks the question, can we design a lightweight 3D encoder-decoder aggrega- tion net to concurrently obtain accurate high-resolution ge- ometry and semantic context?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='02789v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='CV] 7 Jan 2023 (a) Left image (b) CGF-ACV (c) CFNet (d) ACVNet Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Visual comparisons with other methods [24, 31] on KITTI 2012 [8] and 2015 [20] test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Our CGF-ACV performs well in occluded regions and reflective surfaces, and preserves more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Motivated by CoEx [1] that context features from refer- ence image can guide more accurate understanding of stereo geometry than solely on geometry features, this paper iden- tifies the importance of interaction between contextual fea- tures obtained from feature extraction and geometry fea- tures encoded in the cost volume and proposes a novel ag- gregation layer called Context and Geometry Fusion (CGF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' On the one hand, CGF efficiently produces spatial attention weights to adaptively fuse multi-scale context features with geometry features in the cost volume, thus the subsequent 3D decoding layers can decode accurate and high-resolution geometry information with the guidance of context informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' On the other hand, the CGF establishes a direct con- nection between feature extraction and the decoding layers of cost aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' By the connection, the context features can be better learned by directly back-propagating the gra- dients from aggregation and regression to feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We experimentally show that when truncating the gradient back-propagation flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' context features only serve as a guidance for geometry feature aggregation, the performance degrades significantly, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We further pro- pose an informative and concise cost volume, named At- tention Feature Volume (AFV), which exploits a correla- tion volume as attention weights to filter a feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Compared to concatenation volume [2, 10], combined vol- ume [9] and attention concatenation volume [31], our AFV can greatly improve the efficiency while maintaining a com- parable accuracy (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Based on the proposed CGF and AFV, we design an ac- curate and real-time stereo matching network called CGI- Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' At the time of writing, our CGI-Stereo ranks 1st on the popular KITTI 2012 [8] and 2015 [20] benchmarks among all the published real-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' When trained only on synthetic Scene Flow [19] dataset, our CGI-Stereo can generalize very well to real datasets such as KITTI 2012 and 2015, ETH3D [23], and Middlebury [22], outperform- ing all other real-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Our CGI-Stereo has also better generalization ability than DSMNet [39] on KITTI and Middlebury while is 55× faster than it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In summary, our main contributions are: We propose Context and Geometry Fusion (CGF) which enables effective interaction between context features and geometry features to improve accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We propose a new cost volume, name Attention Fea- ture Volume (AFV) to encode both matching and con- tent information, which is more efficient than com- bined volume and attention concatenation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Based on CGF and AFV, we design an accurate and real-time stereo matching network CGI-Stereo, which outperforms all other published real-time methods on KITTI benchmarks and shows better generalization ability than other real-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The proposed CGF can be embedded into many ex- isting stereo matching networks, such as PSMNet [2], GwcNet [9] and ACVNet [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The resulting networks outperform the original models by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Spe- cially, the model CGF-ACV which integrates our CGF with ACVNet ranks 1st on the KITTI 2012 leader- board among all the published methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Related Work Recently, CNNs have shown great potential for stereo matching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' To handle complex real world scenes, es- pecially texture-less regions and occlusions, popular stereo methods [2, 9, 15, 21, 24, 31, 35, 41] usually use 2D convo- lutions to extract robust features based on which a cost vol- ume is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Then they use a large number of 3D convolutions to regularize the cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' GC-Net [10] constructs a 4D cost volume by concatenating the left and 1 32 1 16 1 8 1 4 CGF CGF CGF Expand Add Conv Sigmoid Context and Geometry Fusion (CGF) Add Conv Expand Correlation volume Attention feature volume Context feature Geometry feature Fused feature Spatial attention Multi-scale context features Hadamard product Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Overview of our proposed CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' To improve the effectiveness and efficiency of aggregation, we propose Context and Geometry Fusion (CGF) which produces spatial attention weights to adaptively fuse context information and geometry information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We also construct an informative and concise cost volume, named Attention Feature Volume (AFV), which exploits a correlation volume as attention weights to filter a feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' right CNN features and then utilizes a 3D encoder-decoder network to aggregate and regularize the cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Fol- lowing GC-Net [10], PSMNet [2] and GwcNet [31] ex- ploit the stacked 3D encoder-decoder structure to regular- ize cost volume, achieving great improvement in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' However, the massively stacked 3D convolution layers are computationally expensive and memory-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' To im- prove efficiency of cost aggregation, GANet [38] proposes two guided aggregation layers to replace abundant 3D con- volutions, and AANet [33] proposes intra-scale and inter- scale aggregation layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Without 3D convolutions, RAFT- Stereo [16] exploits GRUs to iteratively update disparity maps from all-pairs correlation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' However, these meth- ods still have difficulties achieving high accuracy and real- time performance concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Several studies [7,26,27,30,32,37] focus on lightweight stereo networks to achieve real-time performance and meanwhile maintain satisfactory accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' They usually construct a low-resolution cost volume or a sparse cost vol- ume to greatly reduce the amount of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For ex- ample, StereoNet [11] and BGNet [30] construct and ag- gregate a low-resolution cost volume, and then use an edge- preserving refinement module and an up-sampling module based on the learned bilateral grid respectively to improve accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' DeepPruner [7] prunes the disparity search range to build a sparse cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' DecNet [37] runs dense matching at a very low resolution and uses sparse match- ing at high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Fast-ACV [32] proposes to exploit a low-resolution correlation volume to generate disparity hy- potheses with high likelihood and the corresponding atten- tion weights, and then construct sparse attention concate- nation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Although these methods achieve real-time performance, they lead to great degradation in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Different from existing methods, our proposed CGF can adaptively fuse context features into geometry features, which helps to decode accurate high-resolution geometry features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In addition, we construct an informative and con- cise cost volume to reduce the amount of parameters and computation for aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Our proposed CGI-Stereo achieves both real-time performance and high accuracy, even surpassing some top-performing methods [2,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Method As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2, our CGI-Stereo consists of multi- scale feature extraction, attention feature volume construc- tion, cost aggregation and disparity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The cost aggregation includes a 3D encoder and a decoder based on context and geometry fusion (CGF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We exploit CGF to de- code accurate and high-resolution geometry features from a low-resolution cost volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In this section, we first describe the design of our CGF (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Then in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2, we present details of the network architecture of CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3, we describe the loss functions used to train our CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Context and Geometry Fusion To decode accurate and high-resolution geometry infor- mation from low-resolution geometry with assistance of context information, we propose context and geometry fu- sion (CGF) for efficient and flexible cost aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Given context features C ∈ RB×C0×H0×W0 (obtained by feature extraction module denoted by light blue squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2) and geometry features G ∈ RB×C0×D0×H0×W0 (obtained by applying 3D convolutions to AFV denoted using gray cubes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2), we expand the size of C to B × C0 × D0 × H0 × W0 (B: batch, C0: channel, D0: disparity, H0: height, W0: width), denoted as Cexpand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Rather than adding Cexpand and G, we use an attention mechanism to fuse features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Geometry features tend to have a unimodal distribution in the disparity dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Thus directly adding expanded context features to geometry fea- tures, that is, context features are shared across the disparity channels, completely ignores the differences in disparity di- mension and in turn yields an ineffective fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Inspired by [28], we generate spatial attention weights by exploring the spatial relationship of context and geometry features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The spatial attention weights can adaptively select ”impor- tant” regions and incorporate context features for enhance- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' To compute the spatial attention weights, we first sum Cexpand and G and then apply a convolution operation as, As = σ(f 5×5(G + Cexpand)), (1) where σ represents the sigmoid function and f 5×5 denotes a convolution operation with the filter size of 1 × 5 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The spatial attention weights As ∈ RB×C0×D0×H0×W0 en- codes where to emphasize or suppress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Accordingly, we fuse Cexpand and G as, Gfused = f 5×5(G + As ⊙ Cexpand), (2) where ⊙ denotes the Hadamard Product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Network Architecture of CGI-Stereo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 Multi-scale Feature Extraction Given an input stereo image pair whose size is H × W × 3, we use a pretrained MobileNetV2 on ImageNet [6] as our backbone to obtain four scales of feature maps whose res- olutions are 1/4, 1/8, 1/16, and 1/32 of the original resolu- tion respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Following CoEx [1], we repeatedly per- form up-sampling on the feature map, until its size reaches H/4 × W/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In more details, each up-sampling block ap- plies a transpose convolution with kernel 4 × 4 and stride 2 to up-sample feature map of coarser resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Features are concatenated with skip-connection, and then a 3 × 3 convolution is applied to merge the skipped and upsampled features for the current resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Finally, we obtain multi- scale context features which are then used for correlation volume construction, AFV construction and CGF as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2 Attention Feature Volume Construction We construct an informative and concise cost volume, named Attention Feature Volume (AFV), which exploits correlation volume as attention weights to filter a feature volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Firstly, we exploit the feature maps at 1/4 resolu- tion to construct a correlation volume Vcorr as, Vcorr(:, d, x, y) = < fl(:, x, y), fr(:, x − d, y) > ||fl(:, x, y)||2 · ||fr(:, x − d, y)||2 , (3) where d is disparity index, and (x, y) represents the pixel coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' fl and fr are the left and right feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The calculated correlation volume by cosine similarity has only one channel, thus we perform a 3 × 3 convolution followed by a BatchNorm and a leaky ReLU to increase its channel to 8, denoted as Acorr ∈ RB×C×D/4×H/4×W/4 (C = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Then we expand the size of the left feature maps at 1/4 res- olution to B × C × D/4 × H/4 × W/4, denoted as Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Finally, the attention feature volume VAF is computed as, VAF = Acorr ⊙ Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' (4) The attention feature volume encodes both matching and context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Compared with combined volume pro- posed by [9] and attention concatenation volume proposed by [31], our attention feature volume is more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As both combined volume and attention concatenation volume require additional cost to construct the concatenation vol- ume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 Cost Aggregation To decode accurate and high-resolution geometry features, we propose CGF to fuse multi-scale context features with geometry features (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' First, we exploit three down- sampling modules to aggregate matching features and in- crease features’ receptive field while reducing computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Each down-sampling module consists of a 3 × 3 × 3 3D convolution with stride 2 and a 3 × 3 × 3 3D convolution with stride 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' After down-sampling, the size of geometry features is B × 6C × D/32 × H/32 × W/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Then we alternately use CGF and an up-sampling module to decode high-resolution geometry features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Each up-sampling mod- ule consists of a 4 × 4 × 4 3D transposed convolution with stride 2 and two 3 × 3 × 3 3D convolutions with stride 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4 Disparity Prediction For the aggregated cost volume, we pick out the top 2 values at every pixel following CoEx [1], and perform softmax on these values to compute the expected disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The com- puted disparity map d0 has the size of B ×1×H/4×W/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Then we exploit ”superpixel” weights surrounding each pixel as [34] to up-sample disparity map d0 to the original resolution d1 ∈ RB×1×H×W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Loss Function The whole network is trained in a supervised end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The final loss function is given by, L = λ0SmoothL1(d0 − dgt) + λ1SmoothL1(d1 − dgt) (5) where d0 is disparity map of 1/4 resolution, d1 is the fi- nal disparity map at full resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The dgt denotes the ground-truth disparity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Method EPE (px) D1 (%) >1px (%) >2px (%) >3px (%) Time (ms) Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='08 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='76 27 AFV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='28 28 CGF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='92 28 AFV+CGF (CGI-Stereo) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='24 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='80 29 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Ablation study on Scene Flow [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Datasets and Evaluation Metrics Scene Flow [19] is a collection of synthetic stereo datasets which provides 35,454 training image pairs and 4,370 testing image pairs with the resolution of 960×540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' This dataset provides dense disparity maps as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For Scene Flow, we utilize the widely-used evaluation met- rics the end point error (EPE) and the percentage of dis- parity outliers D1 as the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The outliers are defined as the pixels whose disparity errors are greater than max(3px, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='05dgt), where dgt denotes the ground- truth disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' KITTI includes KITTI 2012 [8] and KITTI 2015 [20], which are datasets for real-world driving scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' KITTI 2012 contains 194 training pairs and 195 testing pairs, and KITTI 2015 contains 200 training pairs and 200 testing pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Both datasets provide sparse ground-truth disparities obtained with LIDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For KITTI 2012, we report the per- centage of pixels with errors larger than x disparities in both non-occluded (x-noc) and all regions (x-all), as well as the overall EPE in both non occluded (EPE-noc) and all the pix- els (EPE-all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For KITTI 2015, we report the percentage of pixels with EPE larger than 3 pixels in background regions (D1-bg), foreground regions (D1-fg), and all (D1-all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We also use the training sets of KITTI 2012 and KITTI 2015 for generalization performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The >3px (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=', the percentage of points with absolute error larger than 3 pixel) is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Middlebury 2014 [22] is an indoor dataset with 15 train- ing image pairs and 15 testing image pairs with full, half, and quarter resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We use the training image pairs with half resolution to evaluate the cross-domain general- ization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' >2px (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=', the percentage of points with absolute error larger than 2 pixel) is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' ETH3D [23] is a collection of grayscale stereo pairs from indoor and outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' It contains 27 training and 20 testing image pairs with sparse labeled ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The training set is used to evaluate cross-domain generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We report >1px (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=', the percentage of points with absolute error larger than 1 pixel) metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Method EPE (px) D1 (%) >3px (%) Time (ms) Combined volume [9] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='86 36 ACV [31] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='81 35 AFV (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='80 29 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Cost volume analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Compared to combined volume [9] and ACV [31], our AFV is more efficient while maintaining a comparable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Implementation Details We implement our methods with PyTorch and perform our experiments using NVIDIA RTX 3090 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For all the experiments, we use the Adam [12] optimizer, with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The coefficients of two outputs are set as λ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3, λ1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' On Scene Flow [19], we first train CGI-Stereo network for 20 epochs, and then fine-tune it for another 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The initial learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='001 decayed by a factor of 2 after epoch 10, 14, 16, and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For KITTI, we fine-tune the pre-trained model on Scene Flow for 600 epochs on the mixed KITTI 2012 [8] and KITTI 2015 [20] training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The initial learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='001 and decreases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0001 at the 300th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Ablation Study To validate the effectiveness of AFV and CGF proposed in this paper, we conduct ablation experiments on the Scene Flow test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We integrate AFV and CGF into a baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The baseline model constructs a correlation volume and uses a common 3D encoder-decoder structure to reg- ularize the correlation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 1, our proposed AFV and CGF can significantly improve perfor- mance, especially CGF, which improves EPE metric from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='82 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The best performance is obtained by inte- grating AFV and CGF simultaneously, which is denoted as CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 Attention Feature Volume We compare our attention feature volume (AFV) with at- tention concatenation volume (ACV) proposed by ACVNet Position of CGF EPE D1 >3px Time Encoder Decoder (px) (%) (%) (ms) \x17 \x17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='28 28 \x13 \x17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='09 29 \x17 \x13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='80 29 \x13 \x13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='82 30 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Position of CGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Significant improvement is obtained when integrating CGF into the 3D decoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Method EPE (px) D1 (%) >3px (%) Time (ms) Truncating gradient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='10 29 No Truncating 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='80 29 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Interaction analysis of CGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' [31] and combined volume proposed by GwcNet [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' For all comparison models in this study, only the cost volume construction is different, other components remain the same with CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Compared with the combined volume and ACV which construct a concatenation volume to encode context information, our feature volume of AFV already en- codes sufficient context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 2 show that, our AFV is more efficient while maintaining compa- rable accuracy compared with the top-performing cost vol- umes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2 Context and Geometry Fusion Position of CGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We evaluate the performance of CGF at different positions in the 3D encoder-decoder structure, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' When using CGF in the 3D encoder structure, the improvement is very slight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' While significant improvement is obtained when integrating it into a 3D de- coder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We analyze that the 3D encoder is a down- sampling process, which does not require context informa- tion to guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' However, the 3D decoder is difficult to re- cover accurate high-resolution geometry information from low-resolution geometry information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Thus CGF can ex- ploit context knowledge as an effective guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In addi- tion, through CGF operations, context features can be di- rectly supervised, which helps to learn better feature repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Interaction analysis of CGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' The CGF which interacts context and geometry can facilitate the learning of contex- tual and geometric features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4, when truncating the gradient back-propagation flow of contex- tual features in CGF, that is, the contextual features only serve as a guide for geometric features, the performance de- grades significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' While by CGF with the gradient back- propagation flow (without truncating), geometrical features Method KITTI 2012 [8] KITTI 2015 [20] 3-noc 3-all 4-noc 4-all D1-bg D1-all PSMNet [2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='32 CGF-PSM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='80 GwcNet [9] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='11 CGF-Gwc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='71 ACVNet [31] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='65 CGF-ACV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='65 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Performance of CGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Our CGF improves the performance of the state-of-the-art methods [2,9,31] on KITTI benchmarks by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Bold: Best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Method EPE (px) Runtime (ms) PSMNet [2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='09 410 GwcNet [9] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='76 320 GANet [38] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='84 360 LEAStereo [4] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='78 300 CFNet [24] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='97 180 StereoNet [11] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='10 15 BGNet [30] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='17 28 DecNet [37] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='84 50 CoEx [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='68 27 CGI-Stereo (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64 29 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Comparison with the state-of-the-art on Scene Flow [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' can also affect the learning process of contextual features and improve the effectiveness contextual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Performance of CGF To demonstrate the superiority of our CGF, we integrate our CGF into three state-of-the-art models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' PSMNet [2], GwcNet [9], and ACVNet [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We compare the perfor- mance of the original models with those after using our CGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 5, our CGF can improve the per- formance of the original methods on KITTI benchmarks by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Specially, our CGF improves PSMNet by 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8%, GwcNet by 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4%, and ACVNet by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8% on KITTI 2012 [8] for the 3-noc metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' And for D1-all metric on KITTI 2015 [20], our CGF can improve PSMNet by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4%, GwcNet by 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Comparisons with State-of-the-art Scene Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 6, our CGI-Stereo out- performs all other real-time methods [1, 30, 37] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=', the inference time for a stereo pair is less than 50ms), which achieves the remarkable EPE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In addition, our CGI- Stereo also outperforms some complex stereo models, in- cluding PSMNet [2], GwcNet [9], GANet [38], LEASt- ereo [4] and CFNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' KITTI 2012 and 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 7, our CGI- Stereo ranks 1st on KITTI 2012 and 2015 benchmarks (a) Left Image (b) Ground Truth (c) CGF-Stereo (d) CGI-Stereo Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Qualitative results of our CGF-Stereo and CGI-Stereo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Our methods are only trained on Scene Flow and tested on KITTI 2012 [8], KITTI 2015 [20], Middlebury 2014 [22] and ETH3D [23] (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' among all the published real-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Compared to some real-time methods such as DeepPrunerFast [7], AANet [33], and DecNet [37], our CGI-Stereo not only consistently outperforms them by a considerable margin, but is also faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' More importantly, our method also achieves better performance than HITNet [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In order to ensure the fairness of the comparison, the runtime of HIT- Net is tested on our hardware (RTX 3090) using the open- source models in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Our CGF-ACV, which embeds the CGF into ACVNet [31], ranks 1st on the KITTI 2012 leaderboard among all the published methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Qualitative comparisons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 30 B34H0L706皖KA·L1644区区区YAMAHAMiddlebury ALGORITHMSXTarget Method KITTI 2012 [8] KITTI 2015 [20] 3-noc 3-all 4-noc 4-all EPE noc EPE all D1-bg D1-fg D1-all Runtime (ms) Accuracy PSMNet [2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='32 410 GwcNet [9] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='19 32 CoEx [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='13 27 Fast-ACVNet+ [32] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='01 45 HITNet [26] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='98 54∗ CGF-Stereo (Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='04 28 CGI-Stereo (Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='94 29 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Comparison with the state-of-the-art on KITTI benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Bold: Best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' ∗ denotes the runtime is tested on our hardware (RTX 3090).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Generalization Performance We compare our methods with several other stereo meth- ods, including the non-real-time methods and real-time methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In this evaluation, all the comparison methods are only trained on the synthetic Scene Flow [19] training set, and then evaluated on four real-world datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' KITTI 2012 [8] and 2015 [20], Middlebury 2014 [22], and ETH3D [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 8 summarizes the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Among all real- time methods, our CGI-Stereo achieves superior general- ization performance to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Furthermore, compared to domain generalized method DSMNet [39], our method not only has better generalization performance, but also is 55× faster than it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Qualitative results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Conclusion We have proposed CGI-Stereo, a novel neural network architecture that can concurrently achieves real-time perfor- mance, state-of-the-art accuracy, and strong generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We propose CGF to adaptively interact with context and geometry information for more accurate and efficient cost aggregation and meanwhile more effective contextual feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We further propose an informative and Method KITTI Middlebury ETH3D 2012 2015 PSMNet [2] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 GANet [38] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 DSMNet [39] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2 CFNet [24] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 STTR [14] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2 RAFT-Stereo [16] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 FC-PSMNet [40] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 Graft-PSMNet [18] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 DeepPrunerFast [7] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 BGNet [30] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 CoEx [1] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0 CGF-Stereo (Ours) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='6 CGI-Stereo (Ours) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content='3 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Generalization performance on KITTI, Middlebury and ETH3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' All models are only trained on Scene Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' We split methods into two categories: non-real-time and real-time (from top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' concise cost volume, named AFV, which is more efficient while maintaining comparable accuracy compared with the top-performing cost volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' References [1] Antyanta Bangunharcana, Jae Won Cho, Seokju Lee, In So Kweon, Kyung-Soo Kim, and Soohyun Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Correlate-and- excite: Real-time stereo matching via guided cost volume excitation.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' 1 [6] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Imagenet: A large-scale hierarchical image database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' In IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE0T4oBgHgl3EQf8gJ9/content/2301.02789v1.pdf'} +page_content=' Pattern Recog.' metadata={'source': 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Mamduhi, and Sandra Hirche +Abstract—Quantification of the triggering rates of an event- +triggered stochastic system with deterministic thresholds is a +challenging problem due to the non-stationary nature of the +system’s stochastic processes. A typical example is the com- +putation of the average communication rate (ACR) of the +networked event-triggered stochastic control systems (ET-SCS) of +which the communication of the sensor network is scheduled by +whether a system variable of interest exceeds predefined constant +thresholds. For such a system, a closed-loop effect emerges due to +the interdependence between the system variable and the trigger +of communication. This effect, commonly referred to as side +information by related work, distorts the stochastic distribution of +the system variables and makes the ACR computation non-trivial. +Previous work in this area used to over-simplify the computation +by ignoring the side information and misusing a Gaussian +distribution, which leads to approximated results. This paper +proposes both analytical and numerical approaches to predict the +exact ACR for an ET-SCS using a recursive model. Furthermore, +we use theoretical analysis and a numerical study to qualitatively +evaluate the deviation gap of the conventional approach that +ignores the side information. The accuracy of our proposed +method, alongside its comparison with the simplified results of the +conventional approach, is validated by experimental studies. Our +work is promising to benefit the efficient resource planning of +networked control systems with limited communication resources +by providing accurate ACR computation. +I. INTRODUCTION +I +N recent years, emerging networked control systems, such +as intelligent industrial manufacturing [1], smart power +grids [2], and autonomous vehicles [3], are characterized by a +distributed design manner where the plants and the sensors are +located remotely and are connected with a common network. +The development of the closed-loop controllers for these sys- +tems requires sufficient sampling of the system states ensured +by active communication of the network, such that the con- +trollers always have access to the latest system states provided +by the remote sensors through the network. Nevertheless, the +communication resources of the networked systems are often +limited by the power restrictions of the system, especially for +the mobile and portable devices of which the power mainly +rely on batteries. This issue suggests reducing the frequency of +state sampling by properly scheduling the communication of +Z. Zhang is with the Department of Electrical Engineering, Eindhoven +University of Technology, Netherlands (z.zhang3@tue.nl). +Q. Liu is with the Department of Automation, University of Science and +Technology of China, Hefei, China (qingchen liu@ustc.edu.cn). +M. H. Mamduhi is with the Automatic Control Laboratory, ETH Z¨urich, +Switzerland (mmamduhi@ethz.ch). +S. Hirche is with the Chair of Information-Oriented Control, Technical +University of Munich, Munich, Germany (hirche@tum.de). +† Equivalent contribution as the common first authors. +*Corresponding author. +the network. This was conventionally solved with time-based +schemes which later gave way to event-based schemes. +It is argued in [4] that the event-based scheduling schemes +can achieve the same performance as the periodic time-based +ones but with considerably less consumption of commu- +nication resources. This finding motivates the development +of different event-based scheduling schemes, including the +stochastic, periodic, and deterministic event-based ones, for +various networked systems [5]–[7]). The event-based schemes +have shown superior efficiency, high flexibility, and quick +responsiveness incorporating the restriction of communication +resource limitations [8]–[11]. A typical event-based scheduling +scheme is illustrated in Fig. 1, where the sampling of the sys- +tem state or the activation of the communication is governed +by a scheduler that triggers a time-asynchronous event. This +event does not explicitly depend on time but is associated with +a system variable of interest and a predefined threshold. Thus, +the scheduling is performed in a time-asynchronous manner +which activates the communication only when necessary. +The necessity of communication is determined by a certain +triggering event, which intermittently closes the loop between +the controlled system and the sensor. This ensures efficient +consumption of the communication resources [12]–[16]. +Sensor +System +Scheduler +Remote sensing +St +δt +Vt +E(Vt, η) +Figure 1: A typical event-based communication scheduler for +a networked system. The dashed line indicates remote sensing, +such as infra-sensing or visual perception. The switch symbol +denotes the communication status δt (active or inactive) of +the network. St is the sampled system state, Vt is the system +variable of interest, and E (Vt, η) represents the event that +triggers the communication based on a given threshold η. +One of the main purposes of investigating communication +scheduling schemes for networked systems is for the efficient +planning of communication resources. Compared to the time- +based scheduling schemes, the resource planning for the event- +based ones is usually less straightforward and more challeng- +ing due to asynchronous state sampling. A typical index that +facilitates communication resource planning is communication +arXiv:2301.05445v1 [eess.SP] 13 Jan 2023 + +2 +rate (CR) which depicts the likelihood of active commu- +nication at a certain time [17], i.e., the probability of the +communication switch in Fig. 1 being closed. CR provides a +practical insight into how many system resources are allocated +to network communication and supports the efficient planning +of system resources. In most applications, the communication +status is a random variable incorporating stochastic system +uncertainties. Thus, what attracts us is typically the average +communication rate (ACR) which refers to the expected +value of active communication. Particular attention has been +attracted to the stationary ACR [18] which represents the limit +of the ACR as the time approaches infinity. It is often used +to evaluate the consumption of communication resources in +the steady state of a networked system. Conventionally, the +computation of the ACR is solved using statistical methods +via numerical experiments, such as a Monte Carlo experiment. +The analytical solution of the ACR with a closed form is a +challenging problem due to the closed-loop effect of the event- +based scheduling scheme, also known as side information [19]. +In Fig. 1, the closed-loop effect, or side information, refers +to the interdependence between the triggering event and the +system variable of interest, which forms a closed loop between +the system and the scheduler (the blue path). This closed-loop +distorts the distributions of the system variables, such that +they are no more subject to trivial Gaussian distributions even +though the system uncertainties are formulated as Gaussian. +Analyzing the stochastic property of the communication status +is thus challenging since it is difficult to track the probabilistic +propagation of the system variables. In the literature, analytical +computation methods for the ACR are quite sparse. They only +show up in a few papers that mainly focus on the control +and filtering of networked systems [6], [17], [20]–[23]. These +approaches either over-simplify the computation of the ACR +by imposing impractical assumptions or conduct conservative +and coarse approximations. Their results can hardly be used +to predict exact ACR which is important to efficient resource +planning for networked communication systems. In [17], [20], +the ACR is computed by ignoring the closed-loop effect +and approximating the distribution of the concerned system +variable with a Gaussian distribution, which, however, leads +to an accuracy gap with the true ACR. The work in [22] +provides lower and upper bounds for the ACR without giving +its analytical form. In [6], [23], the computation of ACR +is simplified by involving a stochastic triggering threshold +which has obvious shortcomings compared to deterministic +thresholds due to the inferior control performance and the +sensitivity to data loss. To our best knowledge, the accurate +computation of ACR for an event-triggered networked system +with deterministic triggering thresholds has not been addressed +in the literature, although it is a fundamental step towards the +study of the crucial filtering problem [22], [24]–[27]. +As mentioned, the fundamental challenge of accurately +quantifying the ACR for an event-triggered networked system +is the precise tracking of the probabilistic propagation of +the system variables of interest under the closed-loop effect. +From a mathematical perspective, the event-based scheduling +scheme with deterministic thresholds imposes constant trunca- +tion to the support set of the probabilistic distribution functions +(PDF) of the concerned system variables at each sampling +instant. This truncation operation constantly removes the +Gaussianity of the system variables. Thus, the critical technical +point of accurately computing the ACR is to precisely depict +the truncated PDFs of the system variables recursively over +time, which formulates a challenging mathematical problem. +The main goal of this work is to overcome this challenge +and provide accurate computation methods for the ACR of a +typical networked system, an event-triggered stochastic control +system (ET-SCS), where the communication is triggered when +the state estimation error exceeds a deterministic threshold. +To support this, we propose a recursive model that exactly +depicts the temporal evolution of ACR based on the model +raised in [17]. Using this recursive model, we are able to +track the truncated PDF of the system variables at each +sampling time and investigate its influence on the ACR. Based +on this, we are able to compute the ACR at an arbitrary +instant using a finite number of coefficients. We also prove the +existence of the stationary ACR and calculate its value using +these coefficients. Considering the complexity of the proposed +analytical method, we further raise a numerical counterpart +algorithm to calculate the ACR recursively. Both theoretical +analysis and experimental studies are conducted to verify the +accuracy of the proposed methods and qualify the inaccuracy +gap of the conventional methods [6], [17], [20]–[23]. The main +contributions of this article are summarized below. +1) Proposing a novel recursive temporal-evolution model for +the ACR of a generic ET-SCS. +2) Proving the existence of the stationary ACR for a generic +ET-SCS with deterministic event-triggered thresholds. +3) Providing an analytical method and a numerical algorithm +for the computation of ACR for a generic ET-SCS. +4) Conducting theoretical and numerical studies to qualify +the accuracy gap of the conventional method. +5) Validating the feasibility and accuracy of the proposed +methods using a case study. +The rest of this article is organized as follows. Sec. II formu- +lates the problem, with mathematical preliminaries provided. +In Sec. III, we introduce the recursive model of the ACR +and investigate the existence of the stationary ACR. Sec. IV +presents the methods to precisely compute the ACR. In Sec. V, +we use theoretical analysis and a numerical example to verify +the accuracy of the proposed method and evaluate the accuracy +gap of the conventional method. In Sec. VI, a numerical +experiment on a simple vehicle-following case is conducted to +validate our methods. Finally, Sec. VII concludes the article. +Notation: The rest of this article obeys the following nota- +tions. The sets of real and natural numbers are denoted by +R and N. The superscript + sued after them indicates the +subsets only containing the positive elements. A Gaussian +distribution with mean value µ ∈ R and variance σ2, σ ∈ R+ +is represented by N(µ, σ2). For a stochastic event E defined +on a probability space, P(E ) denotes the occurring probability +of E . For a stochastic variable z ∈ R, P(z), pz(·), Fz(·), E(z), +and Var(z) denote, respectively, its probability, PDF, cumula- +tive distribution function (CDF), expectation, and variance. + +3 +II. PROBLEM STATEMENT AND PRELIMINARIES +In this section, we describe the main mathematical problem +of this paper and provide the preliminaries for its solution. +First, we introduce the dynamical model of an ET-SCS and +the definition of the ACR, followed by the problem statement. +Then, we revisit the Jury’s stability criterion and the stochastic +properties of truncated random variables which are important +to the analysis of the ACR in the next sections. +A. Dynamic Model of ET-SCS and State Estimation Error +As shown in Fig. 2, the ET-SCS considered in this article +is composed of a plant and a series of sensors which are con- +nected with a common network. In this paper, we investigate +a scalar ET-SCS, where the dynamic model of the plant is +assumed to be linear time-invariant (LTI) and depicted by the +following scalar stochastic difference equation (SDE), +xk+1 = Axk + Buk + wk, +(1) +where k ∈ N denotes the discrete sampling time of the system, +xk, uk ∈ R are, respectively, the state and control input of the +system at time k, A, B ∈ R are constant parameters, and +wk ∈ R is the stochastic noise of the system. For simplicity, +we assume that the initial state of the system x0 is a known +deterministic variable. Note that the stochastic process wk, +k ∈ N, is subject to the following assumption. +Assumption 1. The Gaussian stochastic process wk is inde- +pendent and identically distributed (i.i.d.) for all k ∈ N, i.e., +1) wk ∼ N +� +0, σ2� +, ∀ k ∈ N, with σ ∈ R+. +2) pw,w(wi, wj) = pw(wi)pw(wj) holds for all i, j ∈ N, +i̸=j, where pw(·) is the PDF of stochastic variable wk, k ∈ N, +and pw,w(·, ·) is the joint PDF of wi and wj, i, j ∈ N. +Sensor +State Estimator +Controller +Plant Dynamics +Scheduler +Memory +rk +uk +ek +− +Network +Ik−1:ι +Plant +δk +xk +− +Ek +ˆxk +Figure 2: The block diagram of an ET-SCS. +Remark 1. Our work in this paper only deals with a scalar +ET-SCS. Note that the exact computation of the ACR for a +multi-dimensional ET-SCS is even more challenging due to +the probabilistic coupling among the individual dimensions of +the non-Gaussian system variables. Also, various triggering +options for multi-dimensional system variables make it tedious +to determine their analytical distribution functions. The study +on a scalar ET-SCS is sufficient to draw essential quantitative +and qualitative conclusions that can be extended to multi- +dimensional systems with additional efforts in future work. +For system (1), its state xk at any time k ∈ N+ can be mea- +sured by one or multiple sensors. However, the state is sampled +for the system plant only when the network communication +is active, indicated by a closed switch in Fig. 2. At the time +k, whether the status of the communication is active or not is +represented as a binary event δk ∈ {0, 1}, namely δk = 1 for +active and δk = 0 for inactive. For brevity, we represent these +conditions as δ{1} +k +and δ{0} +k +. The communication status δk is +affected by an event Ek which is produced by an event-based +scheduler which will be introduced in Sec II-B. When the sys- +tem state is not sampled, an estimated value ˆxk is provided by +a state estimator utilizing the system model (1) and the history +data Ik−1:ι = {δk−1, · · · , δι+1, δι, xι, uk−1, · · · , uι} [7], +ˆxι = xι +ˆxι+1 = Axι + Buι, +· · · +ˆxi = Aˆxi−1 + Bui−1, +(2) +i = ι + 2, · · · , k, where ι ∈ N refers to the last instant of +state sampling and equation ˆxι = xι indicates a state sampling +operation. Since the initial state x0 is deterministic and known, +we have ˆx0 = x0 and e0 = 0. The history data Ik−1:ι is stored +in a memory on the plant. All the system data before the last +instant, i.e., all {δi, xi, ui} where i < ι, is timely abandoned. +Thus, a feedback controller uk = u(ˆxk−rk) can be designed to +achieve the desired closed-loop performance. The performance +of the feedback controller is not within the scope of this paper. +We make a correspondence between the ET-SCS in Fig. 2 +and the general event-based networked system in Fig. 1 by +recognizing xk as the system state and the estimation error +ek = xk − ˆxk as the system variable of interest. +By subtracting (2) from (1), we obtain the dynamic model +of the state estimation error as +eι = 0, +eι+1 = wι, +· · · +ei = Aei−1 + wi−1, +(3) +where i = ι + 2, · · · , k. Therefore, for all i = ι + 1, · · · , k, +ei is a random variable. Substituting ek−1, ek−2, · · · , eι to ek +recursively, we obtain +ek = �k−1 +i=ι Ak−i−1wi, k > ι. +(4) +The dynamic model (3) indicates that the estimation error +accumulates over the time interval {ι + 1, ι + 2, · · · , k} due +to the lack of state sampling. The error accumulation may +lead to the degradation of the system control performance. +Meanwhile, equation (4) shows that ek is subject to a Gaussian +distribution N +� +0, �k−1 +i=ι A2(k−i−1)� +considering the property +of the linear combination of Gaussian stochastic variables. +Nevertheless, we should note that (3) and (4) only hold when +the state estimation error and the triggering event are subject +to an open-loop configuration. In Fig. 2, this pertains to the +scenario where the Scheduler is designed such that its output +Ek is independent of its input ek. Next, we will show that (3) +and (4) do not generally hold and ek is no more a Gaussian- +distributed stochastic variable for an event-triggered scheduler. + +4 +B. Event-Triggered Scheduling and Closed-Loop Effect +To restrict the accumulation of the state estimation error ek, +k ∈ N+, by fully exploiting the communication resources, the +scheduler performs a least-necessary principle meaning that +the communication is only activated when either the last state +estimation error ek−1 exceeds a predefined threshold η ∈ R+, +or the consecutive inactive period is beyond a time limit T ∈ +N+, i.e., for all k ∈ N+ and ι ∈ N, ι < k, +δk = +� +1, +if +|ek−1| ≥ η or k − ι > T +0, +otherwise, +(5) +where the condition that triggers active communication δ{1} +k +refers to a positive event Ek, otherwise a negative event +E k. The scheduling scheme (5) maintains a decent system +performance by ensuring low resource usage by limiting the +frequency of communication while restricting the estimation +errors. Such an event-based scheduling model is widely used +in various networked control systems, such as platooning of +a group of vehicles [28], [29], power systems [30], [31] and +cooperative manipulation in robotics systems [32], [33]. +The result of the event-triggering scheduling scheme (5) +is a closed loop between the triggering event Ek, or the +communication status δk and the state estimation error ek (the +blue path in Fig. 2). This loop has a significant influence +on the probabilistic distribution of the estimation error ek. +Specifically, the error accumulation depicted in (3) only occurs +when |ei| ≤ η, for any i = ι, ι + 1, · · · , k − 1. Otherwise, any +|ei| > η will immediately activate the state sampling and lead +to δi+1 = 1 and ei+1 = 0. Thus, given the last communication +instant ι and the history data Ik−1:ι, the estimation error under +the event-triggered scheduling scheme (5) becomes +ˆeι = 0, +ˆeι+1 = wι, +· · · +ˆei = +� Aˆei−1 + wi−1, +if |ˆei−1| < η, +0, +else, +(6) +where i = ι + 2, · · · , k, for all k ≤ ι + T. Here, we +use a new symbol ˆei to represent this closed-loop state +estimation error, or closed-loop error under the event-triggered +scheduling scheme (5) to distinguish it from the open-loop +error ei in (3). Similar to ei, ˆei is also a random variable, +for i = ι + 1, ι + 2, · · · , k. Differently, the closed-loop error +(6) contains additional side information |ˆei−1| < η compared +to the open-loop error (3). This side information imposes a +truncation operation on the probabilistic distribution of the +closed-loop error at every sampling instant. It breaks the +linearity of the dynamic model of the state estimation error +and distorts its stochastic propagation. As a result, the closed- +loop error is hardly subject to a Gaussian distribution as +time increases, even though the noise is a stationary Gaussian +process according to Assumption 1. Also, it is difficult to bring +up a brief overall analytical form to represent ˆek, similar to +(4), which makes it difficult to track the distorted stochastic +propagation. In this paper, we refer to the fact that the side +information distorts the stochastic propagation of the system +variable of interest as the closed-loop effect. In Sec. II-E, we +will briefly introduce the effect of a truncation operation on a +random variable. +C. Stationary and Transient ACR of an ET-SCS +For the ET-SCS in (1) with the state estimator (2) and the +event-triggered scheduler (5), the ACR is defined as +E(δk) = P(δ{1} +k +), k ∈ N +(7) +which depicts the likelihood of the active status of the com- +munication, or, equivalently, the realization of the event δ{1} +k +. +Meanwhile, the ACR denotes the probability of the state +sampling. Since the initial state x0 is deterministic and known, +we have E(δ0) = P(δ{1} +0 +) = 1. Also, given ˆe0 = x0 − ˆx0 = 0, +we know E(δ1) = P(δ{1} +1 +) = 0. For any k ∈ N+, k ≥ 2, +however, the value of δk is usually random, and the ACR is +computed as +E(δk) = 1 − P(δ{0} +k +) = 1 − +� η +−η pˆek−1(z)dz, +(8) +where pˆek(·) is the PDF of the state estimation error and z ∈ +R is an auxiliary variable. Note that the integration interval +[ −η, η ] corresponds to the constraint |ˆek−1| < η in (5) which +blocks state sampling at time k. +Remark 2. The above statements are based on our assumption +that the initial system state x0 is known. Otherwise, the values +of E(δ0) and E(δ1) are neither 1 nor 0. Instead, they are +dependent on the distribution of x0 and should be valued +within the interval (0, 1). +The limit E(δ∞) = lim +k→∞ E(δk), if it exists, is defined as the +stationary ACR, while E(δk) for a finite k ∈ N is referred to as +the transient ACR. The main issue of exactly calculating the +stationary and the transient ACR is that the analytical form +of the PDF pˆek(·), k ∈ N, is difficult to derive due to the +challenge of capturing the nontrivial stochastic propagation +of the closed-loop errors, as introduced in Sec. II-B. In some +existing work [17], [23], pˆek(·) is approximated by a Gaussian +distribution by ignoring the closed-loop effect, which leads +to the approximated calculation of ACR. We show in this +article that such approximation results in a larger value of +ACR compared to the truth. Alongside this, accurate ACR +computation methods are also provided. +D. Existence of Steady State of A Discrete-Time LTI System +To verify the existence of the stationary ACR for an ET- +SCS, in this paper, we construct a recursive model to depict +the timed evolution of the ACR. The recursive ACR model is +indeed a discrete-time LTI (dt-LTI) system and the stationary +ACR is equivalent to its steady state. This allows us to solve +the stationary ACR by investigating the asymptotic stability of +a general dt-LTI system, which can be examined by the well- +known Jury stability criterion [34]. Consider a characteristic +polynomial with variable z ∈ R in the following form, +D(z) = a0 + a1z + a2z2 + . . . + aNzN, + +5 +where N ∈ N+ is the degree of the characteristic polynomial +and ai ∈ R, i = 1, 2, · · · , N, are coefficients. The following +tests determine whether the system represented by D(z) has +any pole outside the unit circle (the instability region). A sys- +tem must conform to all the following rules to be considered +stable. +Rule 1: If z = 1, D(z) > 0 must hold. +Rule 2: If z = −1, zND(z) > 0 must hold. +Rule 3: |a0| < |aN| must hold. +If all rules satisfied, we expand the Jury Array as follows. +1) +a0 +a1 +a2 +a3 +... +aN +2) +aN +... +a3 +a2 +a1 +a0 +3) +b0 +b1 +b2 +... +bN−1 +4) +bN−1 +... +b2 +... +b0 +... +... +... +... +2N − 3) +v0 +v1 +v2 +Once we reach to a row with 2 members, we stop constructing +further arrays. To calculate the values of the odd-number rows, +we can use the following formula. The even number rows are +equal to the previous row in reverse order. We will use k as +an arbitrary subscript value. These formulas are reusable for +all elements in the array: +bk = +���� +a0 +aN−k +aN +ak +���� , ck = +���� +b0 +bN−1−k +bN−1 +bk +���� , dk = +���� +c0 +cN−2−k +cN−2 +ck +���� . +This pattern can be carried out to all lower rows of the array, +if necessary. +Rule 4: Once the Jury array has been formed, all the following +relationships must be satisfied until the last row of the array +|b0| > |bN−1|, +|c0| > |cN−2|, +|d0| > |dN−3|. +The system is stable if all these conditions are satisfied. +E. Truncated Stochastic Variables +As mentioned in Sec. II-C, the side information |ek−1| < η +in (6) changes the support of pˆek(·) at every sampling instant +k. With a constant threshold η ∈ R+, the change is specifically +a symmetric truncation operation to the PDF pˆek(·). Consider +a scalar stochastic variable ζ ∈ R with an infinite-support +PDF pζ(·). We use ζ[a,b] to represent the truncated stochastic +variable derived from ζ by trimming its support set with a fixed +interval ζ ∈ [ a, b ], a, b ∈ R. Due to this truncation operation, +the derived variable ζ[a,b] has different stochastic properties +compared to its original ζ. Specifically, its PDF reads +pζ[a,b](z) = +� +ρζ(a, b)pζ(z), +a ≤ z ≤ b, +0, +otherwise, +(9) +where z ∈ R is an auxiliary variable, pζ[a,b](·) denotes the PDF +of ζ[a,b] subject to a truncation interval [ a, b ], and ρζ(a, b) is +a scalar calculated as +ρζ(a, b) = 1/(Fζ(b) − Fζ(a)) +where Fζ(·) is the cumulative distribution function (CDF) of +ζ. Also, the expected value and the variance of ζ[a,b] are +E +� +ζ[a,b]� += +� b +a zpζ[a,b](z)dz = ρζ(a, b) +� b +a zpζ(z)dz, +(10) +Var +� +ζ[a,b]� += +� b +a z2pζ[a,b](z)dz − E2� +ζ[a,b]� += ρζ(a, b) +� b +a z2pζ(z)dz − E2� +ζ[a,b]� +. +(11) +Note that the difference between the mean values and the +variances of a stochastic variable ζ and its truncated coun- +terpart ζ[a,b] is reflected not only by the additional multiplier +ρζ(a, b) but also by the changed upper and lower limits of +the integrals, a and b. If ζ is a Gaussian variable, ζ[a,b] is not +necessarily Gaussian. This means that all the properties that +are proposed for Gaussian variables, such as the linear combi- +nation properties, may not hold for their truncated variables. +Ignoring this effect may lead to the inaccurate characterization +of the truncated stochastic variable. +III. COMMUNICATION RATE ANALYSIS +In this section, we conduct a comprehensive analysis of +the transient and the stationary ACR for an ET-SCS defined +in (7). Based on the introduction of the predictive indexes +and the predictive coefficients, we derive a recursive model +for the transient ACR of an ET-SCS. Then, by showing +the equivalence between the recursive model and a dt-LTI +system, we prove the existence of the stationary ACR using +the Jury stability criterion recalled in Sec. II-D. As a result, +the transient and the stationary ACR can be calculated using +a finite number of predictive coefficients. +A. The Predictive Indexes and The Predictive Coefficients +In this section, we introduce the predictive indexes and +the predictive coefficients which are important to analyze the +ACR. We first define a compound event for k, n ∈ N+, n ≤ k, +Ek:k−n = δ{0} +k +∩ δ{0} +k−1 ∩ · · · ∩ δ{0} +k−n+1 ∩ δ{1} +k−n, +(12) +which represents the conjunction of n successive inactive +events after an active event δ{1} +k−n. It is straightforward to show +that the compound event satisfies the following property. +Property 1. For any n, k ∈ N+, n ≤ k, event Ek:k−n satisfies +the following conditions. +1) Ek:k−n ̸= ∅, ∀ n ≤ T, and Ek:k−n = ∅, ∀ n > T. +2) Ek:k−i ∩ Ek:k−j = ∅, for any i, j ≤ k, i ̸= j. +3) �k +n=1 Ek:k−n = δ{0} +k +. +In Property 1, condition 1) is met considering that the non- +communication period of the system should not be larger than +the limit T, according to the event-triggered scheduler (5). +Condition 2) is verified by the mutual exclusion between the +compound events. Condition 3) is justified by taking the union +of all compound events Ek:k−n, for all n = 1, 2, · · · , k. + +6 +1) Predictive indexes: For particular communication vari- +ables δ0, δ1, · · · , δn, we define n-step predictive non- +communication index Pn, or predictive index as +Pn = P +� +δ{0} +n +, δ{0} +n−1, · · · , δ{0} +1 +��� δ{1} +0 +� +, n ∈ N+, +(13) +which denotes the probability that no communication is acti- +vated for n sampling instants given an active communication +event δ{1} +0 +. We will later generalize the predictive index to +arbitrary-time communication status δk, k ∈ N+. The n-step +predictive index Pn satisfies the following property. +Property 2. For any n, T ∈ N+, the predictive index Pn +satisfies the following conditions. +1) For all n > T, Pn = 0. +2) For all n ≤ T, 0 < Pn < 1. +Property 2-1) is justified by that the communication is +activated by force after n > T, according to (5). For n ≤ T, +neither activation nor deactivation of the communication is a +certain event, since the support of the PDF of the noise wk is +infinite, ∀ k = 1, 2, · · · , n. This addresses property 2-2). +2) Predictive coefficients: Also for communication sta- +tus variables δ0, δ1, · · · , δn, the n-stacked predictive non- +communication coefficient P n, or predictive coefficient is +defined as +P n = P +� +δ{0} +n +��� En−1:0 +� +, n = 1, 2, · · · , T, +(14) +which denotes the probability of a single non-communication +event δ{0} +n +given the history compound event En−1:0. Similar +to the predictive index, the predictive coefficient has the +following property due to the infinite support of the PDF of +the stochastic noise. +Property 3. 0 < P n < 1 holds for all n = 1, 2, · · · , T. +3) Relation between the indexes and coefficients: From the +definitions of the predictive index in (13) and the predictive +coefficient (14), we have the following relation, +Pn = P +� +δ{0} +n +, δ{0} +n−1, · · · , δ{0} +1 +��� δ{1} +0 +� += P +� +δ{0} +n +��� δ{0} +n−1, · · · , δ{0} +1 +, δ{1} +0 +� +× P +� +δ{0} +n−1, · · · , δ{0} +1 +��� δ{1} +0 +� += P nPn−1, n = 2, · · · , T, +(15) +with P1 = P 1, which renders the following property. +Property 4. (Relation of the predictive index and coefficient) +The following relation holds. +Pn = �n +i=1 P i, n = 1, 2, · · · , T. +(16) +Meanwhile, applying Property 2 and Property 3 to (15) +recursively, we have the following property. +Property 5. (Monotonic decrease of predictive index) The +condition 0 < PT < . . . < P2 < P1 < 1 always holds. +4) Time-Invariance of the indexes and coefficients: Ensured +by Assumption 1, the system noise wk, k +∈ +N+ is a +stationary stochastic process. Thus, its stochastic properties +are time-invariant. As a result, the dynamic model of the state +estimation error in (6) and the communication status in (5) are +also invariant to the last communication instant ι. This justifies +the following property. +Property 6. (Time-invariance of communication probabilities) +The following conditions hold ∀ n = 1, 2, · · · , T and ∀ ι ∈ N. +P +� +δ{0} +ι+n, δ{0} +ι+n−1, · · · , δ{0} +ι+1 +��� δ{1} +ι +� += Pn, +P +� +δ{0} +ι+n +��� Eι+n−1:ι +� += P n. +(17) +Property 6 proposes a very important claim for our work. It +indicates that the n-step predictive indexes Pn and coefficients +P n can be used to depict the communication probabilities +(17) for an arbitrary state-sampling time ι ∈ N, even though +they are originally defined specifically for ι = 0. Nevertheless, +we should keep in mind that this property only holds when +Assumption 1 is ensured. +B. The Recursive Model of The Transient ACR +Having introduced the predictive indexes and coefficients, +we are ready to present the recursive model for the transient +ACR. According to Property 1-3) and 1-2), we know +P +� +δ{0} +k +� += P +� k� +n=1 +Ek:k−n +� += +k +� +n=1 +P(Ek:k−n) , k ∈ N+, +which leads to +P +� +δ{0} +k +� += +k +� +n=1 +P +� +δ{0} +k +, · · · , δ{0} +k−n+1 +��� δ{1} +k−n +� +� +�� +� +=Pn +P +� +δ{1} +k−n +� +, +(18) +where we used Property 6. Note that Pn = 0 holds ∀ n > T +according to Property 1-1). Thus, (18) can be rewritten as +P +� +δ{0} +k +� += �min(k,T ) +n=1 +PnP +� +δ{1} +k−n +� +, k ∈ N+. +(19) +According to the definition of ACR in (7), (19) leads to the +following recursive model, +E (δk) = 1 − �min(k,T ) +n=1 +PnE (δk−n) , k ∈ N+, +(20) +with an initial condition E(δ0) = 1. Model (20) depicts the +recursive evolution of ACR at an arbitrary sampling instant as +time increases. Using Property 4, model (20) can be rewritten +as +E (δk) = 1 − �min(k,T ) +n=1 +�n +i=1 P iE (δk−n) , k ∈ N+. +(21) +The recursive model (21) indicates that the transient ACR at +any time k ∈ N+ can be recursively calculated using a finite +number of predictive coefficients P 1, P 2, · · · , P T . Therefore, +how to obtain the values of these coefficients is a critical +technical point for the exact computation of the transient ACR. +We explore the solution to this problem in Sec. IV. + +7 +C. The Existence of The Stationary ACR +The recursive model (20), for k ≥ T, is equivalent to +a T-order dt-LTI system. This offers us a solution to study +the existence of the stationary ACR using the Jury Criterion +recalled in Sec. II-D, which renders the following theorem. +Theorem 1. The stationary ACR E(δ∞) = lim +k→∞ E(δk) derived +from the recursive model (20) exists and its value reads +E(δ∞) = 1/(1 + �T +n=1 +�n +i=1 P i). +(22) +Proof. We can rewrite the recursive model (20) as the follow- +ing matrix-vector form +ξk = +� +0⊤ +I +PT +p +� +ξk−1 + β, ∀ k ∈ N+, +(23) +where I is a (T − 1)-dimensional identity matrix, 0 ∈ RT −1 +is a zero vector, and +ξk = [ E(δk+T −1) . . . E(δk+1) E(δk) ]⊤ , +p = [ PT −1 . . . P1 ] , β = [ 0⊤ 1 ]⊤, +with an initial condition +ξ0 = [ E(δT −1) . . . E(δ1) E(δ0) ]⊤ , +(24) +Therefore, (23) can be recognized as a dt-LTI system, where +ξk, k ∈ N, is the system state, β is the constant input, and Pn, +n = 1, 2, · · · , T, are the constant parameters. In this sense, the +existence of the stationary ACR E(δ∞) can be determined by +the stability of the dt-LTI system using the Jury’s criterion. +For any z ∈ R, the characteristic polynomial of the dt-LTI +(23) is +D(z) = PT + PT −1z + . . . + P1zT −1 + zT . +(25) +Given the polynomial (25), we investigate the state conver- +gence of the dt-LTI system (23) using the Jury’s criterion +recalled in Sec. II-D. It is straightforward to verify that Rules +1-3 in Sec. II-D hold for (25). We then use the coefficients in +(25) to construct the Jury array. The elements in the first row +of the Jury array then become +a0 = PT , a1 = PT −1, . . . , aT = 1. +Having the elements on row i and row 2 of Jury array, +the elements bk and bk+1 in the row 3 and row 4, with +k ∈ {0, . . . , T − 1}, can be constructed as +bk = +���� +a0 +aT −k +aT +ak +���� = a0ak − aT −kaT , +bk+1 = +���� +a0 +aT −k−1 +aT +ak+1 +���� = a0ak+1 − aT −k−1aT . +From Lemma 5, we obtain 0 < a0 < a1 < . . . < aT = +1, which implies a0ak < a0ak+1 < aT −k−1aT < aT −kaT . +This inequality further implies −1 < bk < bk+1 < 0 and +|b0| > |bT −1|. These results reveal the relationship among +the elements bk. Similarly, we can construct ck and ck+1, as +follows +ck = +���� +b0 +bT −1−k +bT −1 +bk +���� = b1bk − bT +1−kbT , +ck+1 = +���� +b0 +bT −2−k +bT −1 +bk+1 +���� = b0bk+1 − bT −2−kbT −1. +Similarly, we can readily conclude that 1 > ck > ck+1, which +also implies |c1| > |cT − 1|. Similar analysis can be carried +out to show that Rule 4 of Jury stability criteria always holds. +Therefore, the characteristic polynomial (25) meets Jury’s +stability criteria, which means that all the eigenvalues of the +state transition matrix in (23) are less than or equal to 1, i.e. +the system presented in (23) is asymptotically stable. This +indicates that the limit E(δ∞) = limk→∞ E(δk) exists. By +taking the limit of both sides of (20), we obtain +limk→∞ E (δk) = 1 − �T +n=1 Pn limk→∞ E (δk−n) , +(26) +which leads to (22) and proves this theorem. +Based on a general recursive model (20), Theorem 1 proves +the existence of the stationary ACR for any ET-SCS with an +event-triggered communication scheduler and a deterministic +constant threshold. This claim does not require any additional +conditions, meaning that the stationary ACR in general exists +for any ET-SCS defined in this paper. Equation (22) indicates +that the stationary ACR can also be calculated using a finite +number of predictive coefficients P n, n = 1, 2, · · · , T, similar +to the transient ACR explained in Sec. III-B. +IV. COMPUTATION OF THE PREDICTIVE COEFFICIENTS +As shown in Sec. III, the computation of both the transient +and the stationary ACR requires the predictive coefficients P n, +n = 1, 2, · · · , T. This section provides both analytical and +numerical approaches to compute these coefficients. Then, we +compare our methods with the previous results which apply +the restricted Gaussianity assumption. Finally, we present a +numerical example to demonstrate our theoretical claims. +A. The Analytical Form of The Predictive Coefficients +This section explores the analytical method to exactly +compute the predictive coefficients. According to the definition +of the predictive coefficients in Sec. III-A, we have +P i = P +� +δ{0} +i +��� Ei−1:0 +� += +� η +−η pˆei−1(z) dz, +(27) +for i = 2, · · · , T, with an initial condition P 1 = 1. Thus, +each coefficient P i is the integration of the PDF of the state +estimation error ˆei−1 on a finite support set [ −η, η ], where +the error recursively evolves following (6). Then, the critical +technical point is to obtain the analytical form of these PDFs. +For each i = 1, 2, · · · , T − 1, the PDF of ˆei reads +pˆei(z) = +� η +−η pˆeη +i−1(ξ)pw(z − Aξ)dξ, +(28) +where pˆeη +i−1(·) denotes the PDF of the truncated stochastic +variable ˆeη +i−1 of ˆei−1 with a symmetric truncation interval +[ −η, η ] and pw(·) is the PDF of the disturbance wk, k ∈ N+. +Note that ˆe0 = 0, and ˆe1, wk ∼ N(0, σ), which yields +pˆe0(z)=δ(z), pˆe1(z)=pw(z)= +1 +√ +2πσ exp +� +− z2 +2σ2 +� +, (29) + +8 +where δ(·) is the Dirac delta function. Thus, the distribution +pˆei(z) in (28) can be obtained as +pˆei(z) = +� η +−η +pˆeη +i−1(ξ) +√ +2πσ exp +� +−(z − Aξ)2 +2σ2 +� +dξ. +(30) +According to the PDF of a truncated random variable in +Sec. II-E, we have +pˆeη +i−1(z) = Gˆei−1(z)pˆei−1(z), +(31) +where pˆei−1(·) is the PDF of the non-truncated variable ˆei−1 +and Gˆei−1(·) is a piece-wise constant function defined as +Gˆei−1(z) = +� 1/ +� η +−η pˆei−1(ξ)dξ, +−η ≤ z ≤ η, +0, +otherwise. +(32) +Substituting (32) and (31) to the PDF (30), we obtain +pˆei(z) = Gˆei−1(z) +√ +2πσ +� η +−η +pˆei−1(ξ) exp +� +−(z − Aξ)2 +2σ2 +� +dξ. +(33) +Thus, equations (29) and (33) form a complete recursive model +to solve the analytical forms of the PDFs of the closed-loop +errors, pˆei(·), for all i = 1, 2, · · · , T. Then, (27) can be +used to accurately calculate the predictive coefficients P i, for +i = 2, 3, · · · , T − 1. Note that, for any i = 2, 3, · · · , T, the +PDF pˆei(·) is not necessarily Gaussian due to the recursive +truncation operations. Also, the analytical form of pˆei(·) +becomes increasingly complicated and challenging to solve +as i gets larger. To resolve this issue, in the next section, we +propose a numerical algorithm to approximate the predictive +coefficients using the recursive stochastic sampling technique. +B. Approximating The Predictive Coefficients Numerically +Considering the difficulty of analytically computing the +coefficients P i for large i, we propose a numerical algorithm +to approximate them using the recursive stochastic sampling +method, as shown in Algorithm 1. The computation of P 1 +and P 2 in Line 1 is straightforward since the analytical forms +of pˆe0(·) and pˆe1(·) are trivial and simple. In Line 2, N +particles are initialized from the distribution pˆe1(·), i.e., a +Gaussian distribution N(0, σ). From Line 3, the particles are +used to approximate the nontrivial PDFs pˆei(·) for i ≥ 2. +The particles are a group of real scalars independently drawn +from a certain distribution. Consider that N ∈ N+ particles +Z = {z(1), z(2), · · · , z(N)}, z(i) ∈ R, i = 1, 2, · · · , N, are +independently drawn from a distribution depicted by a PDF +p(·). Then, the unbiased estimation of p(·) can be obtained +using a Gaussian kernel method as +ˆp(z, Z) = +1 +√ +2πˆσN +N +� +j=1 +exp +� +− +� +z − z(j)�2 +2ˆσ2 +� +, +(34) +where ˆσ ∈ R+ is a variance parameter. Here, we use the +symbol ˆp(·) to represent the PDFs approximated using parti- +cles. Based on the approximated PDFs ˆpˆei(·), the predictive +coefficients P i+1 are calculated recursively, following the flow +ˆpˆei−1(·) → ˆpˆeη +i−1(·) → ˆpˆei(·) → P i+1. +The approximation in each iteration is described as follows. +In line 4, the particles exceeding the threshold η are removed, +which simulates the truncation operation to the PDF pˆei−1(·). +Then, in line 5, the PDF pˆeη +i−1(·) of the truncated stochastic +variable ˆeη +i−1 is approximated with the remaining particles. In +line 6, N particles are resampled from the approximated PDF +ˆpˆeη +i−1(·). The particles then perform the stochastic propagation +according to the error dynamics (6), as shown in lines 7-10. +In line 11, the particle approximation method (34) is used +again to approximate the PDF pˆei(·). Finally, the predictive +coefficient P i+1 are calculated in line 14. +Algorithm 1: Approximation of the predictive coeffi- +cients using particles +Inputs : noise variance σ and particle number N +Outputs: P i, ∀ i = 1, 2, · · · , T, T > 2 +1 Calculate P 1, P 2 using (27) with pˆe0(·), pˆe1(·) in (29); +2 Sample particles z(j) +1 +∼ N(0, σ), j = 1, 2, · · · , N; +3 for i ← 2 to T − 1 do +4 +Remove all particles +���z(j) +i−1 +��� ≥ η; +5 +Approximate ˆpˆeη +i−1(·) with z(j) +i−1 using (34); +6 +Re-sample particles z(j) +i−1 ∼ ˆpˆeη +i−1(·); +j = 1, 2, · · · , N; +7 +for j ← 1 to N do +8 +Draw ϵ(j) +i−1 ∼ N(0, σ); +9 +z(j) +i += Az(j) +i−1 + ϵ(j) +i−1; +10 +end +11 +Approximate pˆei(·) with z(j) +i +using (34); +12 +Calculate P i+1 with (27) using PDF pˆei(·); +13 end +Note that Algorithm 1 may lead to approximation errors in +the predictive coefficients. The main source of the errors is the +deviation between the PDFs pˆei(·) and their estimations ˆpˆei(·). +In fact, the unbiasedness of the approximation only holds in +the statistical sense. To reduce the approximation errors, N +should be selected sufficiently large and ˆσ should be small. +Remark 3. In this paper, our theoretical claims and numerical +methods target at a specific class of ET-SCS, where the +network communication is triggered by an asynchronous event +associated with state estimation errors. In fact, the state +estimation error ek, k ∈ N, can be recognized as a variable +that depends on the internal states of the joint dynamic model +of the system plant and the state estimator, namely the plant +state xk and the estimator state ˆxk. Thus, our results can also +be extended to a generic ET-SCS of which the triggering event +may be assigned to an arbitrary state-dependent variable. In +this case, the recursive model of the ACR is still effective. +What changes is that the predictive coefficients are calculated +using the PDF of this state-dependent variable. The challenge +of such an extension depends on the complexity of this PDF. +V. COMPARISON WITH THE CONVENTIONAL METHOD +Based on Sec. III and Sec. IV, we are able to calculate the +stationary and the transient ACR for an ET-SCS using a finite +number of predictive coefficients. The analytical and numerical + +9 +methods to compute these coefficients are also provided. In +this section, we make a comparison between our approaches +and the conventional method [17] that intentionally ignores the +side information for simplification. Both theoretical analysis +and a numerical study are conducted to validate the accuracy +of our approaches and qualitatively verify the accuracy gap +between the conventional method and the ground truth. +A. Deviation Analysis of The Conventional Method +As mentioned above, the computation of ACR without +considering the closed-loop effect leads to an oversimplified +distribution model for the state estimation error and eventually +returns approximated results. Assume that the open-loop state +estimation error is subject to the dynamic model (3). Then, +the error has a fully Gaussian PDF, and, similar to (21), the +open-loop ACR can be recursively computed as +˘E (δk) = 1 − �min(k,T ) +n=1 +�n +j=1 ˘P j˘E (δk−n) , k ∈ N+, +(35) +with ˘E(δ0) = 1, where ˘P i, i = 1, 2, · · · , T, are the coeffi- +cients obtained by +˘P i = +� η +−η pei−1(z)dz, +i = 1, 2, · · · , T, +(36) +where pei(·) is the PDF of the open-loop state estimation +error ei subject to the dynamic model (3) with ι = 0. Hence, +according to (3), for all i > 1, we have +pei(z) = +� ∞ +−∞ pei−1(ξ) pw(z − Aξ)dξ += +� ∞ +−∞ +pei−1(ξ) +√ +2πσ +exp +� +−(z − Aξ)2 +2σ2 +� +dξ, +(37) +with the initial conditions +pe0(z) = δ(z), +pe1(z) = +1 +√ +2πσ exp +� +− z2 +2σ2 +� +. +(38) +Comparing (33) and (37), one notices that ˆe1 and e1 have the +same distribution N(0, σ), while for each i > 1, pˆei(·) has an +additional multiplier Gˆei−1(·), compared to pei(·). Also, the +integration intervals are also different. +Now, we compare the mean values and the variances of +the two stochastic variables ˆei and ei for i = 1, 2, · · · , T. +From (4), we know that the state estimation error ei is a linear +combination of the Gaussian-distributed stochastic variables +w0, · · · , wi−1. Hence, ei is also Gaussian-distributed and has +the following property. +Property 7. Given that w0, · · · , wT −1 are i.i.d. stochastic +variables (Assumption 1), the following statements hold for +all ei, i = 1, 2, · · · , T. +1) For any z ∈ R, pei(z) = pei(−z). +2) E(ei) = �i−1 +n=ι Ai−n−1E(wn) = 0. +3) Var(ei) = �i−1 +n=ι Ai−n−1Var(wn) = �i−1 +n=ι Ai−n−1σ2. +Property 7 is easy to verify using the linear properties +of Gaussian stochastic variables. Nevertheless, the stochastic +properties of the closed-loop state estimation error ˆei are not +that straightforward due to the recursive truncation operations. +Before proceeding with the study on the stochastic properties +of ˆei, it is necessary to propose the following proposition for +truncated stochastic variables. +Proposition 1. Let ζ ∈ R be an arbitrary stochastic variable +of which the PDF pζ(z) has infinite support. E(ζ) and Var(ζ) +are respectively its mean value and variance. Also, let ζη ∈ R +be a truncated stochastic variable by trimming the support of +ζ to be within the symmetrically bilateral interval [ −η, η ], +η > 0. If E(ζ) = 0, and pζ(z) = pζ(−z) holds for all z ∈ R, +then the following conditions are valid. +1) E(ζη) = 0, and pζη(z) = pζη(−z), ∀ z ∈ R. +2) Var(ζη) < Var(ζ). +Proof. If E(ζ) = 0 and pζ(z) = pζ(−z) hold, according to the +definition of the PDF of truncated stochastic variables in (9), +we have +pζη(z) = +pζ(z) +Fζ(η) − Fζ(−η) = +pζ(−z) +Fζ(η) − Fζ(−η) = pζη(−z). +Utilizing this property, we further have +E(ζη) = +� η +−η +zpζη(z)dz = 0. +Therefore, condition 1) is proved. Furthermore, the variance +of ζη reads +Var(ζη) = +� η +−η z2pζη(z)dz − E2(ζη) += +� η +−η z2pζ(z)dz +�� η +−η pζ(z)dz . +Note that Var(ζη) is indeed a function of the truncation +interval η. Thus, we represent it as Var(η). It can be verified +that Var(η) is continuous and continuously differential for η. +Moreover, we know +lim +η→∞ Var(η) = +� ∞ +−∞ +z2pζ(z)dz = Var(ζ), lim +η→0 Var(η) = 0. +(39) +By taking the derivative of Var(η) to η, we obtain +Var′(η) = +��� η +−η z2pζ(z)dz +�′ � η +−η pζ(z)dz +− +�� η +−η pζ(z)dz +�′ � η +−η z2pζ(z)dz +���� η +−η pζ(z)dz +�2 +. +Note that +�� η +−η z2pζ(z)dz +�′ += η2[pζ(η) + pζ(−η)] = 2η2pζ(η), +�� η +−η pζ(z)dz +�′ += pζ(η) + pζ(−η) = 2pζ(η). +Thus, +Var′(η) = 2pζ(η) +� η +−η +� +η2 − ζ2� +pζ(z)dz +� � η +−η +pζ(z)dz. +Since pζ(·) is a non-negative, we conclude V ′(η) > 0, for all +η > 0. This implies that V (η) is a monotonically increasing +function in the interval η ∈ (0, ∞). Therefore, we can write +Var(ζη) = Var(η) < Var(∞) = Var(ζ), for any 0 < η < ∞. +Thus, condition 2) is proved. +Proposition 1 indicates that a truncated stochastic variable +has the same expected value but a smaller variance than its + +10 +original counterpart if the latter has an even PDF around zero +and the truncation interval is symmetric. We now present the +following theorem that characterizes the relation between the +mean values and variances of the closed-loop and open-loop +state estimation errors. +Theorem 2. Given state estimation errors ˆei and ei de- +picted by the dynamic models (6) and (3), respectively, i = +1, 2, · · · , T, the following conditions hold. +1) pˆei(z) = pˆei(−z), for all z ∈ R. +2) E(ˆei) = E(ei) = 0. +3) Var(ˆe1) = Var(e1), Var(ˆei) < Var(ei), for all i ≥ 2. +Proof. We first consider the case k = 1. Since ˆe1, e1 ∼ +N(0, σ), we have pˆe1(z) += +pˆe1(−z), for all z +∈ +R, +E(ˆe1) = E(e1) = 0, and Var(ˆe1) = Var(e1) = σ2. Then, +for a truncated stochastic variable ˆeη +i with threshold η > 0, +according to Proposition 1, given any i = 1, 2, · · · , T −1, such +that pˆei(z) = pˆei(−z), we have pˆeη +i (z) = pˆeη +i (−z), E(ˆeη +i ) = 0, +and Var(ˆeη +i ) < Var(ˆei). +According to (6), we know ˆei+1 = Aˆeη +i + wi, from which +we conclude +pˆei+1 (z) = +� η +−η pˆeη +i (ξ) pw(z − Aξ)dξ. +Considering that pˆeη +1(·) is an even PDF, and pw(z) = pw(−z), +for all z ∈ R, we obtain +pˆei+1 (z) = +� ξ=η +ξ=−η pˆeη +i (−ξ) pw(−z + Aξ)dξ. +Set ˆz = −ξ, then we will have +pˆei+1 (z) = +� −ˆz=η +−ˆz=−η pˆeη +i (ˆz) pw(−z − Aˆz)d(−ˆz) += − +� −ˆz=η +−ˆz=−η pˆeη +i (ˆz) pw(−z − Aˆz)dˆz += +� ˆz=η +ˆz=−η pˆeη +i (ˆz) pw(−z − Aˆz)dˆz = pˆei+1(−z) . +This property leads to +E(ˆei+1) = +� η +−η +zpˆei+1(z) dz = 0. +Also, from Assumption 1, we know that eη +i +and ei are +independent from wi−1. Thus we conclude +Var(ˆei+1) = A2Var(ˆeη +i ) + σ2, +Var(ei+1) = A2Var(ei) + σ2. +According to Proposition 1, we have Var(ˆeη +i ) < Var(ˆei). +Therefore, for any i such that Var(ˆei) ≤ Var(ei), we have +Var(ˆei+1) = A2Var(ˆeη +i ) + σ2 < A2Var(ˆei) + σ2 +≤ A2Var(ei) + σ2 = Var(ei+1). +Finally, we proved pˆei(z) = pˆei(−z), E(ˆei) = 0, and +Var(ˆei) ≤ Var(ei) hold for all i = 1, 2, · · · , T. Note that +Var(ˆei) = Var(ei) only when i = 1. +Theorem 2 indicates the qualitative difference between the +PDFs, the mean values, and the variances of the closed-loop +error ˆei and the open-loop error ei for i = 1, 2, · · · , T. Both of +them have even PDFs and zero mean values. Nevertheless, the +closed-loop error ˆei has a smaller variance than the open-loop +one ei, for i > 1. This indicates that the recursive truncation +operations in (6) result in a shrink in the PDF pˆei(z) along the +z-axis compared to the infinite support Gaussian PDF pei(z). +Therefore, for any i > 1, Theorem 2 results in +� η +−η pˆei(z)dz > +� η +−η pei(z)dz. +(40) +This can be explained in an intuitive manner that the shape of +pˆei(·) is more narrow than pei(·). Based on this, we can infer +that the conventional method using pei(·) instead of pˆei(·) +leads to smaller results for the coefficients, i.e., ˘P i+1 < P i+1 +for i = 3, · · · , T, according to (27), and then larger values of +the transient ACR, i.e., ˘E(δk) < E(δk) for k = 3, · · · , T. +Extending this claim to k → ∞, we also have a similar +conclusion for the stationary ACR, i.e., ˘E(δ∞) < E(δ∞). +The analysis in this section not only proves the accuracy +gap of the conventional method in theory but also qualitatively +points out that it always leads to larger computation results. +B. Accuracy Comparison: A Numerical Example +Here we present a numerical example to verify the ac- +curacy of our proposed analytical and numerical methods, +in Sec. IV-A and Sec. IV-B, respectively. We also validate +the accuracy gap of the conventional method that ignores +the close-loop effect. Consider an ET-SCS as in (1) with +parameters A = 1.25, B = 1, an initial state x0 = −2, +a stochastic process wk ∼ N(0, 1), k ∈ N+, and a state- +feedback controller uk = −ˆxk, where ˆxk is estimated us- +ing (2). The threshold and the maximum triggering interval +of the event-triggered scheduler (5) are η = 1, T = 5. As +addressed in Sec. V-A, the major difference between our work +and the existing works is that the latter ignores the closed-loop +effects of an ET-SCS and use the open-loop estimation error +ek to compute ACR, instead of the closed-loop error ˆek. To +provide a fair and clear comparison study, we use five manners +to compute the transient ACR. +1) The Proposed Analytical Method (PAM): The recursive +expressions (29) and (33) are used to obtain the PDFs pˆei(·) of +the closed-loop state estimation errors ˆei for i = 0, 1, · · · , 4. +Then, the coefficients P i+1 are calculated using (27). Finally, +(21) is recursively used to compute the transient ACR E(δk) +for k = 1, 2, · · · , 5. +2) The Proposed Numerical Method (PNM): Algorithm 1 +is used to approximate the PDFs pˆei(·) of the closed-loop state +estimation errors ˆei for i = 0, 1, · · · , 4, with parameters ˆσ = +0.1 and N = 104. Then, the coefficients P i+1 are calculated +using (27). Finally, (21) is recursively used to compute the +transient ACR E(δk) for k = 1, 2, · · · , 5. +3) The Conventional Analytical Method (CAM): +[17] The +recursive expressions (37) and (38) are used to obtain the +PDFs pei(·) of the open-loop state estimation errors ei for +i = 0, 1, · · · , 4. Then, the open-loop predictive coefficients +˘P i+1 are calculated using (36). Finally, (35) is recursively +used to compute the transient ACR ˘E(δk) for k = 1, 2, · · · , 5. +4) The Conventional Numerical Method (CNM): This ap- +proach is merely used to provide a numerical counterpart of the +CAM approach for the completeness of our work. We first use +Algorithm 1, with the same parameters ˆσ = 0.1 and N = 104 + +11 +as PNM but with the lines 4-6 removed, to calculate the open- +loop predictive coefficients ˘P i+1. Then, (35) is recursively +used to compute the transient ACR ˘E(δk) for k = 1, 2, · · · , 5. +5) Ground Truth (GT): We conduct a Monte-Carlo exper- +iment of the ET-SCS with the same initial state repeated for +104 trials to approximate the true value of ACR, +EGT (δk) = #(δk = 1)/104, +where #(δk = 1) is the total number of trials of which δk = 1. +The following Example 1 provides an instruction to compute +the transient ACR using PAM. Note that we only give the +results for E(δk), k = 1, 2, 3. The results for larger k values +are omitted due to the complexity of analytical computation. +Example 1. (Computation of ACR Using PAM) The compu- +tation procedure of E(δk) for k = 1, 2, 3 is as follows. +• For k = 1, according to (29), we have +P 1 = +� η +−η δ(z)dz = 1, and E(δ1) = 1 − P 1 = 0. +• For k = 2, using (27), we can calculate +P 2 = +� η +−η pˆe1(z) dz = 0.6827, +where the analytical form of pˆe1(·) is provided in (29). Then, +according to (21), we have +E(δ2) = 1 − P 1E(δ1) − P 2P 1E(δ0) = 0.3173. +• For k = 3, we have ˆe2 = Aˆeη +1 + w1 according to (6). +Thus, the analytical form of pˆe2(·) reads +pˆe2(z) = +� η +−η pˆeη +1(ξ) pw(z − Aξ)dξ. +(41) +Note that pˆeη +1(·) is a truncated Gaussian PDF and pw(·) +is a Gaussian PDF, which makes the calculation of pˆe2(·) +nontrivial. According to the formulations in Sec. II-E, we have +pˆeη +1(z) = +1 +√ +2πσ erf +� +η +√ +2σ2 +� exp +� +− z2 +2σ2 +� +, +(42) +where erf(x) = +2 +√π +� x +0 exp(−x2)dx is the Gaussian error +function. Substituting (42) to the integral term in (41), we get +pˆe2(z) = +� η +−η +exp +� +− (z−Aξ)2+ξ2 +2σ2 +� +2πσ2erf +� +η +√ +2σ +� +dξ += +exp +� +− +z2 +2σ2(A2+1) +� +2πσ2erf +� +η +√ +2σ +� +� η +−η +exp +� +� +�− +� +ξ − +Az +A2+1 +�2 +2¯σ2 +� +� +�dξ += +exp +� +− +z2 +2σ2(A2+1) +� +2σ +� +2π(A2 + 1)erf +� +η +√ +2σ +� +(43) +× +� +erf +�ηA2 + η − Az +√ +2σ +√ +A2 + 1 +� ++ erf +�ηA2 + η + Az +√ +2σ +√ +A2 + 1 +�� +. +Therefore, we calculate +P 3 = 1 − +� η +−η +pˆe2(z)dz = 0.5872. +According to (21), we have +E(δ3) = 1−P 1E(δ2)−P 2P 1E(δ1)−P 3P 2P 1E(δ0) = 0.2818. +It has been noticed that the analytical form of pˆe2(·) in +(43) becomes very complicated. The computation of E(δk) +for k > 3 is even more difficult due to the complicated form +of pˆek−1(·). Therefore, we only provide the results for k ≤ 3. +The computation results of the numerical study are reported +in Table I. Slight deviations are seen between PAM and PNM +or between CAM and CNM. Note that these deviations reflect +the inevitable approximation errors between the analytical +methods and their numerical counterparts due to the approx- +imation bias of the Gaussian kernel method. Incorporating +these errors, we can see that the results of PAM and PNM are +very close to the ground truth (with absolute errors smaller +than 0.005), which validates the effectiveness and accuracy of +the proposed methods. On the contrary, the results of CAM +and CNM present large calculation errors. Moreover, they are +all large than the GT results, in general, which verifies our +theoretical arguments in Sec. V-A that the conventional method +overapproximates the ACR values. +Table I: The GT and the computed ACR values for ET-SCS +k +GT +PAM +PNM +CAM +CNM +1 +0 +0 +0 +0 +0 +2 +0.3175 +0.3173 +0.3129 +0.3173 +0.3161 +3 +0.2826 +0.2818 +0.2877 +0.3633 +0.3668 +4 +0.2650 +— +0.2609 +0.3098 +0.3082 +5 +0.2801 +— +0.2797 +0.3117 +0.3126 +More details can be found by taking a deeper look into the +stochastic properties of the state estimation errors. Table II +shows the mean values E(·) and the variances Var(·) of the +closed-loop error ˆek and open-loop error ek for k = 1, · · · , 5. +It can be seen that their mean values are very close to zero, +despite small errors due to the numerical approximation. Also, +we witness Var(e1) = Var(ˆe1) = 0 and Var(ek) > Var(ˆek), +for all k = 2, · · · , 5. This coincides with our theoretical state- +ments in Theorem 2 that the open-loop errors have the same +mean values as the closed-loop errors but larger variances. +Table II: The mean values and the variances of the closed-loop +and the open-loop state estimation errors +k +E(ˆek) +E(ek) +Var(ˆek) +Var(ek) +1 +0 +0 +0 +0 +2 +0.0000 +−0.0000 +1.4549 +2.5625 +3 +−0.0037 +−0.0000 +1.4497 +5.0031 +4 +0.0220 +−0.0000 +1.5108 +8.7302 +5 +0.0149 +0.0000 +1.5288 +13.601 +The PDFs calculated using the closed-loop and the open- +loop errors, pˆek(·) and pek(·), for k = 2, 3, 4, 5, are illustrated +in Fig. 3, using the red line and the blue line, respectively. +The GT PDF of the state estimation errors, drawn as the gray +area, obtained by conducting a Monte Carlo experiment, is +also presented for comparison. We observe that our proposed +method accurately follows the GT. On the contrary, the con- +ventional method obviously deviates from the GT results. The + +12 +deviation becomes larger as k increases. This also verifies our +theoretical claims in Sec. V-A. +(a) k = 2 +(b) k = 3 +(c) k = 4 +(d) k = 5 +Figure 3: The approximated PDFs of the state estimation errors +for k=2, 3, 4, 5. The Red line denotes ˆpˆek(·) calculated using +our proposed methods (k = 2 using PAM and k = 3, 4, 5 +using PNM) and the blue line is ˆpek(·) obtained from the +conventional approach (CAM). The gray area represents the +GT PDF using Monte Carlo sampling. +VI. EXPERIMENTAL STUDY +In this section, we conduct an experimental study of a +leader-follower autonomous driving scenario to validate our +theoretical results interpreted so far. The leader-follower sce- +nario is a simplified case of the widely-used platooning model +in autonomous driving [35]. As illustrated in Fig. 4, the +system contains a leader vehicle that maneuvers according to +a certain trajectory. A follower vehicle is dedicated to keeping +a constant distance from the leader vehicle. The positions and +velocities of the vehicles are measured using a series of remote +sensors. Both the vehicles and the sensors are connected using +a common communication network that allows data exchange +and state sampling. The switches installed on remote sensors, +subject to the triggering scheme (5), determine whether to +transmit the most recent vehicle states to the network. +The position of the leader vehicle follows a predefined +trajectory pL(t) = − cos(t) + 1.2t. The follower is required +to maintain a distance d = 3 m with the leader. The kinematic +model of the follower vehicle is +˙p(t) = v(t), +˙v(t) = u(t), +(44) +where p(t), v(t), u(t) ∈ R are respectively the position, the ve- +locity, and the acceleration of the follower. In this experiment, +the parameters are selected as γ = 1, Q = 1, and K = 1. The +objective of the problem is to design a control law u(t), such +d +Remote Sensor +Communication Network +Plant +Figure 4: Illustration of a leader-follower system. +that p(t) → pL(t) + d and v(t) → ˙pL(t) as time t increases. +We define the following feedback control law, +u(t) = −γQ−1v(t) − Q−1Kp(t) + γQ−1 ˙pL(t) ++ Q−1KpL(t) + ¨pL(t) + Q−1Kd, +(45) +where K, Q, γ ∈ R+ are positive parameters. It can be verified +using a Lyapunov method that p(t) − pL(t) = d and v(t) − +˙pL(t) = 0 render a globally asymptotic equilibrium of the +closed-loop system, which indicates the achievement of the +desired control performance. The proof is omitted in this paper. +In our experiment, we consider the discrete-time version of +the follower vehicle (44), +pk+1 = pk + ∆tvk, +vk+1 = vk + ∆tuk + wk, +(46) +where ∆t is the sampling period, and wk is an i.i.d. noise +process. Accordingly, we discretize the reference trajectory +pL(t) to pL +k using discrete sampling t = k · ∆t. In corre- +spondence with the ET-SCS model in Fig. 2, the leader’s +trajectory pL +k is the reference signal of the overall system. +Each follower is a plant with the state xk = vk. The limited +communication bandwidth motivates the application of the +event-triggered scheduler in (5), for which we set η = 1, +and T = 20 in this experiment. The state estimator (2) +is used to obtain ˆxk = ˆvk, with A = 1, and B = ∆t. +The discrete-time controller based on the estimated state is +uk = u(k · ∆t), for which ˆpk is obtained using the recursive +model ˆpk+1 = ˆpk + ∆tˆvk. The simulation runs for 104 trials +with the same initial conditions p0 = 0 and v0 = 0. Each +trial lasts for t = 40 s with a sampling time ∆t = 0.1s. +The overall control performance is shown in Fig. 5. It is +observed that the average following distance E(p(t)) − pL(t) +slightly fluctuates around d = 3 m, which indicates satisfactory +distance keeping. Also, the average velocity E(v(t)) is very +close to the reference velocity ˙pL(t). This shows that the +configuration of the state estimator (2) and the event-triggered +scheduler (5) successfully achieves the control objectives. +The computed values of the ACR E(δk), using Algorithm 1 +(PNM), with various triggering thresholds η, are shown in +Fig. 6 (in red). To verify the validity of our proposed +method, we also show the GT-ACR obtained from Monte- +Carlo simulation (in black), and the ACR computed according + +0.4 +GT +0.3 +pe, () +-pe, () +L +00.2 +P +0.1 +0 +-5 +0 +50.4 +GT +0.3 +pe, () +-pe) +L +00.2 +P +0.1 +-5 +0 +50.4 +GT +0.3 +pa.() +-pe() +L +00.2 +P +0.1 +0 +-5 +0 +50.4 +GT +0.3 +pe.() +pe,) +00.2 +P +0.1 +0 +-5 +0 +513 +0 +10 +20 +30 +40 +time +1 +2 +3 +4 +distance +(a) tracking distance +0 +10 +20 +30 +40 +time +0 +1 +2 +3 +4 +velocity +(b) velocity tracking +Figure 5: Average performance of the platoon controller (45). +Plot (a) depicts the tracking distance E(p(t)) − pL(t). Plot (b) +shows the leading velocity ˙pL(t) (in red) and the mean of the +actual velocity E(v(t)) (in blue). +to the conventional method (CAM), i.e., ˘E(δk) (in blue). The +information delivered by Fig. 6 can be summarized as follows. +1) The general existence of the stationary ACR: It is +noticed that all ACR values, E(δk), ˘E(δk), and the GT-ACR, +ultimately converge to their respective stationary points for all +triggering threshold values η = 1, 2, 3, 4. This validates our +result on the existence of the stationary ACR in Sec. III-C. +2) The accuracy of the proposed method: It is observed +that the computed ACR E(δk) closely follows the GT-ACR +at all time, indicating the accuracy of our proposed method. +On the contrary, ˘E(δk) shows deviations from the GT-ACR +suggesting inaccuracy of computing the ACR by this method. +Also, ˘E(δk) is in general larger than E(δk) in the steady state, +which validates our theoretical statement in Sec. V-A that this +method overestimates the stationary ACR. +3) The influence of the triggering threshold: The stationary +ACR values tend to be smaller as the triggering threshold η +increases. The intuition behind this observation is that, higher +threshold means higher estimation errors are tolerable, hence +less events will be triggered to reset the estimation error, +which consequently leads to lower ACR. Similar observation is +depicted in Fig. 7, where the change of stationary ACRs E(δ∞) +and ˘E(δk), and their ratios are plotted versus the changes +of the threshold η. It van be seen that E(δ∞) < ˘E(δk), for +all values of η. However, the scale of the deviation between +the two approaches is not monotone with respect to η, i.e., +larger triggering thresholds do not necessarily lead to larger +deviations . The largest deviation occurs around η = 3, with +more than 25%, which is noticeable. +VII. CONCLUSION +Motivated by the conservativeness of the conventional exist- +ing methods, in this article we provide comprehensive analyt- +ical formulations to accurately compute the average commu- +nication rate for networked control systems under the event- +triggered sampling model. By incorporating the distribution +truncation operations that correspond to the side information +generated by the triggering decisions, we prove the existence +of stationary ACR using a novel recursive model. Afterwards, +we propose analytical and numerical approaches to accurately +calculate ACR at any arbitrary time and demonstrate the +noticeable ACR over-estimation when the triggering-induced +0 +10 +20 +30 +40 +time +0 +0.1 +0.2 +0.3 +0.4 +ACR +(a) η = 1 +0 +10 +20 +30 +40 +time +0 +0.1 +0.2 +0.3 +0.4 +ACR +(b) η = 2 +0 +10 +20 +30 +40 +time +0 +0.1 +0.2 +0.3 +0.4 +ACR +(c) η = 3 +0 +10 +20 +30 +40 +time +0 +0.1 +0.2 +0.3 +0.4 +ACR +(d) η = 4 +Figure 6: ACRs computed using our proposed method E(δk) +(in red), the existing method ˘E(δk) (in blue), and GT-ACR (in +black), for various triggering thresholds η = 1, 2, 3, 4. +1 +2 +3 +4 +5 +0 +0.1 +0.2 +0.3 +stationary ACR +(a) Stationary ACRs +1 +2 +3 +4 +5 +0.6 +0.7 +0.8 +0.9 +ACR ratio +(b) ACR ratio +Figure 7: Comparison between the stationary ACRs vs. trigger- +ing threshold. Plot (a), our proposed method E(δ∞) (in red), +and the existing method ˘E(δ∞) (in blue). Plot (b) shows the +ratio of the two stationary ACRs vs. triggering threshold. +truncations are ignored in computing the ACR. Our proposed +method and the theoretical claims are validated with a nu- +merical example and an experimental study on a platooning +scenario, showing that our ACR computation model precisely +follows the ground truth case. +ACKNOWLEDGEMENT +The authors would like to thank Prof. Biqiang Mu and Prof. +Hongsheng Qi from the Chinese Academy of Sciences for their +valuable discussions on truncation analysis. The authors would +also like to thank Prof. Yirui Cong for helpful discussions on +the communication rate analysis. +REFERENCES +[1] S. Cai and V. K. Lau, “Zero MAC latency sensor networking for cyber- +physical systems,” IEEE Transactions on Signal Processing, vol. 66, +no. 14, pp. 3814–3823, 2018. +[2] F. Mager, D. Baumann, R. Jacob, L. Thiele, S. Trimpe, and M. 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Liu, “Event-based communication and control +for task-space consensus of networked Euler–Lagrange systems,” IEEE +Transactions on Control of Network Systems, vol. 8, no. 2, pp. 555–565, +2021. +[34] E. I. Jury, Theory and Application of the z-Transform Method. Wiley, +1964. +[35] A. Liu, L. Lian, V. Lau, G. Liu, and M.-J. Zhao, “Cloud-assisted +cooperative localization for vehicle platoons: A turbo approach,” IEEE +Transactions on Signal Processing, vol. 68, pp. 605–620, 2020. + diff --git a/ndE5T4oBgHgl3EQfIA5T/content/tmp_files/load_file.txt b/ndE5T4oBgHgl3EQfIA5T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..63f1627c6a9d9828e3597f3cfba4c850af621379 --- /dev/null +++ b/ndE5T4oBgHgl3EQfIA5T/content/tmp_files/load_file.txt @@ -0,0 +1,1023 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf,len=1022 +page_content='1 Average Communication Rate for Event-Triggered Stochastic Control Systems Zengjie Zhang†, Qingchen Liu†*, Mohammad H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Mamduhi, and Sandra Hirche Abstract—Quantification of the triggering rates of an event- triggered stochastic system with deterministic thresholds is a challenging problem due to the non-stationary nature of the system’s stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A typical example is the com- putation of the average communication rate (ACR) of the networked event-triggered stochastic control systems (ET-SCS) of which the communication of the sensor network is scheduled by whether a system variable of interest exceeds predefined constant thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For such a system, a closed-loop effect emerges due to the interdependence between the system variable and the trigger of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This effect, commonly referred to as side information by related work, distorts the stochastic distribution of the system variables and makes the ACR computation non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Previous work in this area used to over-simplify the computation by ignoring the side information and misusing a Gaussian distribution, which leads to approximated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This paper proposes both analytical and numerical approaches to predict the exact ACR for an ET-SCS using a recursive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Furthermore, we use theoretical analysis and a numerical study to qualitatively evaluate the deviation gap of the conventional approach that ignores the side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The accuracy of our proposed method, alongside its comparison with the simplified results of the conventional approach, is validated by experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Our work is promising to benefit the efficient resource planning of networked control systems with limited communication resources by providing accurate ACR computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' INTRODUCTION I N recent years, emerging networked control systems, such as intelligent industrial manufacturing [1], smart power grids [2], and autonomous vehicles [3], are characterized by a distributed design manner where the plants and the sensors are located remotely and are connected with a common network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The development of the closed-loop controllers for these sys- tems requires sufficient sampling of the system states ensured by active communication of the network, such that the con- trollers always have access to the latest system states provided by the remote sensors through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Nevertheless, the communication resources of the networked systems are often limited by the power restrictions of the system, especially for the mobile and portable devices of which the power mainly rely on batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This issue suggests reducing the frequency of state sampling by properly scheduling the communication of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Zhang is with the Department of Electrical Engineering, Eindhoven University of Technology, Netherlands (z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='zhang3@tue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='nl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Liu is with the Department of Automation, University of Science and Technology of China, Hefei, China (qingchen liu@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Mamduhi is with the Automatic Control Laboratory, ETH Z¨urich, Switzerland (mmamduhi@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='ch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Hirche is with the Chair of Information-Oriented Control, Technical University of Munich, Munich, Germany (hirche@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' † Equivalent contribution as the common first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This was conventionally solved with time-based schemes which later gave way to event-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It is argued in [4] that the event-based scheduling schemes can achieve the same performance as the periodic time-based ones but with considerably less consumption of commu- nication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This finding motivates the development of different event-based scheduling schemes, including the stochastic, periodic, and deterministic event-based ones, for various networked systems [5]–[7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The event-based schemes have shown superior efficiency, high flexibility, and quick responsiveness incorporating the restriction of communication resource limitations [8]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A typical event-based scheduling scheme is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1, where the sampling of the sys- tem state or the activation of the communication is governed by a scheduler that triggers a time-asynchronous event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This event does not explicitly depend on time but is associated with a system variable of interest and a predefined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, the scheduling is performed in a time-asynchronous manner which activates the communication only when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The necessity of communication is determined by a certain triggering event, which intermittently closes the loop between the controlled system and the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This ensures efficient consumption of the communication resources [12]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Sensor System Scheduler Remote sensing St δt Vt E(Vt, η) Figure 1: A typical event-based communication scheduler for a networked system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The dashed line indicates remote sensing, such as infra-sensing or visual perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The switch symbol denotes the communication status δt (active or inactive) of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' St is the sampled system state, Vt is the system variable of interest, and E (Vt, η) represents the event that triggers the communication based on a given threshold η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' One of the main purposes of investigating communication scheduling schemes for networked systems is for the efficient planning of communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Compared to the time- based scheduling schemes, the resource planning for the event- based ones is usually less straightforward and more challeng- ing due to asynchronous state sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A typical index that facilitates communication resource planning is communication arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='05445v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='SP] 13 Jan 2023 2 rate (CR) which depicts the likelihood of active commu- nication at a certain time [17], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', the probability of the communication switch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1 being closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' CR provides a practical insight into how many system resources are allocated to network communication and supports the efficient planning of system resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In most applications, the communication status is a random variable incorporating stochastic system uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, what attracts us is typically the average communication rate (ACR) which refers to the expected value of active communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Particular attention has been attracted to the stationary ACR [18] which represents the limit of the ACR as the time approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It is often used to evaluate the consumption of communication resources in the steady state of a networked system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Conventionally, the computation of the ACR is solved using statistical methods via numerical experiments, such as a Monte Carlo experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The analytical solution of the ACR with a closed form is a challenging problem due to the closed-loop effect of the event- based scheduling scheme, also known as side information [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1, the closed-loop effect, or side information, refers to the interdependence between the triggering event and the system variable of interest, which forms a closed loop between the system and the scheduler (the blue path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This closed-loop distorts the distributions of the system variables, such that they are no more subject to trivial Gaussian distributions even though the system uncertainties are formulated as Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Analyzing the stochastic property of the communication status is thus challenging since it is difficult to track the probabilistic propagation of the system variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In the literature, analytical computation methods for the ACR are quite sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' They only show up in a few papers that mainly focus on the control and filtering of networked systems [6], [17], [20]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' These approaches either over-simplify the computation of the ACR by imposing impractical assumptions or conduct conservative and coarse approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Their results can hardly be used to predict exact ACR which is important to efficient resource planning for networked communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In [17], [20], the ACR is computed by ignoring the closed-loop effect and approximating the distribution of the concerned system variable with a Gaussian distribution, which, however, leads to an accuracy gap with the true ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The work in [22] provides lower and upper bounds for the ACR without giving its analytical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In [6], [23], the computation of ACR is simplified by involving a stochastic triggering threshold which has obvious shortcomings compared to deterministic thresholds due to the inferior control performance and the sensitivity to data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To our best knowledge, the accurate computation of ACR for an event-triggered networked system with deterministic triggering thresholds has not been addressed in the literature, although it is a fundamental step towards the study of the crucial filtering problem [22], [24]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' As mentioned, the fundamental challenge of accurately quantifying the ACR for an event-triggered networked system is the precise tracking of the probabilistic propagation of the system variables of interest under the closed-loop effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' From a mathematical perspective, the event-based scheduling scheme with deterministic thresholds imposes constant trunca- tion to the support set of the probabilistic distribution functions (PDF) of the concerned system variables at each sampling instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This truncation operation constantly removes the Gaussianity of the system variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, the critical technical point of accurately computing the ACR is to precisely depict the truncated PDFs of the system variables recursively over time, which formulates a challenging mathematical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The main goal of this work is to overcome this challenge and provide accurate computation methods for the ACR of a typical networked system, an event-triggered stochastic control system (ET-SCS), where the communication is triggered when the state estimation error exceeds a deterministic threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To support this, we propose a recursive model that exactly depicts the temporal evolution of ACR based on the model raised in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Using this recursive model, we are able to track the truncated PDF of the system variables at each sampling time and investigate its influence on the ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Based on this, we are able to compute the ACR at an arbitrary instant using a finite number of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We also prove the existence of the stationary ACR and calculate its value using these coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Considering the complexity of the proposed analytical method, we further raise a numerical counterpart algorithm to calculate the ACR recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Both theoretical analysis and experimental studies are conducted to verify the accuracy of the proposed methods and qualify the inaccuracy gap of the conventional methods [6], [17], [20]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The main contributions of this article are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) Proposing a novel recursive temporal-evolution model for the ACR of a generic ET-SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) Proving the existence of the stationary ACR for a generic ET-SCS with deterministic event-triggered thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) Providing an analytical method and a numerical algorithm for the computation of ACR for a generic ET-SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 4) Conducting theoretical and numerical studies to qualify the accuracy gap of the conventional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 5) Validating the feasibility and accuracy of the proposed methods using a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The rest of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II formu- lates the problem, with mathematical preliminaries provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III, we introduce the recursive model of the ACR and investigate the existence of the stationary ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' IV presents the methods to precisely compute the ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' V, we use theoretical analysis and a numerical example to verify the accuracy of the proposed method and evaluate the accuracy gap of the conventional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' VI, a numerical experiment on a simple vehicle-following case is conducted to validate our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' VII concludes the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Notation: The rest of this article obeys the following nota- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The sets of real and natural numbers are denoted by R and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The superscript + sued after them indicates the subsets only containing the positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A Gaussian distribution with mean value µ ∈ R and variance σ2, σ ∈ R+ is represented by N(µ, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For a stochastic event E defined on a probability space, P(E ) denotes the occurring probability of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For a stochastic variable z ∈ R, P(z), pz(·), Fz(·), E(z), and Var(z) denote, respectively, its probability, PDF, cumula- tive distribution function (CDF), expectation, and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' PROBLEM STATEMENT AND PRELIMINARIES In this section, we describe the main mathematical problem of this paper and provide the preliminaries for its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' First, we introduce the dynamical model of an ET-SCS and the definition of the ACR, followed by the problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, we revisit the Jury’s stability criterion and the stochastic properties of truncated random variables which are important to the analysis of the ACR in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Dynamic Model of ET-SCS and State Estimation Error As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2, the ET-SCS considered in this article is composed of a plant and a series of sensors which are con- nected with a common network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this paper, we investigate a scalar ET-SCS, where the dynamic model of the plant is assumed to be linear time-invariant (LTI) and depicted by the following scalar stochastic difference equation (SDE), xk+1 = Axk + Buk + wk, (1) where k ∈ N denotes the discrete sampling time of the system, xk, uk ∈ R are, respectively, the state and control input of the system at time k, A, B ∈ R are constant parameters, and wk ∈ R is the stochastic noise of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For simplicity, we assume that the initial state of the system x0 is a known deterministic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that the stochastic process wk, k ∈ N, is subject to the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The Gaussian stochastic process wk is inde- pendent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=') for all k ∈ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', 1) wk ∼ N � 0, σ2� , ∀ k ∈ N, with σ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) pw,w(wi, wj) = pw(wi)pw(wj) holds for all i, j ∈ N, i̸=j, where pw(·) is the PDF of stochastic variable wk, k ∈ N, and pw,w(·, ·) is the joint PDF of wi and wj, i, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Sensor State Estimator Controller Plant Dynamics Scheduler Memory rk uk ek − Network Ik−1:ι Plant δk xk − Ek ˆxk Figure 2: The block diagram of an ET-SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Our work in this paper only deals with a scalar ET-SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that the exact computation of the ACR for a multi-dimensional ET-SCS is even more challenging due to the probabilistic coupling among the individual dimensions of the non-Gaussian system variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, various triggering options for multi-dimensional system variables make it tedious to determine their analytical distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The study on a scalar ET-SCS is sufficient to draw essential quantitative and qualitative conclusions that can be extended to multi- dimensional systems with additional efforts in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For system (1), its state xk at any time k ∈ N+ can be mea- sured by one or multiple sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' However, the state is sampled for the system plant only when the network communication is active, indicated by a closed switch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' At the time k, whether the status of the communication is active or not is represented as a binary event δk ∈ {0, 1}, namely δk = 1 for active and δk = 0 for inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For brevity, we represent these conditions as δ{1} k and δ{0} k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The communication status δk is affected by an event Ek which is produced by an event-based scheduler which will be introduced in Sec II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' When the sys- tem state is not sampled, an estimated value ˆxk is provided by a state estimator utilizing the system model (1) and the history data Ik−1:ι = {δk−1, · · · , δι+1, δι, xι, uk−1, · · · , uι} [7], ˆxι = xι ˆxι+1 = Axι + Buι, · · ˆxi = Aˆxi−1 + Bui−1, (2) i = ι + 2, · · · , k, where ι ∈ N refers to the last instant of state sampling and equation ˆxι = xι indicates a state sampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Since the initial state x0 is deterministic and known, we have ˆx0 = x0 and e0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The history data Ik−1:ι is stored in a memory on the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' All the system data before the last instant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', all {δi, xi, ui} where i < ι, is timely abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, a feedback controller uk = u(ˆxk−rk) can be designed to achieve the desired closed-loop performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The performance of the feedback controller is not within the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We make a correspondence between the ET-SCS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2 and the general event-based networked system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1 by recognizing xk as the system state and the estimation error ek = xk − ˆxk as the system variable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' By subtracting (2) from (1), we obtain the dynamic model of the state estimation error as eι = 0, eι+1 = wι, · · ei = Aei−1 + wi−1, (3) where i = ι + 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, for all i = ι + 1, · · · , k, ei is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Substituting ek−1, ek−2, · · · , eι to ek recursively, we obtain ek = �k−1 i=ι Ak−i−1wi, k > ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (4) The dynamic model (3) indicates that the estimation error accumulates over the time interval {ι + 1, ι + 2, · · · , k} due to the lack of state sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The error accumulation may lead to the degradation of the system control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Meanwhile, equation (4) shows that ek is subject to a Gaussian distribution N � 0, �k−1 i=ι A2(k−i−1)� considering the property of the linear combination of Gaussian stochastic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Nevertheless, we should note that (3) and (4) only hold when the state estimation error and the triggering event are subject to an open-loop configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2, this pertains to the scenario where the Scheduler is designed such that its output Ek is independent of its input ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Next, we will show that (3) and (4) do not generally hold and ek is no more a Gaussian- distributed stochastic variable for an event-triggered scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Event-Triggered Scheduling and Closed-Loop Effect To restrict the accumulation of the state estimation error ek, k ∈ N+, by fully exploiting the communication resources, the scheduler performs a least-necessary principle meaning that the communication is only activated when either the last state estimation error ek−1 exceeds a predefined threshold η ∈ R+, or the consecutive inactive period is beyond a time limit T ∈ N+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', for all k ∈ N+ and ι ∈ N, ι < k, δk = � 1, if |ek−1| ≥ η or k − ι > T 0, otherwise, (5) where the condition that triggers active communication δ{1} k refers to a positive event Ek, otherwise a negative event E k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The scheduling scheme (5) maintains a decent system performance by ensuring low resource usage by limiting the frequency of communication while restricting the estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Such an event-based scheduling model is widely used in various networked control systems, such as platooning of a group of vehicles [28], [29], power systems [30], [31] and cooperative manipulation in robotics systems [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The result of the event-triggering scheduling scheme (5) is a closed loop between the triggering event Ek, or the communication status δk and the state estimation error ek (the blue path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This loop has a significant influence on the probabilistic distribution of the estimation error ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Specifically, the error accumulation depicted in (3) only occurs when |ei| ≤ η, for any i = ι, ι + 1, · · · , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Otherwise, any |ei| > η will immediately activate the state sampling and lead to δi+1 = 1 and ei+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, given the last communication instant ι and the history data Ik−1:ι, the estimation error under the event-triggered scheduling scheme (5) becomes ˆeι = 0, ˆeι+1 = wι, · · ˆei = � Aˆei−1 + wi−1, if |ˆei−1| < η, 0, else, (6) where i = ι + 2, · · · , k, for all k ≤ ι + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Here, we use a new symbol ˆei to represent this closed-loop state estimation error, or closed-loop error under the event-triggered scheduling scheme (5) to distinguish it from the open-loop error ei in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Similar to ei, ˆei is also a random variable, for i = ι + 1, ι + 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Differently, the closed-loop error (6) contains additional side information |ˆei−1| < η compared to the open-loop error (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This side information imposes a truncation operation on the probabilistic distribution of the closed-loop error at every sampling instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It breaks the linearity of the dynamic model of the state estimation error and distorts its stochastic propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' As a result, the closed- loop error is hardly subject to a Gaussian distribution as time increases, even though the noise is a stationary Gaussian process according to Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, it is difficult to bring up a brief overall analytical form to represent ˆek, similar to (4), which makes it difficult to track the distorted stochastic propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this paper, we refer to the fact that the side information distorts the stochastic propagation of the system variable of interest as the closed-loop effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-E, we will briefly introduce the effect of a truncation operation on a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Stationary and Transient ACR of an ET-SCS For the ET-SCS in (1) with the state estimator (2) and the event-triggered scheduler (5), the ACR is defined as E(δk) = P(δ{1} k ), k ∈ N (7) which depicts the likelihood of the active status of the com- munication, or, equivalently, the realization of the event δ{1} k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Meanwhile, the ACR denotes the probability of the state sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Since the initial state x0 is deterministic and known, we have E(δ0) = P(δ{1} 0 ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, given ˆe0 = x0 − ˆx0 = 0, we know E(δ1) = P(δ{1} 1 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For any k ∈ N+, k ≥ 2, however, the value of δk is usually random, and the ACR is computed as E(δk) = 1 − P(δ{0} k ) = 1 − � η −η pˆek−1(z)dz, (8) where pˆek(·) is the PDF of the state estimation error and z ∈ R is an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that the integration interval [ −η, η ] corresponds to the constraint |ˆek−1| < η in (5) which blocks state sampling at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The above statements are based on our assumption that the initial system state x0 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Otherwise, the values of E(δ0) and E(δ1) are neither 1 nor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Instead, they are dependent on the distribution of x0 and should be valued within the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The limit E(δ∞) = lim k→∞ E(δk), if it exists, is defined as the stationary ACR, while E(δk) for a finite k ∈ N is referred to as the transient ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The main issue of exactly calculating the stationary and the transient ACR is that the analytical form of the PDF pˆek(·), k ∈ N, is difficult to derive due to the challenge of capturing the nontrivial stochastic propagation of the closed-loop errors, as introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In some existing work [17], [23], pˆek(·) is approximated by a Gaussian distribution by ignoring the closed-loop effect, which leads to the approximated calculation of ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We show in this article that such approximation results in a larger value of ACR compared to the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Alongside this, accurate ACR computation methods are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Existence of Steady State of A Discrete-Time LTI System To verify the existence of the stationary ACR for an ET- SCS, in this paper, we construct a recursive model to depict the timed evolution of the ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The recursive ACR model is indeed a discrete-time LTI (dt-LTI) system and the stationary ACR is equivalent to its steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This allows us to solve the stationary ACR by investigating the asymptotic stability of a general dt-LTI system, which can be examined by the well- known Jury stability criterion [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Consider a characteristic polynomial with variable z ∈ R in the following form, D(z) = a0 + a1z + a2z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' + aNzN, 5 where N ∈ N+ is the degree of the characteristic polynomial and ai ∈ R, i = 1, 2, · · · , N, are coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The following tests determine whether the system represented by D(z) has any pole outside the unit circle (the instability region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A sys- tem must conform to all the following rules to be considered stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Rule 1: If z = 1, D(z) > 0 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Rule 2: If z = −1, zND(z) > 0 must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Rule 3: |a0| < |aN| must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' If all rules satisfied, we expand the Jury Array as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) a0 a1 a2 a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' aN 2) aN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' a3 a2 a1 a0 3) b0 b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' bN−1 4) bN−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' b0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2N − 3) v0 v1 v2 Once we reach to a row with 2 members, we stop constructing further arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To calculate the values of the odd-number rows, we can use the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The even number rows are equal to the previous row in reverse order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We will use k as an arbitrary subscript value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' These formulas are reusable for all elements in the array: bk = ���� a0 aN−k aN ak ���� , ck = ���� b0 bN−1−k bN−1 bk ���� , dk = ���� c0 cN−2−k cN−2 ck ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This pattern can be carried out to all lower rows of the array, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Rule 4: Once the Jury array has been formed, all the following relationships must be satisfied until the last row of the array |b0| > |bN−1|, |c0| > |cN−2|, |d0| > |dN−3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The system is stable if all these conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Truncated Stochastic Variables As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-C, the side information |ek−1| < η in (6) changes the support of pˆek(·) at every sampling instant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' With a constant threshold η ∈ R+, the change is specifically a symmetric truncation operation to the PDF pˆek(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Consider a scalar stochastic variable ζ ∈ R with an infinite-support PDF pζ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We use ζ[a,b] to represent the truncated stochastic variable derived from ζ by trimming its support set with a fixed interval ζ ∈ [ a, b ], a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Due to this truncation operation, the derived variable ζ[a,b] has different stochastic properties compared to its original ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Specifically, its PDF reads pζ[a,b](z) = � ρζ(a, b)pζ(z), a ≤ z ≤ b, 0, otherwise, (9) where z ∈ R is an auxiliary variable, pζ[a,b](·) denotes the PDF of ζ[a,b] subject to a truncation interval [ a, b ], and ρζ(a, b) is a scalar calculated as ρζ(a, b) = 1/(Fζ(b) − Fζ(a)) where Fζ(·) is the cumulative distribution function (CDF) of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, the expected value and the variance of ζ[a,b] are E � ζ[a,b]� = � b a zpζ[a,b](z)dz = ρζ(a, b) � b a zpζ(z)dz, (10) Var � ζ[a,b]� = � b a z2pζ[a,b](z)dz − E2� ζ[a,b]� = ρζ(a, b) � b a z2pζ(z)dz − E2� ζ[a,b]� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (11) Note that the difference between the mean values and the variances of a stochastic variable ζ and its truncated coun- terpart ζ[a,b] is reflected not only by the additional multiplier ρζ(a, b) but also by the changed upper and lower limits of the integrals, a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' If ζ is a Gaussian variable, ζ[a,b] is not necessarily Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This means that all the properties that are proposed for Gaussian variables, such as the linear combi- nation properties, may not hold for their truncated variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Ignoring this effect may lead to the inaccurate characterization of the truncated stochastic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' COMMUNICATION RATE ANALYSIS In this section, we conduct a comprehensive analysis of the transient and the stationary ACR for an ET-SCS defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Based on the introduction of the predictive indexes and the predictive coefficients, we derive a recursive model for the transient ACR of an ET-SCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, by showing the equivalence between the recursive model and a dt-LTI system, we prove the existence of the stationary ACR using the Jury stability criterion recalled in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' As a result, the transient and the stationary ACR can be calculated using a finite number of predictive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The Predictive Indexes and The Predictive Coefficients In this section, we introduce the predictive indexes and the predictive coefficients which are important to analyze the ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We first define a compound event for k, n ∈ N+, n ≤ k, Ek:k−n = δ{0} k ∩ δ{0} k−1 ∩ · · · ∩ δ{0} k−n+1 ∩ δ{1} k−n, (12) which represents the conjunction of n successive inactive events after an active event δ{1} k−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It is straightforward to show that the compound event satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For any n, k ∈ N+, n ≤ k, event Ek:k−n satisfies the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) Ek:k−n ̸= ∅, ∀ n ≤ T, and Ek:k−n = ∅, ∀ n > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) Ek:k−i ∩ Ek:k−j = ∅, for any i, j ≤ k, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) �k n=1 Ek:k−n = δ{0} k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Property 1, condition 1) is met considering that the non- communication period of the system should not be larger than the limit T, according to the event-triggered scheduler (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Condition 2) is verified by the mutual exclusion between the compound events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Condition 3) is justified by taking the union of all compound events Ek:k−n, for all n = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 6 1) Predictive indexes: For particular communication vari- ables δ0, δ1, · · · , δn, we define n-step predictive non- communication index Pn, or predictive index as Pn = P � δ{0} n , δ{0} n−1, · · · , δ{0} 1 ��� δ{1} 0 � , n ∈ N+, (13) which denotes the probability that no communication is acti- vated for n sampling instants given an active communication event δ{1} 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We will later generalize the predictive index to arbitrary-time communication status δk, k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The n-step predictive index Pn satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For any n, T ∈ N+, the predictive index Pn satisfies the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) For all n > T, Pn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) For all n ≤ T, 0 < Pn < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 2-1) is justified by that the communication is activated by force after n > T, according to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For n ≤ T, neither activation nor deactivation of the communication is a certain event, since the support of the PDF of the noise wk is infinite, ∀ k = 1, 2, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This addresses property 2-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) Predictive coefficients: Also for communication sta- tus variables δ0, δ1, · · · , δn, the n-stacked predictive non- communication coefficient P n, or predictive coefficient is defined as P n = P � δ{0} n ��� En−1:0 � , n = 1, 2, · · · , T, (14) which denotes the probability of a single non-communication event δ{0} n given the history compound event En−1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Similar to the predictive index, the predictive coefficient has the following property due to the infinite support of the PDF of the stochastic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 0 < P n < 1 holds for all n = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) Relation between the indexes and coefficients: From the definitions of the predictive index in (13) and the predictive coefficient (14), we have the following relation, Pn = P � δ{0} n , δ{0} n−1, · · · , δ{0} 1 ��� δ{1} 0 � = P � δ{0} n ��� δ{0} n−1, · · · , δ{0} 1 , δ{1} 0 � × P � δ{0} n−1, · · · , δ{0} 1 ��� δ{1} 0 � = P nPn−1, n = 2, · · · , T, (15) with P1 = P 1, which renders the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (Relation of the predictive index and coefficient) The following relation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Pn = �n i=1 P i, n = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (16) Meanwhile, applying Property 2 and Property 3 to (15) recursively, we have the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (Monotonic decrease of predictive index) The condition 0 < PT < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' < P2 < P1 < 1 always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 4) Time-Invariance of the indexes and coefficients: Ensured by Assumption 1, the system noise wk, k ∈ N+ is a stationary stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, its stochastic properties are time-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' As a result, the dynamic model of the state estimation error in (6) and the communication status in (5) are also invariant to the last communication instant ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This justifies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (Time-invariance of communication probabilities) The following conditions hold ∀ n = 1, 2, · · · , T and ∀ ι ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' P � δ{0} ι+n, δ{0} ι+n−1, · · · , δ{0} ι+1 ��� δ{1} ι � = Pn, P � δ{0} ι+n ��� Eι+n−1:ι � = P n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (17) Property 6 proposes a very important claim for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It indicates that the n-step predictive indexes Pn and coefficients P n can be used to depict the communication probabilities (17) for an arbitrary state-sampling time ι ∈ N, even though they are originally defined specifically for ι = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Nevertheless, we should keep in mind that this property only holds when Assumption 1 is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The Recursive Model of The Transient ACR Having introduced the predictive indexes and coefficients, we are ready to present the recursive model for the transient ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' According to Property 1-3) and 1-2), we know P � δ{0} k � = P � k� n=1 Ek:k−n � = k � n=1 P(Ek:k−n) , k ∈ N+, which leads to P � δ{0} k � = k � n=1 P � δ{0} k , · · · , δ{0} k−n+1 ��� δ{1} k−n � � �� � =Pn P � δ{1} k−n � , (18) where we used Property 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that Pn = 0 holds ∀ n > T according to Property 1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, (18) can be rewritten as P � δ{0} k � = �min(k,T ) n=1 PnP � δ{1} k−n � , k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (19) According to the definition of ACR in (7), (19) leads to the following recursive model, E (δk) = 1 − �min(k,T ) n=1 PnE (δk−n) , k ∈ N+, (20) with an initial condition E(δ0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Model (20) depicts the recursive evolution of ACR at an arbitrary sampling instant as time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Using Property 4, model (20) can be rewritten as E (δk) = 1 − �min(k,T ) n=1 �n i=1 P iE (δk−n) , k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (21) The recursive model (21) indicates that the transient ACR at any time k ∈ N+ can be recursively calculated using a finite number of predictive coefficients P 1, P 2, · · · , P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, how to obtain the values of these coefficients is a critical technical point for the exact computation of the transient ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We explore the solution to this problem in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The Existence of The Stationary ACR The recursive model (20), for k ≥ T, is equivalent to a T-order dt-LTI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This offers us a solution to study the existence of the stationary ACR using the Jury Criterion recalled in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-D, which renders the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The stationary ACR E(δ∞) = lim k→∞ E(δk) derived from the recursive model (20) exists and its value reads E(δ∞) = 1/(1 + �T n=1 �n i=1 P i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We can rewrite the recursive model (20) as the follow- ing matrix-vector form ξk = � 0⊤ I PT p � ξk−1 + β, ∀ k ∈ N+, (23) where I is a (T − 1)-dimensional identity matrix, 0 ∈ RT −1 is a zero vector, and ξk = [ E(δk+T −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' E(δk+1) E(δk) ]⊤ , p = [ PT −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' P1 ] , β = [ 0⊤ 1 ]⊤, with an initial condition ξ0 = [ E(δT −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' E(δ1) E(δ0) ]⊤ , (24) Therefore, (23) can be recognized as a dt-LTI system, where ξk, k ∈ N, is the system state, β is the constant input, and Pn, n = 1, 2, · · · , T, are the constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this sense, the existence of the stationary ACR E(δ∞) can be determined by the stability of the dt-LTI system using the Jury’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For any z ∈ R, the characteristic polynomial of the dt-LTI (23) is D(z) = PT + PT −1z + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' + P1zT −1 + zT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (25) Given the polynomial (25), we investigate the state conver- gence of the dt-LTI system (23) using the Jury’s criterion recalled in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It is straightforward to verify that Rules 1-3 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-D hold for (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We then use the coefficients in (25) to construct the Jury array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The elements in the first row of the Jury array then become a0 = PT , a1 = PT −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' , aT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Having the elements on row i and row 2 of Jury array, the elements bk and bk+1 in the row 3 and row 4, with k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' , T − 1}, can be constructed as bk = ���� a0 aT −k aT ak ���� = a0ak − aT −kaT , bk+1 = ���� a0 aT −k−1 aT ak+1 ���� = a0ak+1 − aT −k−1aT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' From Lemma 5, we obtain 0 < a0 < a1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' < aT = 1, which implies a0ak < a0ak+1 < aT −k−1aT < aT −kaT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This inequality further implies −1 < bk < bk+1 < 0 and |b0| > |bT −1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' These results reveal the relationship among the elements bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Similarly, we can construct ck and ck+1, as follows ck = ���� b0 bT −1−k bT −1 bk ���� = b1bk − bT +1−kbT , ck+1 = ���� b0 bT −2−k bT −1 bk+1 ���� = b0bk+1 − bT −2−kbT −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Similarly, we can readily conclude that 1 > ck > ck+1, which also implies |c1| > |cT − 1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Similar analysis can be carried out to show that Rule 4 of Jury stability criteria always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, the characteristic polynomial (25) meets Jury’s stability criteria, which means that all the eigenvalues of the state transition matrix in (23) are less than or equal to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' the system presented in (23) is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This indicates that the limit E(δ∞) = limk→∞ E(δk) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' By taking the limit of both sides of (20), we obtain limk→∞ E (δk) = 1 − �T n=1 Pn limk→∞ E (δk−n) , (26) which leads to (22) and proves this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Based on a general recursive model (20), Theorem 1 proves the existence of the stationary ACR for any ET-SCS with an event-triggered communication scheduler and a deterministic constant threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This claim does not require any additional conditions, meaning that the stationary ACR in general exists for any ET-SCS defined in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Equation (22) indicates that the stationary ACR can also be calculated using a finite number of predictive coefficients P n, n = 1, 2, · · · , T, similar to the transient ACR explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' COMPUTATION OF THE PREDICTIVE COEFFICIENTS As shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III, the computation of both the transient and the stationary ACR requires the predictive coefficients P n, n = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This section provides both analytical and numerical approaches to compute these coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, we compare our methods with the previous results which apply the restricted Gaussianity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, we present a numerical example to demonstrate our theoretical claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The Analytical Form of The Predictive Coefficients This section explores the analytical method to exactly compute the predictive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' According to the definition of the predictive coefficients in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III-A, we have P i = P � δ{0} i ��� Ei−1:0 � = � η −η pˆei−1(z) dz, (27) for i = 2, · · · , T, with an initial condition P 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, each coefficient P i is the integration of the PDF of the state estimation error ˆei−1 on a finite support set [ −η, η ], where the error recursively evolves following (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, the critical technical point is to obtain the analytical form of these PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For each i = 1, 2, · · · , T − 1, the PDF of ˆei reads pˆei(z) = � η −η pˆeη i−1(ξ)pw(z − Aξ)dξ, (28) where pˆeη i−1(·) denotes the PDF of the truncated stochastic variable ˆeη i−1 of ˆei−1 with a symmetric truncation interval [ −η, η ] and pw(·) is the PDF of the disturbance wk, k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that ˆe0 = 0, and ˆe1, wk ∼ N(0, σ), which yields pˆe0(z)=δ(z), pˆe1(z)=pw(z)= 1 √ 2πσ exp � − z2 2σ2 � , (29) 8 where δ(·) is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, the distribution pˆei(z) in (28) can be obtained as pˆei(z) = � η −η pˆeη i−1(ξ) √ 2πσ exp � −(z − Aξ)2 2σ2 � dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (30) According to the PDF of a truncated random variable in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-E, we have pˆeη i−1(z) = Gˆei−1(z)pˆei−1(z), (31) where pˆei−1(·) is the PDF of the non-truncated variable ˆei−1 and Gˆei−1(·) is a piece-wise constant function defined as Gˆei−1(z) = � 1/ � η −η pˆei−1(ξ)dξ, −η ≤ z ≤ η, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (32) Substituting (32) and (31) to the PDF (30), we obtain pˆei(z) = Gˆei−1(z) √ 2πσ � η −η pˆei−1(ξ) exp � −(z − Aξ)2 2σ2 � dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (33) Thus, equations (29) and (33) form a complete recursive model to solve the analytical forms of the PDFs of the closed-loop errors, pˆei(·), for all i = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, (27) can be used to accurately calculate the predictive coefficients P i, for i = 2, 3, · · · , T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that, for any i = 2, 3, · · · , T, the PDF pˆei(·) is not necessarily Gaussian due to the recursive truncation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, the analytical form of pˆei(·) becomes increasingly complicated and challenging to solve as i gets larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To resolve this issue, in the next section, we propose a numerical algorithm to approximate the predictive coefficients using the recursive stochastic sampling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Approximating The Predictive Coefficients Numerically Considering the difficulty of analytically computing the coefficients P i for large i, we propose a numerical algorithm to approximate them using the recursive stochastic sampling method, as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The computation of P 1 and P 2 in Line 1 is straightforward since the analytical forms of pˆe0(·) and pˆe1(·) are trivial and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In Line 2, N particles are initialized from the distribution pˆe1(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', a Gaussian distribution N(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' From Line 3, the particles are used to approximate the nontrivial PDFs pˆei(·) for i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The particles are a group of real scalars independently drawn from a certain distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Consider that N ∈ N+ particles Z = {z(1), z(2), · · · , z(N)}, z(i) ∈ R, i = 1, 2, · · · , N, are independently drawn from a distribution depicted by a PDF p(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, the unbiased estimation of p(·) can be obtained using a Gaussian kernel method as ˆp(z, Z) = 1 √ 2πˆσN N � j=1 exp � − � z − z(j)�2 2ˆσ2 � , (34) where ˆσ ∈ R+ is a variance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Here, we use the symbol ˆp(·) to represent the PDFs approximated using parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Based on the approximated PDFs ˆpˆei(·), the predictive coefficients P i+1 are calculated recursively, following the flow ˆpˆei−1(·) → ˆpˆeη i−1(·) → ˆpˆei(·) → P i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The approximation in each iteration is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In line 4, the particles exceeding the threshold η are removed, which simulates the truncation operation to the PDF pˆei−1(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, in line 5, the PDF pˆeη i−1(·) of the truncated stochastic variable ˆeη i−1 is approximated with the remaining particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In line 6, N particles are resampled from the approximated PDF ˆpˆeη i−1(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The particles then perform the stochastic propagation according to the error dynamics (6), as shown in lines 7-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In line 11, the particle approximation method (34) is used again to approximate the PDF pˆei(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, the predictive coefficient P i+1 are calculated in line 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Algorithm 1: Approximation of the predictive coeffi- cients using particles Inputs : noise variance σ and particle number N Outputs: P i, ∀ i = 1, 2, · · · , T, T > 2 1 Calculate P 1, P 2 using (27) with pˆe0(·), pˆe1(·) in (29);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2 Sample particles z(j) 1 ∼ N(0, σ), j = 1, 2, · · · , N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3 for i ← 2 to T − 1 do 4 Remove all particles ���z(j) i−1 ��� ≥ η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 5 Approximate ˆpˆeη i−1(·) with z(j) i−1 using (34);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 6 Re-sample particles z(j) i−1 ∼ ˆpˆeη i−1(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' j = 1, 2, · · · , N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 7 for j ← 1 to N do 8 Draw ϵ(j) i−1 ∼ N(0, σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 9 z(j) i = Az(j) i−1 + ϵ(j) i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 10 end 11 Approximate pˆei(·) with z(j) i using (34);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 12 Calculate P i+1 with (27) using PDF pˆei(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 13 end Note that Algorithm 1 may lead to approximation errors in the predictive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The main source of the errors is the deviation between the PDFs pˆei(·) and their estimations ˆpˆei(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In fact, the unbiasedness of the approximation only holds in the statistical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To reduce the approximation errors, N should be selected sufficiently large and ˆσ should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this paper, our theoretical claims and numerical methods target at a specific class of ET-SCS, where the network communication is triggered by an asynchronous event associated with state estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In fact, the state estimation error ek, k ∈ N, can be recognized as a variable that depends on the internal states of the joint dynamic model of the system plant and the state estimator, namely the plant state xk and the estimator state ˆxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, our results can also be extended to a generic ET-SCS of which the triggering event may be assigned to an arbitrary state-dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this case, the recursive model of the ACR is still effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' What changes is that the predictive coefficients are calculated using the PDF of this state-dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The challenge of such an extension depends on the complexity of this PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' COMPARISON WITH THE CONVENTIONAL METHOD Based on Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' IV, we are able to calculate the stationary and the transient ACR for an ET-SCS using a finite number of predictive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The analytical and numerical 9 methods to compute these coefficients are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this section, we make a comparison between our approaches and the conventional method [17] that intentionally ignores the side information for simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Both theoretical analysis and a numerical study are conducted to validate the accuracy of our approaches and qualitatively verify the accuracy gap between the conventional method and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Deviation Analysis of The Conventional Method As mentioned above, the computation of ACR without considering the closed-loop effect leads to an oversimplified distribution model for the state estimation error and eventually returns approximated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Assume that the open-loop state estimation error is subject to the dynamic model (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, the error has a fully Gaussian PDF, and, similar to (21), the open-loop ACR can be recursively computed as ˘E (δk) = 1 − �min(k,T ) n=1 �n j=1 ˘P j˘E (δk−n) , k ∈ N+, (35) with ˘E(δ0) = 1, where ˘P i, i = 1, 2, · · · , T, are the coeffi- cients obtained by ˘P i = � η −η pei−1(z)dz, i = 1, 2, · · · , T, (36) where pei(·) is the PDF of the open-loop state estimation error ei subject to the dynamic model (3) with ι = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Hence, according to (3), for all i > 1, we have pei(z) = � ∞ −∞ pei−1(ξ) pw(z − Aξ)dξ = � ∞ −∞ pei−1(ξ) √ 2πσ exp � −(z − Aξ)2 2σ2 � dξ, (37) with the initial conditions pe0(z) = δ(z), pe1(z) = 1 √ 2πσ exp � − z2 2σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (38) Comparing (33) and (37), one notices that ˆe1 and e1 have the same distribution N(0, σ), while for each i > 1, pˆei(·) has an additional multiplier Gˆei−1(·), compared to pei(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, the integration intervals are also different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Now, we compare the mean values and the variances of the two stochastic variables ˆei and ei for i = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' From (4), we know that the state estimation error ei is a linear combination of the Gaussian-distributed stochastic variables w0, · · · , wi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Hence, ei is also Gaussian-distributed and has the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Given that w0, · · · , wT −1 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' stochastic variables (Assumption 1), the following statements hold for all ei, i = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) For any z ∈ R, pei(z) = pei(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) E(ei) = �i−1 n=ι Ai−n−1E(wn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) Var(ei) = �i−1 n=ι Ai−n−1Var(wn) = �i−1 n=ι Ai−n−1σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Property 7 is easy to verify using the linear properties of Gaussian stochastic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Nevertheless, the stochastic properties of the closed-loop state estimation error ˆei are not that straightforward due to the recursive truncation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Before proceeding with the study on the stochastic properties of ˆei, it is necessary to propose the following proposition for truncated stochastic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Let ζ ∈ R be an arbitrary stochastic variable of which the PDF pζ(z) has infinite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' E(ζ) and Var(ζ) are respectively its mean value and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, let ζη ∈ R be a truncated stochastic variable by trimming the support of ζ to be within the symmetrically bilateral interval [ −η, η ], η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' If E(ζ) = 0, and pζ(z) = pζ(−z) holds for all z ∈ R, then the following conditions are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) E(ζη) = 0, and pζη(z) = pζη(−z), ∀ z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) Var(ζη) < Var(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' If E(ζ) = 0 and pζ(z) = pζ(−z) hold, according to the definition of the PDF of truncated stochastic variables in (9), we have pζη(z) = pζ(z) Fζ(η) − Fζ(−η) = pζ(−z) Fζ(η) − Fζ(−η) = pζη(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Utilizing this property, we further have E(ζη) = � η −η zpζη(z)dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, condition 1) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Furthermore, the variance of ζη reads Var(ζη) = � η −η z2pζη(z)dz − E2(ζη) = � η −η z2pζ(z)dz �� η −η pζ(z)dz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that Var(ζη) is indeed a function of the truncation interval η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, we represent it as Var(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It can be verified that Var(η) is continuous and continuously differential for η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Moreover, we know lim η→∞ Var(η) = � ∞ −∞ z2pζ(z)dz = Var(ζ), lim η→0 Var(η) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (39) By taking the derivative of Var(η) to η, we obtain Var′(η) = ��� η −η z2pζ(z)dz �′ � η −η pζ(z)dz − �� η −η pζ(z)dz �′ � η −η z2pζ(z)dz ���� η −η pζ(z)dz �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that �� η −η z2pζ(z)dz �′ = η2[pζ(η) + pζ(−η)] = 2η2pζ(η), �� η −η pζ(z)dz �′ = pζ(η) + pζ(−η) = 2pζ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, Var′(η) = 2pζ(η) � η −η � η2 − ζ2� pζ(z)dz � � η −η pζ(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Since pζ(·) is a non-negative, we conclude V ′(η) > 0, for all η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This implies that V (η) is a monotonically increasing function in the interval η ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, we can write Var(ζη) = Var(η) < Var(∞) = Var(ζ), for any 0 < η < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, condition 2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Proposition 1 indicates that a truncated stochastic variable has the same expected value but a smaller variance than its 10 original counterpart if the latter has an even PDF around zero and the truncation interval is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We now present the following theorem that characterizes the relation between the mean values and variances of the closed-loop and open-loop state estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Given state estimation errors ˆei and ei de- picted by the dynamic models (6) and (3), respectively, i = 1, 2, · · · , T, the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) pˆei(z) = pˆei(−z), for all z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) E(ˆei) = E(ei) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) Var(ˆe1) = Var(e1), Var(ˆei) < Var(ei), for all i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We first consider the case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Since ˆe1, e1 ∼ N(0, σ), we have pˆe1(z) = pˆe1(−z), for all z ∈ R, E(ˆe1) = E(e1) = 0, and Var(ˆe1) = Var(e1) = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, for a truncated stochastic variable ˆeη i with threshold η > 0, according to Proposition 1, given any i = 1, 2, · · · , T −1, such that pˆei(z) = pˆei(−z), we have pˆeη i (z) = pˆeη i (−z), E(ˆeη i ) = 0, and Var(ˆeη i ) < Var(ˆei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' According to (6), we know ˆei+1 = Aˆeη i + wi, from which we conclude pˆei+1 (z) = � η −η pˆeη i (ξ) pw(z − Aξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Considering that pˆeη 1(·) is an even PDF, and pw(z) = pw(−z), for all z ∈ R, we obtain pˆei+1 (z) = � ξ=η ξ=−η pˆeη i (−ξ) pw(−z + Aξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Set ˆz = −ξ, then we will have pˆei+1 (z) = � −ˆz=η −ˆz=−η pˆeη i (ˆz) pw(−z − Aˆz)d(−ˆz) = − � −ˆz=η −ˆz=−η pˆeη i (ˆz) pw(−z − Aˆz)dˆz = � ˆz=η ˆz=−η pˆeη i (ˆz) pw(−z − Aˆz)dˆz = pˆei+1(−z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This property leads to E(ˆei+1) = � η −η zpˆei+1(z) dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, from Assumption 1, we know that eη i and ei are independent from wi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus we conclude Var(ˆei+1) = A2Var(ˆeη i ) + σ2, Var(ei+1) = A2Var(ei) + σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' According to Proposition 1, we have Var(ˆeη i ) < Var(ˆei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, for any i such that Var(ˆei) ≤ Var(ei), we have Var(ˆei+1) = A2Var(ˆeη i ) + σ2 < A2Var(ˆei) + σ2 ≤ A2Var(ei) + σ2 = Var(ei+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, we proved pˆei(z) = pˆei(−z), E(ˆei) = 0, and Var(ˆei) ≤ Var(ei) hold for all i = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that Var(ˆei) = Var(ei) only when i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Theorem 2 indicates the qualitative difference between the PDFs, the mean values, and the variances of the closed-loop error ˆei and the open-loop error ei for i = 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Both of them have even PDFs and zero mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Nevertheless, the closed-loop error ˆei has a smaller variance than the open-loop one ei, for i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This indicates that the recursive truncation operations in (6) result in a shrink in the PDF pˆei(z) along the z-axis compared to the infinite support Gaussian PDF pei(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, for any i > 1, Theorem 2 results in � η −η pˆei(z)dz > � η −η pei(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (40) This can be explained in an intuitive manner that the shape of pˆei(·) is more narrow than pei(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Based on this, we can infer that the conventional method using pei(·) instead of pˆei(·) leads to smaller results for the coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', ˘P i+1 < P i+1 for i = 3, · · · , T, according to (27), and then larger values of the transient ACR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', ˘E(δk) < E(δk) for k = 3, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Extending this claim to k → ∞, we also have a similar conclusion for the stationary ACR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', ˘E(δ∞) < E(δ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The analysis in this section not only proves the accuracy gap of the conventional method in theory but also qualitatively points out that it always leads to larger computation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Accuracy Comparison: A Numerical Example Here we present a numerical example to verify the ac- curacy of our proposed analytical and numerical methods, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' IV-A and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' IV-B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We also validate the accuracy gap of the conventional method that ignores the close-loop effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Consider an ET-SCS as in (1) with parameters A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='25, B = 1, an initial state x0 = −2, a stochastic process wk ∼ N(0, 1), k ∈ N+, and a state- feedback controller uk = −ˆxk, where ˆxk is estimated us- ing (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The threshold and the maximum triggering interval of the event-triggered scheduler (5) are η = 1, T = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' As addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' V-A, the major difference between our work and the existing works is that the latter ignores the closed-loop effects of an ET-SCS and use the open-loop estimation error ek to compute ACR, instead of the closed-loop error ˆek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To provide a fair and clear comparison study, we use five manners to compute the transient ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) The Proposed Analytical Method (PAM): The recursive expressions (29) and (33) are used to obtain the PDFs pˆei(·) of the closed-loop state estimation errors ˆei for i = 0, 1, · · · , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, the coefficients P i+1 are calculated using (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, (21) is recursively used to compute the transient ACR E(δk) for k = 1, 2, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) The Proposed Numerical Method (PNM): Algorithm 1 is used to approximate the PDFs pˆei(·) of the closed-loop state estimation errors ˆei for i = 0, 1, · · · , 4, with parameters ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 and N = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, the coefficients P i+1 are calculated using (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, (21) is recursively used to compute the transient ACR E(δk) for k = 1, 2, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) The Conventional Analytical Method (CAM): [17] The recursive expressions (37) and (38) are used to obtain the PDFs pei(·) of the open-loop state estimation errors ei for i = 0, 1, · · · , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, the open-loop predictive coefficients ˘P i+1 are calculated using (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Finally, (35) is recursively used to compute the transient ACR ˘E(δk) for k = 1, 2, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 4) The Conventional Numerical Method (CNM): This ap- proach is merely used to provide a numerical counterpart of the CAM approach for the completeness of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We first use Algorithm 1, with the same parameters ˆσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 and N = 104 11 as PNM but with the lines 4-6 removed, to calculate the open- loop predictive coefficients ˘P i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, (35) is recursively used to compute the transient ACR ˘E(δk) for k = 1, 2, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 5) Ground Truth (GT): We conduct a Monte-Carlo exper- iment of the ET-SCS with the same initial state repeated for 104 trials to approximate the true value of ACR, EGT (δk) = #(δk = 1)/104, where #(δk = 1) is the total number of trials of which δk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The following Example 1 provides an instruction to compute the transient ACR using PAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that we only give the results for E(δk), k = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The results for larger k values are omitted due to the complexity of analytical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (Computation of ACR Using PAM) The compu- tation procedure of E(δk) for k = 1, 2, 3 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For k = 1, according to (29), we have P 1 = � η −η δ(z)dz = 1, and E(δ1) = 1 − P 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For k = 2, using (27), we can calculate P 2 = � η −η pˆe1(z) dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='6827, where the analytical form of pˆe1(·) is provided in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Then, according to (21), we have E(δ2) = 1 − P 1E(δ1) − P 2P 1E(δ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' For k = 3, we have ˆe2 = Aˆeη 1 + w1 according to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Thus, the analytical form of pˆe2(·) reads pˆe2(z) = � η −η pˆeη 1(ξ) pw(z − Aξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (41) Note that pˆeη 1(·) is a truncated Gaussian PDF and pw(·) is a Gaussian PDF, which makes the calculation of pˆe2(·) nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' According to the formulations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' II-E, we have pˆeη 1(z) = 1 √ 2πσ erf � η √ 2σ2 � exp � − z2 2σ2 � , (42) where erf(x) = 2 √π � x 0 exp(−x2)dx is the Gaussian error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Substituting (42) to the integral term in (41), we get pˆe2(z) = � η −η exp � − (z−Aξ)2+ξ2 2σ2 � 2πσ2erf � η √ 2σ � dξ = exp � − z2 2σ2(A2+1) � 2πσ2erf � η √ 2σ � � η −η exp � � �− � ξ − Az A2+1 �2 2¯σ2 � � �dξ = exp � − z2 2σ2(A2+1) � 2σ � 2π(A2 + 1)erf � η √ 2σ � (43) × � erf �ηA2 + η − Az √ 2σ √ A2 + 1 � + erf �ηA2 + η + Az √ 2σ √ A2 + 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, we calculate P 3 = 1 − � η −η pˆe2(z)dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='5872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' According to (21), we have E(δ3) = 1−P 1E(δ2)−P 2P 1E(δ1)−P 3P 2P 1E(δ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It has been noticed that the analytical form of pˆe2(·) in (43) becomes very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The computation of E(δk) for k > 3 is even more difficult due to the complicated form of pˆek−1(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Therefore, we only provide the results for k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The computation results of the numerical study are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Slight deviations are seen between PAM and PNM or between CAM and CNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Note that these deviations reflect the inevitable approximation errors between the analytical methods and their numerical counterparts due to the approx- imation bias of the Gaussian kernel method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Incorporating these errors, we can see that the results of PAM and PNM are very close to the ground truth (with absolute errors smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='005), which validates the effectiveness and accuracy of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' On the contrary, the results of CAM and CNM present large calculation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Moreover, they are all large than the GT results, in general, which verifies our theoretical arguments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' V-A that the conventional method overapproximates the ACR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Table I: The GT and the computed ACR values for ET-SCS k GT PAM PNM CAM CNM 1 0 0 0 0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3161 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3668 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2650 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3082 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2801 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3126 More details can be found by taking a deeper look into the stochastic properties of the state estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Table II shows the mean values E(·) and the variances Var(·) of the closed-loop error ˆek and open-loop error ek for k = 1, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It can be seen that their mean values are very close to zero, despite small errors due to the numerical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, we witness Var(e1) = Var(ˆe1) = 0 and Var(ek) > Var(ˆek), for all k = 2, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This coincides with our theoretical state- ments in Theorem 2 that the open-loop errors have the same mean values as the closed-loop errors but larger variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Table II: The mean values and the variances of the closed-loop and the open-loop state estimation errors k E(ˆek) E(ek) Var(ˆek) Var(ek) 1 0 0 0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0000 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4549 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='5625 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0037 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4497 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0031 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0220 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='5108 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='7302 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='5288 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='601 The PDFs calculated using the closed-loop and the open- loop errors, pˆek(·) and pek(·), for k = 2, 3, 4, 5, are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3, using the red line and the blue line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The GT PDF of the state estimation errors, drawn as the gray area, obtained by conducting a Monte Carlo experiment, is also presented for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We observe that our proposed method accurately follows the GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' On the contrary, the con- ventional method obviously deviates from the GT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The 12 deviation becomes larger as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This also verifies our theoretical claims in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' (a) k = 2 (b) k = 3 (c) k = 4 (d) k = 5 Figure 3: The approximated PDFs of the state estimation errors for k=2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The Red line denotes ˆpˆek(·) calculated using our proposed methods (k = 2 using PAM and k = 3, 4, 5 using PNM) and the blue line is ˆpek(·) obtained from the conventional approach (CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The gray area represents the GT PDF using Monte Carlo sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' EXPERIMENTAL STUDY In this section, we conduct an experimental study of a leader-follower autonomous driving scenario to validate our theoretical results interpreted so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The leader-follower sce- nario is a simplified case of the widely-used platooning model in autonomous driving [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 4, the system contains a leader vehicle that maneuvers according to a certain trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' A follower vehicle is dedicated to keeping a constant distance from the leader vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The positions and velocities of the vehicles are measured using a series of remote sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Both the vehicles and the sensors are connected using a common communication network that allows data exchange and state sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The switches installed on remote sensors, subject to the triggering scheme (5), determine whether to transmit the most recent vehicle states to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The position of the leader vehicle follows a predefined trajectory pL(t) = − cos(t) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The follower is required to maintain a distance d = 3 m with the leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The kinematic model of the follower vehicle is ˙p(t) = v(t), ˙v(t) = u(t), (44) where p(t), v(t), u(t) ∈ R are respectively the position, the ve- locity, and the acceleration of the follower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In this experiment, the parameters are selected as γ = 1, Q = 1, and K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The objective of the problem is to design a control law u(t), such d Remote Sensor Communication Network Plant Figure 4: Illustration of a leader-follower system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' that p(t) → pL(t) + d and v(t) → ˙pL(t) as time t increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' We define the following feedback control law, u(t) = −γQ−1v(t) − Q−1Kp(t) + γQ−1 ˙pL(t) + Q−1KpL(t) + ¨pL(t) + Q−1Kd, (45) where K, Q, γ ∈ R+ are positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It can be verified using a Lyapunov method that p(t) − pL(t) = d and v(t) − ˙pL(t) = 0 render a globally asymptotic equilibrium of the closed-loop system, which indicates the achievement of the desired control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The proof is omitted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In our experiment, we consider the discrete-time version of the follower vehicle (44), pk+1 = pk + ∆tvk, vk+1 = vk + ∆tuk + wk, (46) where ∆t is the sampling period, and wk is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' noise process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Accordingly, we discretize the reference trajectory pL(t) to pL k using discrete sampling t = k · ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' In corre- spondence with the ET-SCS model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2, the leader’s trajectory pL k is the reference signal of the overall system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Each follower is a plant with the state xk = vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The limited communication bandwidth motivates the application of the event-triggered scheduler in (5), for which we set η = 1, and T = 20 in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The state estimator (2) is used to obtain ˆxk = ˆvk, with A = 1, and B = ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The discrete-time controller based on the estimated state is uk = u(k · ∆t), for which ˆpk is obtained using the recursive model ˆpk+1 = ˆpk + ∆tˆvk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The simulation runs for 104 trials with the same initial conditions p0 = 0 and v0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Each trial lasts for t = 40 s with a sampling time ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The overall control performance is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It is observed that the average following distance E(p(t)) − pL(t) slightly fluctuates around d = 3 m, which indicates satisfactory distance keeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, the average velocity E(v(t)) is very close to the reference velocity ˙pL(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This shows that the configuration of the state estimator (2) and the event-triggered scheduler (5) successfully achieves the control objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The computed values of the ACR E(δk), using Algorithm 1 (PNM), with various triggering thresholds η, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 6 (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' To verify the validity of our proposed method, we also show the GT-ACR obtained from Monte- Carlo simulation (in black), and the ACR computed according 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 GT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 pe, () pe, () L 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0 5 0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 GT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 pe, () pe) L 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 5 0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 GT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='() pe() L 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0 5 0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 GT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' () pe,) 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0 5 0 513 0 10 20 30 40 time 1 2 3 4 distance (a) tracking distance 0 10 20 30 40 time 0 1 2 3 4 velocity (b) velocity tracking Figure 5: Average performance of the platoon controller (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Plot (a) depicts the tracking distance E(p(t)) − pL(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Plot (b) shows the leading velocity ˙pL(t) (in red) and the mean of the actual velocity E(v(t)) (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' to the conventional method (CAM), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', ˘E(δk) (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The information delivered by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 6 can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1) The general existence of the stationary ACR: It is noticed that all ACR values, E(δk), ˘E(δk), and the GT-ACR, ultimately converge to their respective stationary points for all triggering threshold values η = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' This validates our result on the existence of the stationary ACR in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 2) The accuracy of the proposed method: It is observed that the computed ACR E(δk) closely follows the GT-ACR at all time, indicating the accuracy of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' On the contrary, ˘E(δk) shows deviations from the GT-ACR suggesting inaccuracy of computing the ACR by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Also, ˘E(δk) is in general larger than E(δk) in the steady state, which validates our theoretical statement in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' V-A that this method overestimates the stationary ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 3) The influence of the triggering threshold: The stationary ACR values tend to be smaller as the triggering threshold η increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The intuition behind this observation is that, higher threshold means higher estimation errors are tolerable, hence less events will be triggered to reset the estimation error, which consequently leads to lower ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Similar observation is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 7, where the change of stationary ACRs E(δ∞) and ˘E(δk), and their ratios are plotted versus the changes of the threshold η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' It van be seen that E(δ∞) < ˘E(δk), for all values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' However, the scale of the deviation between the two approaches is not monotone with respect to η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=', larger triggering thresholds do not necessarily lead to larger deviations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' The largest deviation occurs around η = 3, with more than 25%, which is noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' CONCLUSION Motivated by the conservativeness of the conventional exist- ing methods, in this article we provide comprehensive analyt- ical formulations to accurately compute the average commu- nication rate for networked control systems under the event- triggered sampling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' By incorporating the distribution truncation operations that correspond to the side information generated by the triggering decisions, we prove the existence of stationary ACR using a novel recursive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Afterwards, we propose analytical and numerical approaches to accurately calculate ACR at any arbitrary time and demonstrate the noticeable ACR over-estimation when the triggering-induced 0 10 20 30 40 time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 ACR (a) η = 1 0 10 20 30 40 time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 ACR (b) η = 2 0 10 20 30 40 time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 ACR (c) η = 3 0 10 20 30 40 time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='4 ACR (d) η = 4 Figure 6: ACRs computed using our proposed method E(δk) (in red), the existing method ˘E(δk) (in blue), and GT-ACR (in black), for various triggering thresholds η = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='3 stationary ACR (a) Stationary ACRs 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content='9 ACR ratio (b) ACR ratio Figure 7: Comparison between the stationary ACRs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' trigger- ing threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Plot (a), our proposed method E(δ∞) (in red), and the existing method ˘E(δ∞) (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Plot (b) shows the ratio of the two stationary ACRs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' triggering threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' truncations are ignored in computing the ACR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Our proposed method and the theoretical claims are validated with a nu- merical example and an experimental study on a platooning scenario, showing that our ACR computation model precisely follows the ground truth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE5T4oBgHgl3EQfIA5T/content/2301.05445v1.pdf'} +page_content=' Biqiang 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a/qNE5T4oBgHgl3EQfJQ54/content/tmp_files/2301.05456v1.pdf.txt b/qNE5T4oBgHgl3EQfJQ54/content/tmp_files/2301.05456v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..acb232cfa79b72197656077421b836fafeba9785 --- /dev/null +++ b/qNE5T4oBgHgl3EQfJQ54/content/tmp_files/2301.05456v1.pdf.txt @@ -0,0 +1,1645 @@ +Data Quality for Software Vulnerability Datasets +Roland Croft∗†, M. Ali Babar∗†, M. Mehdi Kholoosi∗† +∗ School of Computer Science, CREST, University of Adelaide, Australia, {firstname.lastname}@adelaide.edu.au +† Cyber Security Cooperative Research Centre, Australia +Abstract—The use of learning-based techniques to achieve +automated software vulnerability detection has been of long- +standing interest within the software security domain. These +data-driven solutions are enabled by large software vulnerability +datasets used for training and benchmarking. However, we +observe that the quality of the data powering these solutions +is currently ill-considered, hindering the reliability and value of +produced outcomes. Whilst awareness of software vulnerability +data preparation challenges is growing, there has been little +investigation into the potential negative impacts of software +vulnerability data quality. For instance, we lack confirmation that +vulnerability labels are correct or consistent. Our study seeks +to address such shortcomings by inspecting five inherent data +quality attributes for four state-of-the-art software vulnerability +datasets and the subsequent impacts that issues can have on +software vulnerability prediction models. Surprisingly, we found +that all the analyzed datasets exhibit some data quality problems. +In particular, we found 20-71% of vulnerability labels to be +inaccurate in real-world datasets, and 17-99% of data points were +duplicated. We observed that these issues could cause significant +impacts on downstream models, either preventing effective model +training or inflating benchmark performance. We advocate for +the need to overcome such challenges. Our findings will enable +better consideration and assessment of software vulnerability +data quality in the future. +Index Terms—software vulnerability, data quality, machine +learning +I. INTRODUCTION +Software vulnerability detection is a vital task for achieving +secure software systems [1]. However, traditional techniques +for detecting vulnerabilities (e.g., rule-based methods) struggle +in terms of scalability and false positive rates [2]. Hence, +many researchers have been motivated to leverage the technical +advancements of Artificial Intelligence (AI) and Machine +Learning (ML) to support automatic software vulnerability +detection [3]. Recent studies have reported great success in this +direction [4]–[8], with performance that surpasses traditional +approaches [9]. We refer to these learning-based techniques as +Software Vulnerability Prediction (SVP). Like any data-driven +task, SVP is highly data-dependent. In order to learn complex +features of vulnerabilities, we require large code datasets that +have been labelled as either vulnerable or non-vulnerable. +Nonetheless, software vulnerability data collection is not a +trivial task [10]. Labelled examples of software vulnerabilities +are difficult to obtain in the real-world, as they are scarce +[11], poorly documented [12], and limited to reported vulner- +abilities [13]. Consequently, many researchers have conducted +labourious work constructing large-scale software vulnera- +bility datasets [14]–[17]. However, we found that relatively +little counterpart work has been conducted to understand +the software vulnerability data quality. Whilst data quantity +is important, an effective machine learning system requires +adequate data quality [18]. Despite the increasing realisation of +software vulnerability data preparation challenges [10], there +has been relatively little effort made to provide a systematic +understanding of how these challenges can potentially impact +data quality, and subsequently affect the reliability of down- +stream software vulnerability analysis. +A lack of understanding of data quality leads to critical +barriers to advance software assurance against vulnerabilities. +Data quality is an integral component of any data-driven +system: garbage in, garbage out [19]. Certain data biases or +misinformation can make benchmark performance results mis- +leading [20]–[22]. This can cause models to fail to generalise +to real-world scenarios [17], [23], [24], if they have not been +trained with fair and realistic data. +Thus, we set out to understand the nature of data quality for +software vulnerability datasets. To achieve this, we focused +on inherent data quality attributes that are intrinsic to the +data itself. Table I presents the five data quality attributes that +we systematically analyse: accuracy, uniqueness, consistency, +completeness, and currentness. For each attribute, we provide +a measurement of its prevalence in existing datasets and +analysis into the causes of observed issues. Our findings +revealed that even state-of-the-art datasets exhibit considerable +data quality challenges. As expected, we found these issues +to cause significant negative impacts on both training and +benchmarking of state-of-the-art SVP models. Our findings +have substantial implications: +• Data quality issues may constrain the patterns able to be +learnt by SVP models. We found that real-world datasets +can face substantial issues for label correctness of the +vulnerable class. Approximately 20-71% of vulnerability +labels were inaccurate. Furthermore, up to 47% of labels +were inconsistent. These issues cause models to learn +false or insufficient patterns. +• Software vulnerability benchmark datasets may lead to +inflated performance. A few datasets exhibited large data +duplication rates, between 17-99%. Hence, data leakage +causes models to report inflated performance using stan- +dard test setups. Evaluation performance decreased by up +to 82% after removing such duplicates. +To achieve the most reliable results from learning-based +software vulnerability analytics, we must consider and address +the issue of data quality. Whilst some of the observed quality +issues can be solved easily through rule-based detection and +arXiv:2301.05456v1 [cs.SE] 13 Jan 2023 + +TABLE I +INHERENT DATA QUALITY ATTRIBUTES DEFINED BY ISO/IEC 25012 [25]. +Attribute +Definition +Interpretation +Accuracy +The degree to which the data has attributes that correctly represent the true value of the intended attribute of a +concept or event. +Correct labelling. +Uniqueness +The degree to which there is no duplication in records. +No duplicate values. +Consistency +The degree to which data has attributes that are free from contradiction and are coherent with other data. +Consistent labelling. +Completeness +The degree to which subject data associated with an entity has values for all expected attributes and related instances. +No missing values. +Currentness +The degree to which data has attributes that are of the right age. +Not obsolete data. +removal, others cannot be remediated in this manner. As a +community, we must focus on developing knowledge and tools +for constructing high quality software vulnerability datasets. +Our contributions are twofold: +1) We provide understanding of the nature and causes of +observed data quality issues in software vulnerability +datasets. We also demonstrate the corresponding impact +of these issues for software vulnerability prediction. Such +investigation highlights the problem of data quality and +the need to mitigate these challenges. Furthermore, our +insights help overcome data quality issues, for which we +have provided directions in the discussion. +2) We propose and conduct methods for measurement of +data quality for software vulnerability datasets. These +efforts can enable practitioners to consider and perform +data quality assessment. We have provided a reproduction +package of this study to assist with such efforts [26]. +II. BACKGROUND AND MOTIVATION +A. Software Vulnerability Data Preparation +Software Vulnerability Prediction (SVP) models use pro- +gram analysis techniques to learn software vulnerability pat- +terns automatically from historical examples [27], [28]. Due to +the unstructured nature of source code, researchers have found +the most success using Deep Learning (DL) techniques that +learn from program syntax and semantics [3], [7], [8], [24]. +Hence, SVP is a data-hungry process [16]: the models +require a large training dataset of annotated code modules, +labelled as vulnerable or non-vulnerable. However, acquiring +a reliable vulnerability label source is a non-trivial task; there +is no oracle that can unfailingly prove the existence or absence +of vulnerabilities from a codebase [29]. Thus, researchers have +relied on a variety of label sources to account for different +shortcomings. Following prior analysis [10], [24], we outline +four main label categories: +• Security Vendor Provided. Security vendors maintain +vulnerability databases that aggregate information from +various advisories. This provides a standardised collec- +tion of disclosed vulnerabilities. Examples include the +National Vulnerability Database (NVD) [30], or the Snyk +Vulnerability Database [31]. Vulnerability records often +provide links to patches, which can then be traced to +identify real-world source code and vulnerabilities. +• Developer Provided. Vulnerability databases may not +properly document all vulnerabilities of a project [32]. +Hence, researchers may collect vulnerability fixing com- +mits directly from developers via the development history +or via a project’s issue tracking systems. However, this +method requires additional effort to search development +artefacts for security-related defects. +• Tool Created. Developer or security vendor-provided la- +bels have a major limitation of only collecting reported +vulnerabilities, which severely limits the number of ex- +amples that can be collected. In reality, vulnerabilities can +remain latent or undetected [33], which limits the dataset +size and adds considerable label noise to the modules. +To circumvent this, some researchers have utilised static +security analysis tools to automatically produce labels +for the source code [16], [34], [35]. This process relies +heavily on the accuracy of the static analyser used, which +is a source of contention [15]. +• Synthetically Created. Finally, to bypass the limitations +of other label sources, vulnerable code examples and +annotations can be created artificially from known vul- +nerable patterns. Synthetically producing entries ensures +label correctness at the cost of source code diversity [24]. +Despite these caveats, researchers have faced data prepa- +ration challenges regardless of the selected dataset [10]. +Whilst state-of-the-art SVP models report good performance +on benchmark datasets, performance only measures the ability +of a model to fit a particular dataset. A good performance +value does not guarantee that a model will generalise to +real-world scenarios [36]. Hence, data quality issues will +hinder the reliability and trustworthiness of the outcomes. For +instance, previous studies [13], [33] have highlighted inflated +performance due to inaccurate labelling mechanisms for non- +vulnerable modules. Consequently, the industry value and +adoption of SVP models is uncertain [37], [38]. Our study +seeks to shed light on the state of software vulnerability data +quality so that we better understand the reliability and trust- +worthiness of the reported outcomes that use these datasets. +B. Data Quality in Software Engineering Research +Training data is an integral component of ML systems that +heavily influences the produced models. Unlike conventional +software systems, ML systems exhibit both system and data +requirements [18]. As a result, data quality is becoming +an essential component of AI-based Software Engineering +research [39]. Software engineering data and artefacts are +often noisy as they are usually collected post-hoc via mining +software repositories [40]. The data and labels are not gener- + +ated explicitly for the purposes of research. Hence, software +engineering data has been found to exhibit issues with data +accuracy, relevance, and provenance [40]. +Existing studies that investigate data quality characteristics +of software engineering datasets are currently non-systematic +and limited. Data quality can be defined by a large range of +dimensions, like those defined in Table I. Existing studies often +limit their analysis to semantic or syntactic data accuracy and +noise [22], [41]–[45]. Similarly, Jimenez et al. [33] considered +label accuracy within software vulnerability datasets. However, +this approach fails to provide a complete picture. To make +informed data decisions, there is a need for a systematised +and objective investigation of data quality in the software +engineering domain. Croft et al. [10] conducted a systematic +literature review of the data quality issues considered by SVP +researchers. Whilst this work provides a systematised view, +the observations are unsubstantiated with respect to actual +software vulnerability datasets and SVP models. Our work +purports to perform quantitative analysis of data quality within +software vulnerability datasets and explicitly show its impacts +on SVP models. +Additionally, researchers have recently begun constructing +automated cleaning frameworks to reduce the observed data +noise issues and ensure correctness [22], [45]. These frame- +works are grounded in a deep understanding of the data quality +issues afflicting the relevant datasets. We expect the findings +obtained in our study to enable the creation of a cleaning +framework for software vulnerability datasets. +III. STUDY DESIGN +To understand the data quality of the state-of-the-art soft- +ware vulnerability datasets, we address two Research Ques- +tions (RQs) through the analysis of several data quality +attributes. Figure 1 displays the overall workflow used to +conduct this study. We describe each of the three processes +in Sections III-B, III-C and III-D, respectively. +A. Research Questions +Our investigation is guided by the following RQs: +Processes +Data Quality +frameworks +Artefacts +SVP Models +SV Datasets +Rationale +RQ1 +Measure Attributes +Analysis +RQ2 +Validate Attribute +Impact +Impact +Identify Data Quality +Attributes +Fig. 1. The overall study design. +• RQ1: What data quality issues are present in the state- +of-the-art software vulnerability datasets? We firstly +aim to inform practitioners of the state and nature of data +quality for software vulnerability datasets. +• RQ2: To what extent do data quality issues impact +downstream software vulnerability prediction models? +Our second aim is to verify the importance of data qual- +ity. We attempt to demonstrate the potentially negative +impact that observed data quality issues can have on the +downstream tasks for software vulnerability prediction. +B. Identifying Data Quality Attributes +The lack of existing data quality consideration for software +vulnerability datasets has perhaps been caused by the lack of +systematic definitions for data quality and hence measurement. +It is not easy to define data quality, due to its many dimensions +[46]. Not all dimensions may be relevant to a specific scenario +[47], and different organisations would value quality attributes +differently [18]. For instance, not all organisations would +prioritise data confidentiality. There are two main categories +of data quality attributes [25]: inherent data quality, which +intrinsically relates to the data itself, or system-dependent +data quality, which extrinsically arises from external fac- +tors and requirements. For this study, we focused on purely +inherent data quality attributes, as SVP models have not +yet achieved widespread industrial application [37]. System- +dependent attributes cannot be properly measured without +an associated deployment context. In this sense, we focused +on the data rather than how the data is used. Hence, our +findings are not constrained to particular modelling techniques +or features. +To identify inherent data quality attributes, we used the +standardised data quality framework ISO/IEC 25012 [25]. +This framework has been used for data quality assessment +in both the Software Engineering [47] and Machine Learning +[18], [48] domains. ISO/IEC 25012 outlines five inherent data +quality attributes: Accuracy, Consistency, Completeness, Cur- +rentness, and Credibility. We excluded the credibility attribute +as it is difficult to quantify in existing software vulnerability +datasets. Credibility indicates the level of trust that we have +in a dataset; the authenticity of the data source or supplier. We +need to ensure that our data points are free from contamination +or fake information [36]. As datasets have been produced by +peer-reviewed research or respected government organisations +[10], we assume a level of trust in the data source and supplier +of each dataset. Additionally, we considered a uniqueness +data dimension, due to its prevalence in existing software +engineering research [49], and its importance as highlighted +by previous SVP researchers [24]. Table I summarises the +selected inherent data quality attributes for analysis. +C. Measuring Attributes +For the analysis, we collected one dataset of each label +source described in Section II-A. To ensure the collected +datasets represented the state-of-the-art appropriately, we con- +sidered datasets that were created or used by a conference or + +TABLE II +SELECTED STATE-OF-THE-ART DATASETS FOR EXAMINATION. +Dataset +Label source +# Functions +% Vul +Big-Vul [14] +Security vendor provided +188,636 +5.78 +Devign [15] +Developer provided +27,318 +45.61 +D2A [16] +Tool created +1,295,623 +1.44 +Juliet [50] +Synthetically created +253,002 +36.77 +journal paper published in a high quality venue, as indicated +by a CORE1 ranking of A or A*. New datasets continue to be +published in order to improve on previous dataset shortcom- +ings. We selected the most recently published datasets as of +March 2022. We also chose datasets that contained appropriate +metadata about how the labels were obtained. Table II displays +our selected datasets. All four of the examined datasets provide +source code for C/C++ functions. +• Big-Vul [14] scraped software versions prior to a vulner- +ability fix through linked patches from the CVE Details +database. Functions with lines changed in a patch were +labelled as vulnerable. All remaining functions in a file +touched by a commit were labelled as non-vulnerable. +• Devign [15] used a similar data collection method by +scraping vulnerability fixes directly from GitHub com- +mits. A keyword approach was used to separate vul- +nerability and non-vulnerability related commits. The +vulnerability-related commits were then filtered manually +to ensure accuracy. All relevant functions of each commit +were collected for their respective classes. +• D2A [16] collected source code by running a static anal- +ysis tool on project versions before and after bug fixing +commits of six open source repositories. The vulnerable +class was formed from tool warnings that disappeared in +the post-fix version. The non-vulnerable class consists of +the remaining tool warnings. Each data entry indicates +a function containing the original vulnerability location. +We retrieved all such functions to form the D2A dataset. +• Juliet [50] contains synthetically generated examples of +programs demonstrating a variety of known vulnera- +ble code patterns. Programs are generated automatically, +based on pre-defined augmentation rules. Each program +contains a vulnerable version and non-vulnerable version. +Juliet was originally created to test static and dynamic +security tools, but it has also been used for training SVP +models. Source code data is provided in files with func- +tions being annotated as vulnerable or not. We retrieved +all functions from each annotated section of the data. +We followed the measurement practices specified by Naka- +jima and Nakatani [48] for using the ISO/IEC 25012 frame- +work with respect to AI training data requirements. Table I +describes the interpretation of each attribute. We identified the +percentage of samples in a dataset that satisfy the relevant +characteristics to produce an overall measurement. Hence, +the measurement value for each attribute lies between 0 and +1, with 1 indicating no data quality issues are present. We +1http://portal.core.edu.au/conf-ranks/, http://portal.core.edu.au/jnl-ranks/ +formally define this measurement in Equation 1, where N +denotes the number of samples in a dataset and dq(i) returns +1 if a data entry i satisfies the relevant characteristic. Further +details of the measurements for each individual attribute are +provided later in Section IV. +Attribute = +N +� +i=1 +dq(i) +N +(1) +D. Validating Attribute Impact +Finally, to validate the impact of the observed data quality +issues, we investigated the performance impacts on a state-of- +the-art SVP model. Depending on the observed data quality +issues, we either measured the performance change on a +retrained model after mitigating data quality issues or altered +the test setup to highlight the data quality characteristic of +focus. We provide further details in Section IV. +For the benchmark performance, we trained a model on each +dataset, without any pre-processing of the data. However, we +removed inconsistent entries for the D2A benchmark, as we +were otherwise unable to produce an effective classifier for +this dataset. We ran all experiments five times using random +80:10:10 training/validation/test splits unless otherwise speci- +fied, as this is a standard test setup in prior research [6]–[8]. +We selected the LineVul SVP model [7], as it is a re- +cently published model that has been shown to outperform +all previous baselines for both function level and line level +predictions. LineVul [7] relies on CodeBERT [51] to obtain +code feature representations that capture lexical and logical +semantics. CodeBERT is a pre-trained state-of-the-art code +embedding model based on the RoBERTa architecture [52]. +Similar studies have demonstrated the effectiveness of Code- +BERT for SVP [8], [13]. LineVul generates function-level +predictions using a transformer-based architecture. Although +LineVul also has the capability to localise its predictions to +the line-level after performing the function-level prediction, +all of the selected datasets provide labels at the function-level. +Hence, we perform prediction at the function-level granularity. +We evaluated model performance using Recall, Precision +and Matthews Correlation Coefficient (MCC). We opted to +use MCC as an overall indicator of performance, as its use +has been recommended for similar tasks [53]. MCC values +range between -1 and 1, with 1 being the optimal value. +IV. DATA QUALITY ANALYSIS +Table III displays the attribute values for each dataset. +A. Accuracy +Rationale. Accuracy defines the correctness of the data +points that comprise a dataset. This largely relates to the +semantic label correctness; i.e., whether or not data points +labelled as vulnerable or non-vulnerable genuinely align. It +has previously been observed that non-vulnerable labels are +unreliable in real-world datasets as there is no ground truth +label source for this class [10], [13], [33]. No oracle can +reliably ensure the security and absence of exploits in a + +given code snippet. Hence, non-vulnerable labels are usually +collected simply through the absence of a vulnerable label. +Thus, our analysis was constrained to the vulnerable label +source. We focused our investigation on label correctness of +data points labelled as vulnerable. +Analysis. We determined if a label is correct via manual +analysis with respect to each dataset’s labelling mechanism: +whether a vulnerability accurately represents the vulnerability +report or static analysis tool warning that it was derived from. +In this sense, we did not verify whether a vulnerability was +actually exploitable, but rather whether a code snippet is +functionally relevant to the reported vulnerability of each label. +The following steps were taken to assess the label correctness +of each entry: +1) We first extracted information relating to the vulnerability +and fixing commit of each dataset. All datasets provided a +git fixing commit ID except Juliet. Big-Vul also provided +CVE-IDs and D2A contained the static analysis tool trace. +2) We read the fixing commit description and other available +information (i.e., the vulnerability description from NVD +for Big-Vul and the static tool trace for D2A) to gain an +understanding of the vulnerability and the fixing commit +changes. +3) We then examined the changed lines in the fixing commit +for the relevant function, as well as the entire function’s +code to understand the context of the changed lines. +Based on this code comprehension, we made an assess- +ment as to whether the changed lines were functionally +relevant to the information from the previous step. +4) If we did not interpret them as functionally relevant, we +examined all the fixing commit changes to identify where +the root changes were to understand why the flagged +function was not relevant. +5) Afterwards, the authors discussed the labels that were in +disagreement and reached a consensus. +To facilitate our manual review, we examined 70 random +samples of each dataset (90% confidence level +/- 10% [54]). +Two of the authors of this paper conducted this manual +analysis independently; each of them had two to five years of +software security-related experience gained in academia and +industry. The two raters achieved a Cohen Kappa value of +0.627 [55], which implies moderate to strong agreement. +Our findings revealed that label inaccuracy occurred within +the real-world datasets. We obtained accuracy values of 0.8 +(Devign), 0.543 (Big-Vul), and 0.286 (D2A). We found no +TABLE III +MEASURED VALUE OF EACH ATTRIBUTE FOR EACH DATASET. +Attribute +Dataset +Big-Vul +Devign +D2A +Juliet +Accuracy* +0.543 +0.800 +0.286 +1.000 +Uniqueness +0.830 +0.899 +0.021 +0.163 +Consistency +0.999 +0.991 +0.531 +0.750 +Completeness +0.824 +0.944 +0.981 +1.000 +Currentness +0.761 +0.811 +0.844 +- +* Based on a sample of the data. +inaccuracies within the synthetic Juliet dataset, as the vulner- +able cases are crafted specifically for the label rather than +collected post-hoc. Real-world labelling works by tracing a +vulnerability identifier (usually a vulnerability fix or warning) +to the original code snippet. The two authors who conducted +the manual labelling noted their reasoning behind a label being +correct or incorrect. We conducted a thematic analysis [56] of +the label reasoning to identify the causes of dataset inaccuracy. +Table IV displays the proportion of each theme. +• Irrelevant code changes. The real-world datasets largely +assume that code touched by a vulnerability fix is vulner- +able code. However, a vulnerability fixing commit may +not necessarily provide a patch alone. Non-functional +changes, such as style changes, refactoring and code +migration can confuse the data labelling process. For +instance, this example fixing line2 simply converts a +constant value to the equivalent macro. Similarly, tan- +gled commits can implement other irrelevant changes in +parallel [57], which will be misinterpreted as vulnerable +code. +• Cleanup changes. Vulnerability fixes can sometimes +be large and disparate due to the complexity of code. +Tertiary changes can be made in a commit to help better +facilitate a vulnerability fix, such as adding, deleting or +altering variables, functions or parameters. For example, +in this fixing commit3 example a vulnerability occurs +for when read_only is set as True rather than a +protected memory object. The cleanup change converts +False read_only values to a nullptr, simply to +avoid confusion. These are functional changes that relate +to the vulnerability fix, so we do not consider them as +irrelevant. Nonetheless, they do not indicate the location +of the underlying exploitable code, and hence produce +false positive labels. We call these cleanup changes, +although they have also been referred to as casualty +changes by Sejfia et al. [58]. +• Inaccurate vulnerability fix identification. If the la- +belling mechanism fails to identify a vulnerability fix, the +subsequent code snippet will naturally not be a vulnera- +bility. Datasets like Big-Vul that trace vulnerability fixes +from external vulnerability reports can introduce errors +into this process. For instance, we found the majority +of vulnerability reports for the Chromium project to be +improperly traced as this repository is not naturally hosted +via GitHub. Furthermore, datasets that attempt to identify +vulnerability fixes directly from commit history (Devign +2https://github.com/FFmpeg/FFmpeg/commit/8b2fce0d3f5a56c40c28899c9237210ca8f9cf75 +3https://github.com/chromium/chromium/commit/673ce95d481ea9368c4d4d43ac756ba1d6d9e608 +TABLE IV +TYPES OF LABEL INACCURACY IN REAL-WORLD DATASETS. +Dataset +Irrelevant +Cleanup +Inaccurate +Big-Vul +25% +28.1% +46.9% +Devign +42.9% +21.4% +35.7% +D2A +0 +0 +100% + +and D2A) can also be rife with errors. Researchers +usually attempt to identify these commits through inac- +curate and unreliable keyword matching methods. Lastly, +D2A uses additional help from static analysis tools to +identify vulnerability fixes. These tools produce many +false positive vulnerability warnings. +Mainly, we observed tangled commits to cause problems +for current real-world data labelling heuristics [57]. Current +datasets assume vulnerable code to be all code touched in +a vulnerability fix, but commits are messy in practice [59]. +Similarly, vague, generic or unclear commit messages can +make vulnerability fix identification difficult [12]. In contrast, +correct vulnerability labels typically stem from simple, focused +and well-defined vulnerability fixing commits. +Additionally, the datasets included samples for which we +found it difficult to verify or agree upon the label. This +often occurred when the location of a vulnerability falls in +a grey area. For instance, should the caller of a vulnerable +code snippet also be labelled as such? Herbold et al. [60] +encountered similar problems in their investigation of tangled +commits. Alternatively, the label source may not contain +enough information in the bug report to properly trace it. +We tentatively labeled these ambiguous cases as correct. +However, the software security domain should work towards +clear definitions that prevent such ambiguous cases, to help +with ensuring label correctness. +Devign did not exhibit as many issues, as it is the only +dataset for which the creators attempted to perform manual +validation of the fixing commits. However, this accuracy as- +surance comes at the cost of data size. Devign is the smallest of +the datasets, due to the strenuous efforts of manual validation. +Nonetheless, Devign still exhibits some inaccuracies. The +majority of the errors came from irrelevant changes, such as +refactoring or code migration, which may imply the original +authors did not check for such things. +The accuracy for Big-Vul was lower, as many of the +vulnerability fixing commits used during data extraction for +this dataset were large, tangled or noisy. Most errors arose +from inaccuracies in tracing the fixing commits, particularly +for the Chromium project. 36% of the vulnerable entries in +Big-Vul are from the Chromium project. +Over two-thirds of the D2A labels were inaccurate. We +found that this was primarily due to the static analysis tool +warnings being unreliable, as well as the vulnerability commit +identifier being inaccurate. The majority of commits flagged +by the D2A data extractor were not actually vulnerability +fixes, as the context of the security-specific words was often +misinterpreted. For instance, not all commits that contained +the word “memory” were necessarily fixing unsafe memory +operations. The majority of static tool warnings were also false +positives. Static analysis tools often output an indication of the +reliability of a warning, based on how confident the tool is. For +example, a confident integer overflow warning would know the +integer data type and variable values, whereas an unreliable +report may know neither. Over 97% of the static analysis +warnings included in D2A are from the lowest reliability +warning class, making them often inaccurate. However, as +the static analysis tools attempt to infer the location of the +vulnerability directly, there were no false positives caused by +irrelevant or cleanup code changes. +Impact. To evaluate the impact of inaccurate labels, we +retrained each model using our manually-validated samples of +each dataset as a separate holdout test set. We measured model +performance when using the original labels in comparison +with the manually-corrected labels. We could not measure +MCC as the test set had no samples that were originally +labelled as non-vulnerable. The precision decreased by 29%, +50% and 80% for Devign, Big-Vul and D2A, respectively, +which we confirmed to be significant using a Mann-Whitney +U test [61] (p < 0.05). This was because incorrect vulnerable +labels caused the models to infer incorrect patterns for this +class. The models were taught vulnerable patterns that were +actually non-vulnerable. Hence, in terms of model evaluation, +what were previously considered true positives became false +positives. Correspondingly, we found that the model recall +was not significantly affected (using a Mann-Whitney U test +[61]) as we only uncovered label inaccuracy for the vulnerable +class; the number of false negatives was unchanged. These +impacts are still significant however, as they can lead to high +false positive rates in models which would greatly increase +inspection efforts during practical use. +Accuracy is limited for some real-world datasets due to +their reliance on noisy and hard-to-identify vulnerability +fixing commits. Accuracy issues cause SVP models to infer +the wrong patterns between classes. +B. Uniqueness +Rationale. Uniqueness is not necessarily an intrinsic data +property, as a real-world data distribution may contain dupli- +cated samples. However, code duplication has been demon- +strated to have adverse effects on trained models [49]. Dupli- +cates can introduce bias in a model towards certain samples. +Inflated performance values can result when duplication occurs +between the training and test sets [24]. Hence, ensuring +uniqueness of samples within a dataset helps models generalise +towards a true data distribution [62]. Consequently, we treat it +as an inherent attribute and decided to investigate the impacts +that a lack of uniqueness would have for SVP. +Similar or identical code fragments are defined as code +clones [63], of which there are four main types [64]: +1) Type-1: Identical code fragments, except for differences +in white-space, layout and comments. +2) Type-2: Identical code fragments, except for differences +in identifier names and literal values, in addition to Type- +1 clone differences. +3) Type-3: Syntactically similar code fragments that differ at +the statement level. The fragments have statements added, +modified and/or removed with respect to each other, in +addition to Type-1 and Type-2 clone differences. +4) Type-4: Syntactically dissimilar code fragments that im- +plement the same functionality. + +We followed standard practices and considered type-3 code +clones as duplicates [49]. Even functionally similar code +fragments will include duplicated patterns and tokens that +can adversely affect the model performance and evaluation. +However, for software vulnerability datasets, slight functional +changes can form the difference between a vulnerable and +non-vulnerable label. A typical vulnerability fix only alters a +few lines of code [65]. It is important that a model is able +to capture these slight functional differences across prediction +classes to avoid excessive false positive or false negative rates +[24]. Hence, we only considered duplicates with the same +labels (vulnerable or non-vulnerable) as code clones. +Analysis. An entry is not unique if it is a code clone of +any other entry of the same label. To identify code clones, we +reused the code duplicate detector tool produced by Allamanis +[49]. We lowered the minimum token count of a sample to +five, as functions are smaller than the files for which this tool +was originally built. This tool outputs clusters of duplicates, +as there can be more than one duplicate per function. +We observed code duplication to occur within all the +datasets, but less frequently for the Big-Vul and Devign +datasets. We obtained a uniqueness value of 0.830 (Big-Vul), +0.899 (Devign), 0.021 (D2A), and 0.163 (Juliet). We manually +examined a sample of 30 random duplicate clusters for each +dataset (74 functions for Big-Vul, 79 functions for Devign, 210 +functions for Juliet, 2288 functions for D2A) to understand +why duplicate code entries are present. Using thematic analysis +[56], we observed three main causes of code duplication in +real-world datasets: +• Updated code. All real-world datasets collect code from +multiple versions of the same code repository in order to +maximise the number of vulnerabilities observed. Across +the versions, subtle functional or non-functional updates +to the code introduce predominantly duplicated code +snippets. For vulnerable cases, these types of duplicates +can imply that the code update either failed to fix the +vulnerability or introduced a new one. +• Similar function sets. A code file may contain a suite +of simple modular functions. These functions are often +identical in terms of variable names, logic, and layout +but have slight functional differences. For example, two +functions may be implemented to start and stop a process +respectively, or a set of functions may each perform a +unique mathematical operation on a data flow. +• Renamed functions. Identical functions may be dupli- +cated and renamed for use in different files and contexts. +We illustrate these causes in Figure 2. These factors are +inherent in source code datasets due to both the spatial and +temporal repetitiveness of code in software repositories. +We found duplication to be especially significant for D2A. +Each unique function in the dataset had an average of 57 +duplicates. This is because D2A produces label information +at the line level, which is then abstracted to the function +scope. The same function can be included multiple times if +unique lines are flagged. Hence, the D2A labelling process +introduces many additional exact duplicates. Over 94% of the +TABLE V +PERFORMANCE IMPACT OF UNIQUENESS ISSUES. +Dataset +With Duplication +Without duplication +Change +Precision +Recall +MCC +Precision +Recall +MCC +(MCC) +Big-Vul +0.920 +0.765 +0.830 +0.922 +0.762 +0.829 +0.0% ↓ +Devign +0.680 +0.428 +0.284 +0.651 +0.399 +0.244 +13.9% ↓ +D2A +0.961 +0.630 +0.774 +0.741 +0.049 +0.141 +81.7% ↓ +Juliet +0.939 +0.945 +0.909 +0.962 +0.799 +0.814 +10.4% ↓ +D2A dataset were type-1 code clones. Furthermore, two of +the six repositories that comprise D2A are forks of each other +(FFmpeg and Libav), which led to further duplication. The +lack of uniqueness for D2A questions the claim of the dataset’s +size; there is limited information at the function level for this +dataset. +We also found a large number of duplicates in Juliet, due to +the subtlety in the variance of the test cases. New test cases are +produced by making slight changes to the control flow logic, +internal function calls, or literal values. Furthermore, the non- +vulnerable fixed statements can exhibit exact duplication due +to having a constant corrected implementation. +Impact. For SVP, duplicates can appear in the training +set, test set, or across these two sets. Figure 3 illustrates +these duplicate types. In-train duplicates may produce model +biases [62], but it is hard to measure these aspects via model +performance [49]. We focus our analysis on the impact of +uniqueness for model evaluation. We split each dataset into a +training, validation and test set, as specified in Section III-D. +We then compared the evaluation performance of the model +when cross-set duplicates to the test set were either removed +or kept. Allamanis [49] found cross-set duplication to be the +most significant type in software engineering research. +Table V displays the performance change for SVP models +when we removed the identified duplicate entries. We observed +that cross-set duplication is a significant factor for some +datasets as the overall evaluation results (MCC) decreased +for Devign, D2A and Juliet, which we confirmed to be +significant using a Mann-Whitney U test [61] (p < 0.05). The +model trained with Big-Vul data was not significantly affected. +This implies that a lack of uniqueness may not always be +problematic. +Duplicates can allow for data leakage in the evaluation setup +[49]; the models can trivially classify samples in the test set +that are also duplicated in the training set, inflating the true +performance. We observed that duplication had a larger neg- +ative influence on recall rather than precision for all datasets. ++ func view_obj2() ++ func view_obj1() ++ func add_fileY_var() +File Y ++ func do_operation_X() ++ func add_fileX_var() +File X +COMMIT 1 ++ func do_operation_X() +File X +COMMIT 2 += Updated Function += Renamed Functions += Similar Function Sets +Fig. 2. An example of the three main code duplicate causes. + +Train Set +Test Set +Cross-Set +Duplicates +In-Train +Duplicates +In-Test +Duplicates +Fig. 3. Types of duplicates for ML models, adapted from Allamanis [49]. +The removal of cross-set duplicates removed trivial samples +from the test set, primarily lowering true positives. This had +a larger impact on recall, due to the higher ratio of false +negatives in comparison to false positives. Recall significantly +decreased for Devign (7% decrease), D2A (92% decrease) +and Juliet (15% decrease) (confirmed using a Mann-Whitney +U test [61]), whereas precision actually even increased after +duplicate removal for Big-Vul and Juliet. However, the overall +performance (MCC) still decreased for each dataset other than +Big-Vul. +Uniqueness issues are present within all datasets due to the +repetitive and incremental nature of code. Duplicate code +snippets can potentially inflate overall evaluation perfor- +mance due to data leakage. +C. Consistency +Rationale. Consistency denotes that data entries should +not provide conflicting information. For software vulnerability +datasets, this simply implies that similar code snippets should +not have conflicting labels. A piece of code cannot be both +vulnerable and non-vulnerable. Inconsistency can arise in +software vulnerability data however, due to the multiple data +streams that are used to construct a dataset [66]. Consistency is +understandably important for model training and construction, +as conflicting labels confuse any AI-based model that is +attempting to distinguish between two classes. +Consistency is related to the uniqueness attribute as we +again examined duplicated data. However, consistency mea- +sures duplicated entries with conflicting labels. As slight +functional changes can form the functional difference between +a vulnerable and non-vulnerable code snippet, we only con- +sidered type-1 code clones (exact matches). +Analysis. An entry is consistent if it does not have any +duplicates with conflicting labels. We observed high consis- +tency values for Big-Vul (0.999) and Devign (0.991), but +lower values for D2A (0.531) and Juliet (0.75). We manually +examined a random sample of 30 inconsistent clusters to +determine reasons for inconsistent vulnerability labels. We +found that the causes of inconsistent labels were fairly unique +to each data collection approach, which we discuss below. +For Big-Vul, inconsistent labels were produced by latent +vulnerabilities that existed within the source code. The la- +belling heuristic of this dataset assumes that all functions in +the files of a commit that were not explicitly touched are +non-vulnerable. However, these functions can actually contain +vulnerabilities unknown to developers. These vulnerabilities +can be reported and then collected at a later date. Figure 4 +illustrates this process. Although the number of inconsistent +cases is relatively small, these are only the latent vulnera- +bilities we know about. In reality, complete knowledge of +the latent vulnerabilities is unobtainable. Croft et al. [13] +observed at least twice as many latent vulnerabilities as known +vulnerabilities in their dataset. +In the Devign dataset, inconsistencies occurred due to +simultaneous code branches. The vulnerability fixing commit +may only be identified in one branch, leaving the same +commits in other branches to be treated as non-vulnerable. +This primarily occurs due to merging commits on branches, +as merged commits can contain vulnerability fixes but are not +described as such. Like the inconsistent labels of Big-Vul, this +implies there are incorrect labels for the non-vulnerable class, +as the other branch commits are improperly identified. +The static analysis tools that inferred the labels of the +D2A dataset produce an excessive number of warnings. All +functions are scanned over every analysed commit during the +D2A data extraction. Hence, the same function can receive +the same warning from the static analysis tool over different +commits. If one of the commits edits the flagged lines whereas +the others do not, then inconsistent labels will be introduced. +We found this occurred commonly in practice, as demonstrated +by the relatively low consistency value of this dataset. +The Juliet test cases can include tertiary functions that +perform unsafe operations, e.g., writing data to a buffer. +Test cases are set up like this to help test the ability of +vulnerability scanning tools to track data flow across functions. +Although these tertiary functions are vulnerable as they lack +the necessary security checks, an exploit will only occur when +specific values are passed to them. As a result, duplicate copies +of these functions are contained in both the vulnerable and +non-vulnerable annotated sections of this dataset. +From the analysis, we observe that inconsistent samples +primarily point to inaccuracies within the data collection +processes for the non-vulnerable class. This is due to a lack +of proper label sources or checks for this class; it is formed +from the absence of vulnerability labels. +Commit +1 +Commit +2 +Commit +3 +Commit +i +Commit +i+1 +Vulnerability Introducing +Commit for func b() +Vulnerability Fixing +Commit for func a() +Vulnerability Fixing +Commit for func b() ++ func a() +  func b() +Changes +Fixed +Unchanged ++ func b() +Changes +Fixed +Fig. 4. An example of inconsistency introduced from latent vulnerabilities. +Function b is vulnerable in commit 3 until commit i+1, but it is only recorded +as such for the latter. + +TABLE VI +PERFORMANCE IMPACT OF CONSISTENCY ISSUES, WITH COMPARISON TO ORIGINAL DATA SETUPS. +Dataset +All inconsistent (original) +Consistent test set +Consistent train & test set +Precision +Recall +MCC +Precision +Recall +MCC +Precision +Recall +MCC +Big-Vul +0.902 +0.774 +0.826 +0.919 (↑) +0.774 (-) +0.835 (↑) +0.915 (↑) +0.775 (↑) +0.833 (↑) +Devign +0.625 +0.569 +0.285 +0.668 (↑) +0.500 (↓) +0.311 (↑) +0.653 (↑) +0.502 (↓) +0.289 (↑) +D2A +0 +0 +0 +0 (-) +0 (-) +0 (-) +0.948 (↑) +0.599 (↑) +0.748 (↑) +Juliet +0.937 +0.950 +0.910 +0.998 (↑) +0.985 (↑) +0.987 (↑) +0.999 (↑) +0.999 (↑) +0.999 (↑) +Impact. Like uniqueness, inconsistency can appear within +the training set, test set, or across these two sets, as depicted in +Figure 3. Training set inconsistency would affect the patterns +learnt by the model, whereas test set inconsistency would +affect model evaluation. We considered both of these aspects +in our impact analysis experiments. We removed inconsis- +tency via entries from the non-vulnerable class of inconsistent +clusters, as our manual analysis found these non-vulnerable +entries to be incorrect. Using the experimental setup described +in Section III-D, we considered three scenarios: the original +case when all inconsistent examples are retained, a consistent +test set in which all within-test and cross-set inconsistencies +are removed but the training set remains inconsistent, and an +entirely consistent dataset in which all inconsistent entries are +removed. We trained and evaluated a model for each setup. +Table VI displays the performance impact. We observed +inconsistency to potentially have an effect on model evaluation +as MCC performance increased when using consistent test +sets. This is because a model will naturally make the same +prediction for identical inputs, producing wrong predictions +for a portion of the inconsistent entries. Hence, inconsistent +samples hinder performance as the lack of distinguished labels +either prevent the models from inferring important patterns or +causes them to bias toward an incorrect class label. In the case +of D2A, inconsistency was so prevalent that the model fails to +make any correct predictions unless training with a consistent +training set. We observed the model would default predictions +to the most prevalent label of an inconsistent cluster; which is +the non-vulnerable class in the case of D2A. Using a Mann- +Whitney U test [61] (p < 0.05), we confirmed that removing +inconsistency issues significantly improved performance for +the most afflicted datasets (D2A and Juliet). +We observed that increased consistency has a larger positive +influence on precision in comparison to recall. Recall actually +even decreased when using consistent datasets for Devign +(although the overall performance still increased). This is +likely because inconsistent clusters more often produce false +positives, due to the larger number of non-vulnerable samples +in each dataset. +Performance impacts were relatively small for Big-Vul and +Devign, due to the relatively small number of affected entries. +We were unable to confirm whether the performance changes +using these datasets were statistically significant. However, +we expect that these inconsistencies actually point to larger +problems in the non-vulnerable classes of these datasets. There +is likely to be a much larger number of latent vulnerabilities +or misclassified fixing commits, but we only observe a low +TABLE VII +FREQUENCY FOR TYPES OF MISSING VALUES IN DATASETS. +Dataset +Truncation +Empty +Declaration +Total +Start +End +Both +Big-Vul +32,973 +133 +140 +0 +0 +33,246 +Devign +814 +265 +9 +0 +0 +1,088 +D2A +0 +0 +0 +10,824 +13,300 +24,124 +number via inconsistent labels. Both Jimenez et al. [33] and +Croft et al. [13] found mislabelled latent vulnerabilities to +impact downstream SVP models significantly. Data collection +processes must be improved to ensure consistency. +Consistency issues arise due to a lack of label indicators or +checks for non-vulnerable code. Whilst measured values are +small; they may be an indicator of more significant prob- +lems. Consistency can be a significant issue that prevents +the model from learning necessary patterns. +D. Completeness +Rationale. Completeness can either refer to the complete- +ness of information within a dataset, or to the values of indi- +vidual data entries. As the former requires external reference +information, we focus on the latter as it is an inherent property +of the data. For vulnerability datasets, source code can be +missing information if the values do not contain all the code +of the original function. +Analysis. To detect missing information, we automatically +checked for incomplete code snippets by analysing the C/C++ +function syntax. We found that some code entries were missing +or cut off. Overall, we observed completeness values of +0.824, 0.944, 0.981, and 1.0 for Big-Vul, Devign, D2A and +Juliet, respectively. These relatively high values imply that +completeness is less frequently problematic than the other data +quality attributes. Missing information was only present in +three of the four analyzed datasets. Table VII displays the +frequency of the truncation types present in each dataset. We +have excluded Juliet because none of its entries contained +missing information. +We found truncation at the start of functions to occur +predominantly in the Big-Vul dataset. Return types of func- +tion definitions were truncated when they were defined over +multiple lines, as function parsers commonly start on the line +containing the function name. We also found a few functions +in the Big-Vul and Devign dataset to be cut off prematurely, +missing functional lines of code. We were unable to determine +the exact cause for this truncation as we did not have access to + +the scripts used to produce the datasets. We hypothesise that +complexities within the source code confuse the lexicograph- +ical parser being used to extract them. For instance, many of +the early truncated samples contained additional curly brackets +(}) within literals. +D2A was resilient to truncation but it contains empty miss- +ing values, for which no code was provided. These occurred +when the static analysis tools flagged lines in a code file +outside of any containing function. Furthermore, D2A contains +13,300 single line function declarations that do not contain any +functional source code. +Impact. To see the impact of missing information on SVP +models, we set aside a common test set for each dataset +containing no incomplete entries. We then split the remaining +entries of each dataset into equal-sized halves to produce two +training sets: one containing incomplete data values and the +other without. The MCC performance marginally increased +for the complete training sets on all datasets. However, we +were unable to confirm any performance change for MCC, +precision or recall as significant using a Mann-Whitney U +test [61] (p > 0.05). Whilst the amount of information +truncated can be of arbitrary complexity, it appears to be +a relatively small part of the overall functions and occurs +relatively infrequently. However, we still advise practitioners +to ensure the completeness of software vulnerability data in +future, as more severe issues may produce larger impact. +Completeness issues can arise during data collection, but +these issues are easily solvable and do not have a high +impact as they cause relatively little missing information. +E. Currentness +Rationale. Currentness aims to ensure that datasets have +homogeneous temporal characteristics to their application con- +texts [48]. This is known in the machine learning domain +as concept drift [67]: a scenario in which the relationship +between the input data and target variable changes over time. +It is important for vulnerability datasets to stay up to date +as vulnerabilities and source code have an evolving nature +[68], [69]. We denote the date of an entry in a vulnerability +dataset as the date that the vulnerability was reported via the +dataset’s labelling mechanism. Currentness does not relate to +the synthetically created Juliet dataset. +Analysis. Currentness pertains to an entire dataset rather +than individual data points, so we selected a standard non- +contextual method for concept drift detection [67]. We used the +Jensen-Shannon divergence metric [70] to represent current- +ness, as it measures the statistical distance of the original and +current data in a dataset. The formula for this metric is reported +in [70]. For simplicity, we denoted the original and current data +as the oldest and newest half of the dataset, respectively. We +represented the distribution of the vulnerability data through a +Bag-Of-Tokens set. We tokenised all source code in a set using +a lexicographical parser and then normalised the values based +on the total frequency to obtain a probability distribution of +the occurrences of each token. +Training +Validation +Test 1 +Test 2 +Test 3 +Test 4 +Test 5 +Timestep: +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Fig. 5. Currentness impact experiment setup. +As the Jensen-Shannon divergence metric measures dissim- +ilarity, we compute currentness as one minus this value. We +obtained currentness values of 0.761 (Big-Vul), 0.811 (Devign) +and 0.844 (D2A). These values are relatively high for this +attribute and are unlikely to indicate concept drift. +Impact. We used a similar experimental setup to McIntosh +et al. [69] to determine whether vulnerability data is a moving +target. We sorted all entries by date and then split each dataset +into ten equal partitions. The four earliest partitions were +used to train an SVP model, the fifth partition was used for +tuning, and the remaining five were used as individual test +sets. Figure 5 displays the experiment setup. However, using +a Kendall rank correlation test [71] (p > 0.05), we observed no +significant decrease in model performance for MCC, precision +or recall as the time between the training and test set increased. +Currentness issues were not observed for software vulner- +ability datasets. They exhibit good temporal distributions of +data as they are collected over a long time range. +V. DISCUSSION +Software vulnerability datasets are particularly sensitive to +data quality challenges due to the difficulties of data prepa- +ration [10]. Most existing SVP studies focus on advances in +modelling but often overshadow data quality. Consequently, +our systematic analysis of inherent data quality attributes has +revealed critical data issues afflicting the current state of soft- +ware vulnerability datasets. Data preprocessing for software +vulnerability data is currently cursory or inconsistent [10]. +There is a lack of methods to guide data cleaning efforts for +SVP research. We present the following lessons learned from +our quality assessment of existing datasets: +• Be wary of reusing existing datasets without first check- +ing the data quality. +• Uniqueness is poor for software vulnerability data, so +avoid using evaluation setups that lead to significant +duplication across the training and test set. +• Inconsistently labelled data points should be removed, +based on the causes of such inconsistency. +• Source code entries with missing or incomplete informa- +tion should be removed or amended. +Issues in uniqueness, consistency, and completeness can be +detected with rule-based syntactic filters, as we have done +in this study. Hence, we can theoretically solve these issues +through exclusion of noisy samples that do not satisfy the +quality attributes. However, it may not be that easy in practice +as software vulnerability data is very scarce. SVP requires +large datasets [16], so removing noisy samples may make +datasets insufficiently small. Figure 6 displays the ratio of + +clean samples that we can automatically detect for each +dataset. Furthermore, we manually observed the data accuracy +issues to be severe, but there is no existing method to automat- +ically detect such problems. Data inaccuracy could potentially +decrease the number of clean entries by a further 20-71%. +Hence, we need to solve the underlying causes of these +problems. From the findings we have obtained, we summarise +the causes of the current major data quality issues below. We +provide some directions for researchers to investigate, to help +with overcoming these challenges: +• Automatic data collection often leads to data inaccu- +racy. We found the major cause of incorrectly labelled +vulnerabilities to stem from inaccuracies in vulnerability +fix identification. Either incorrect commits or line changes +were selected. Substantial work has been conducted for +ML-based models to identify correct vulnerability patches +[12], [72]. Semantic filters or heuristics for correct vul- +nerability fixing lines is currently lacking. +• Source code duplication may make datasets lack +diversity. An underlying problem for data collection is +a lack of sample diversity and uniqueness. Whilst we are +constrained in the vulnerability samples we can collect, +we have a selection choice for the non-vulnerable class. +Thus, we suggest the need for development of data +collection heuristics that can obtain more diverse non- +vulnerable code samples. Similarly, there is a need for +better synthetic data generation methods. Bug seeding +has shown promising results in this regard [17], but this +technique still relies on the data quality of the real-world +bugs from which the technique infers the seeds. +• Unknown vulnerabilities can introduce label inconsis- +tency. Label inconsistency problems arose from underly- +ing problems for the non-vulnerable class. Big-Vul sam- +ples contained undetected vulnerabilities, and both De- +vign and D2A contained undocumented vulnerabilities. +This is particularly problematic as we lack a label source +for non-vulnerable code. Semi-supervised semantic filters +have shown promise for reducing noise in non-vulnerable +labels [13], [73]. Synthetic datasets need clearly defined +usage guidelines when used as training datasets. +VI. THREATS TO VALIDITY +Construct Validity: Our interpreted data quality analysis +may not perfectly represent the target attributes. We have +formed our analysis using standard practices from relevant +domains [48] and existing knowledge of software vulnerability +data practices [10]. Requirements elicitation using domain +Fig. 6. Ratio of clean to unclean samples in a dataset that can be automatically +detected. +experts would help improve these claims in future [18]. The +need for manual analysis of some attributes is also a potential +limitation, as it may contain bias or inaccuracies. We used two +independent raters to minimize such impacts. We used CORE +rankings as a criteria for our dataset selection, even though +CORE journal rankings have become deprecated. We consider +these ranking still sufficient, as they were only deprecated two +months prior to the date of data collection. +Internal Validity: The outcomes of our impact analysis +experiments may be affected by confounding factors. We +analysed each data attribute individually, so other data quality +issues were present during each experiment. More work is +required to examine data quality attributes cumulatively. +External Validity: We have constrained our analysis to four +state-of-the-art datasets. Measurements are also limited to +datasets that contain appropriate metadata. For instance, we +were unable to investigate the ReVeal dataset [24] due to this +issue. Similarly, we performed impact analysis using a single +SVP model. This model has been demonstrated to be state-of- +the-art [7]. Furthermore, we considered inherent data quality +attributes, so the issues remain, regardless of the model. +VII. CONCLUSION +We have systematically examined five data quality attributes +for four state-of-the-art software vulnerability datasets, to help +improve the validity and trustworthiness of downstream data- +driven tasks that rely on this information. Our findings revealed +that some software vulnerability datasets are prone to data +quality issues, particularly in terms of data accuracy, unique- +ness, and consistency. We found 20-71% of vulnerability +labels were inaccurate in real-world datasets, which altered +performance up to 65%. Furthermore, 0-47% of the labels +were inconsistent, which hindered model training completely +in the most extreme circumstances. +Data quality requires ongoing consideration and analysis. +We advise future researchers and practitioners to consider data +quality in more effective detail through the means that we +have provided. Furthermore, we advocate the importance of +data quality and the need to overcome the quality issues that +we have observed. Lastly, we urge the need for additional +investigation into system-dependent data quality attributes to +help achieve specific operational needs. +VIII. DATA AVAILABILITY +We have made our data and analysis scripts available via a +reproduction package [26]. +ACKNOWLEDGMENT +This work has been supported by the Cyber Security Coop- +erative Research Centre Limited whose activities are partially +funded by the Australian Government’s Cooperative Research +Centre Programme. + +Big-Vul +Devign +D2A +Juliet +Clean +UncleanREFERENCES +[1] G. 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LeTraon, “Learning from what we know: How to perform +vulnerability prediction using noisy historical data,” arXiv preprint +arXiv:2207.11018, 2022. + diff --git a/qNE5T4oBgHgl3EQfJQ54/content/tmp_files/load_file.txt b/qNE5T4oBgHgl3EQfJQ54/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35133272a1ae2946309e08d9ab55d7e7bee896d6 --- /dev/null +++ b/qNE5T4oBgHgl3EQfJQ54/content/tmp_files/load_file.txt @@ -0,0 +1,1249 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf,len=1248 +page_content='Data Quality for Software Vulnerability Datasets Roland Croft∗†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Ali Babar∗†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Mehdi Kholoosi∗† ∗ School of Computer Science, CREST, University of Adelaide, Australia, {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='lastname}@adelaide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='au † Cyber Security Cooperative Research Centre, Australia Abstract—The use of learning-based techniques to achieve automated software vulnerability detection has been of long- standing interest within the software security domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These data-driven solutions are enabled by large software vulnerability datasets used for training and benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, we observe that the quality of the data powering these solutions is currently ill-considered, hindering the reliability and value of produced outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst awareness of software vulnerability data preparation challenges is growing, there has been little investigation into the potential negative impacts of software vulnerability data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, we lack confirmation that vulnerability labels are correct or consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our study seeks to address such shortcomings by inspecting five inherent data quality attributes for four state-of-the-art software vulnerability datasets and the subsequent impacts that issues can have on software vulnerability prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Surprisingly, we found that all the analyzed datasets exhibit some data quality problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In particular, we found 20-71% of vulnerability labels to be inaccurate in real-world datasets, and 17-99% of data points were duplicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed that these issues could cause significant impacts on downstream models, either preventing effective model training or inflating benchmark performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We advocate for the need to overcome such challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our findings will enable better consideration and assessment of software vulnerability data quality in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Index Terms—software vulnerability, data quality, machine learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' INTRODUCTION Software vulnerability detection is a vital task for achieving secure software systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, traditional techniques for detecting vulnerabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=', rule-based methods) struggle in terms of scalability and false positive rates [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, many researchers have been motivated to leverage the technical advancements of Artificial Intelligence (AI) and Machine Learning (ML) to support automatic software vulnerability detection [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Recent studies have reported great success in this direction [4]–[8], with performance that surpasses traditional approaches [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We refer to these learning-based techniques as Software Vulnerability Prediction (SVP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Like any data-driven task, SVP is highly data-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In order to learn complex features of vulnerabilities, we require large code datasets that have been labelled as either vulnerable or non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Nonetheless, software vulnerability data collection is not a trivial task [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Labelled examples of software vulnerabilities are difficult to obtain in the real-world, as they are scarce [11], poorly documented [12], and limited to reported vulner- abilities [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consequently, many researchers have conducted labourious work constructing large-scale software vulnera- bility datasets [14]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, we found that relatively little counterpart work has been conducted to understand the software vulnerability data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst data quantity is important, an effective machine learning system requires adequate data quality [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Despite the increasing realisation of software vulnerability data preparation challenges [10], there has been relatively little effort made to provide a systematic understanding of how these challenges can potentially impact data quality, and subsequently affect the reliability of down- stream software vulnerability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A lack of understanding of data quality leads to critical barriers to advance software assurance against vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data quality is an integral component of any data-driven system: garbage in, garbage out [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Certain data biases or misinformation can make benchmark performance results mis- leading [20]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This can cause models to fail to generalise to real-world scenarios [17], [23], [24], if they have not been trained with fair and realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Thus, we set out to understand the nature of data quality for software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To achieve this, we focused on inherent data quality attributes that are intrinsic to the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table I presents the five data quality attributes that we systematically analyse: accuracy, uniqueness, consistency, completeness, and currentness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For each attribute, we provide a measurement of its prevalence in existing datasets and analysis into the causes of observed issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our findings revealed that even state-of-the-art datasets exhibit considerable data quality challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As expected, we found these issues to cause significant negative impacts on both training and benchmarking of state-of-the-art SVP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our findings have substantial implications: Data quality issues may constrain the patterns able to be learnt by SVP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found that real-world datasets can face substantial issues for label correctness of the vulnerable class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Approximately 20-71% of vulnerability labels were inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, up to 47% of labels were inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These issues cause models to learn false or insufficient patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Software vulnerability benchmark datasets may lead to inflated performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A few datasets exhibited large data duplication rates, between 17-99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, data leakage causes models to report inflated performance using stan- dard test setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Evaluation performance decreased by up to 82% after removing such duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To achieve the most reliable results from learning-based software vulnerability analytics, we must consider and address the issue of data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst some of the observed quality issues can be solved easily through rule-based detection and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='05456v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='SE] 13 Jan 2023 TABLE I INHERENT DATA QUALITY ATTRIBUTES DEFINED BY ISO/IEC 25012 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Attribute Definition Interpretation Accuracy The degree to which the data has attributes that correctly represent the true value of the intended attribute of a concept or event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Correct labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Uniqueness The degree to which there is no duplication in records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' No duplicate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency The degree to which data has attributes that are free from contradiction and are coherent with other data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistent labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Completeness The degree to which subject data associated with an entity has values for all expected attributes and related instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' No missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness The degree to which data has attributes that are of the right age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Not obsolete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' removal, others cannot be remediated in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As a community, we must focus on developing knowledge and tools for constructing high quality software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our contributions are twofold: 1) We provide understanding of the nature and causes of observed data quality issues in software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We also demonstrate the corresponding impact of these issues for software vulnerability prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Such investigation highlights the problem of data quality and the need to mitigate these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, our insights help overcome data quality issues, for which we have provided directions in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 2) We propose and conduct methods for measurement of data quality for software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These efforts can enable practitioners to consider and perform data quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We have provided a reproduction package of this study to assist with such efforts [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' BACKGROUND AND MOTIVATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Software Vulnerability Data Preparation Software Vulnerability Prediction (SVP) models use pro- gram analysis techniques to learn software vulnerability pat- terns automatically from historical examples [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Due to the unstructured nature of source code, researchers have found the most success using Deep Learning (DL) techniques that learn from program syntax and semantics [3], [7], [8], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, SVP is a data-hungry process [16]: the models require a large training dataset of annotated code modules, labelled as vulnerable or non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, acquiring a reliable vulnerability label source is a non-trivial task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' there is no oracle that can unfailingly prove the existence or absence of vulnerabilities from a codebase [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Thus, researchers have relied on a variety of label sources to account for different shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Following prior analysis [10], [24], we outline four main label categories: Security Vendor Provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Security vendors maintain vulnerability databases that aggregate information from various advisories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This provides a standardised collec- tion of disclosed vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Examples include the National Vulnerability Database (NVD) [30], or the Snyk Vulnerability Database [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Vulnerability records often provide links to patches, which can then be traced to identify real-world source code and vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Developer Provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Vulnerability databases may not properly document all vulnerabilities of a project [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, researchers may collect vulnerability fixing com- mits directly from developers via the development history or via a project’s issue tracking systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, this method requires additional effort to search development artefacts for security-related defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Tool Created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Developer or security vendor-provided la- bels have a major limitation of only collecting reported vulnerabilities, which severely limits the number of ex- amples that can be collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In reality, vulnerabilities can remain latent or undetected [33], which limits the dataset size and adds considerable label noise to the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To circumvent this, some researchers have utilised static security analysis tools to automatically produce labels for the source code [16], [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This process relies heavily on the accuracy of the static analyser used, which is a source of contention [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Synthetically Created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Finally, to bypass the limitations of other label sources, vulnerable code examples and annotations can be created artificially from known vul- nerable patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Synthetically producing entries ensures label correctness at the cost of source code diversity [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Despite these caveats, researchers have faced data prepa- ration challenges regardless of the selected dataset [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst state-of-the-art SVP models report good performance on benchmark datasets, performance only measures the ability of a model to fit a particular dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A good performance value does not guarantee that a model will generalise to real-world scenarios [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, data quality issues will hinder the reliability and trustworthiness of the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, previous studies [13], [33] have highlighted inflated performance due to inaccurate labelling mechanisms for non- vulnerable modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consequently, the industry value and adoption of SVP models is uncertain [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our study seeks to shed light on the state of software vulnerability data quality so that we better understand the reliability and trust- worthiness of the reported outcomes that use these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data Quality in Software Engineering Research Training data is an integral component of ML systems that heavily influences the produced models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Unlike conventional software systems, ML systems exhibit both system and data requirements [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As a result, data quality is becoming an essential component of AI-based Software Engineering research [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Software engineering data and artefacts are often noisy as they are usually collected post-hoc via mining software repositories [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The data and labels are not gener- ated explicitly for the purposes of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, software engineering data has been found to exhibit issues with data accuracy, relevance, and provenance [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Existing studies that investigate data quality characteristics of software engineering datasets are currently non-systematic and limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data quality can be defined by a large range of dimensions, like those defined in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Existing studies often limit their analysis to semantic or syntactic data accuracy and noise [22], [41]–[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similarly, Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [33] considered label accuracy within software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, this approach fails to provide a complete picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To make informed data decisions, there is a need for a systematised and objective investigation of data quality in the software engineering domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Croft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [10] conducted a systematic literature review of the data quality issues considered by SVP researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst this work provides a systematised view, the observations are unsubstantiated with respect to actual software vulnerability datasets and SVP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our work purports to perform quantitative analysis of data quality within software vulnerability datasets and explicitly show its impacts on SVP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Additionally, researchers have recently begun constructing automated cleaning frameworks to reduce the observed data noise issues and ensure correctness [22], [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These frame- works are grounded in a deep understanding of the data quality issues afflicting the relevant datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We expect the findings obtained in our study to enable the creation of a cleaning framework for software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' STUDY DESIGN To understand the data quality of the state-of-the-art soft- ware vulnerability datasets, we address two Research Ques- tions (RQs) through the analysis of several data quality attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Figure 1 displays the overall workflow used to conduct this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We describe each of the three processes in Sections III-B, III-C and III-D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Research Questions Our investigation is guided by the following RQs: Processes Data Quality frameworks Artefacts SVP Models SV Datasets Rationale RQ1 Measure Attributes Analysis RQ2 Validate Attribute Impact Impact Identify Data Quality Attributes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The overall study design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' RQ1: What data quality issues are present in the state- of-the-art software vulnerability datasets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We firstly aim to inform practitioners of the state and nature of data quality for software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' RQ2: To what extent do data quality issues impact downstream software vulnerability prediction models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our second aim is to verify the importance of data qual- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We attempt to demonstrate the potentially negative impact that observed data quality issues can have on the downstream tasks for software vulnerability prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Identifying Data Quality Attributes The lack of existing data quality consideration for software vulnerability datasets has perhaps been caused by the lack of systematic definitions for data quality and hence measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' It is not easy to define data quality, due to its many dimensions [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Not all dimensions may be relevant to a specific scenario [47], and different organisations would value quality attributes differently [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, not all organisations would prioritise data confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' There are two main categories of data quality attributes [25]: inherent data quality, which intrinsically relates to the data itself, or system-dependent data quality, which extrinsically arises from external fac- tors and requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For this study, we focused on purely inherent data quality attributes, as SVP models have not yet achieved widespread industrial application [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' System- dependent attributes cannot be properly measured without an associated deployment context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In this sense, we focused on the data rather than how the data is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, our findings are not constrained to particular modelling techniques or features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To identify inherent data quality attributes, we used the standardised data quality framework ISO/IEC 25012 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This framework has been used for data quality assessment in both the Software Engineering [47] and Machine Learning [18], [48] domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' ISO/IEC 25012 outlines five inherent data quality attributes: Accuracy, Consistency, Completeness, Cur- rentness, and Credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We excluded the credibility attribute as it is difficult to quantify in existing software vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Credibility indicates the level of trust that we have in a dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' the authenticity of the data source or supplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We need to ensure that our data points are free from contamination or fake information [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As datasets have been produced by peer-reviewed research or respected government organisations [10], we assume a level of trust in the data source and supplier of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Additionally, we considered a uniqueness data dimension, due to its prevalence in existing software engineering research [49], and its importance as highlighted by previous SVP researchers [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table I summarises the selected inherent data quality attributes for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Measuring Attributes For the analysis, we collected one dataset of each label source described in Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To ensure the collected datasets represented the state-of-the-art appropriately, we con- sidered datasets that were created or used by a conference or TABLE II SELECTED STATE-OF-THE-ART DATASETS FOR EXAMINATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Dataset Label source # Functions % Vul Big-Vul [14] Security vendor provided 188,636 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='78 Devign [15] Developer provided 27,318 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='61 D2A [16] Tool created 1,295,623 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='44 Juliet [50] Synthetically created 253,002 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='77 journal paper published in a high quality venue, as indicated by a CORE1 ranking of A or A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' New datasets continue to be published in order to improve on previous dataset shortcom- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We selected the most recently published datasets as of March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We also chose datasets that contained appropriate metadata about how the labels were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table II displays our selected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' All four of the examined datasets provide source code for C/C++ functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Big-Vul [14] scraped software versions prior to a vulner- ability fix through linked patches from the CVE Details database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Functions with lines changed in a patch were labelled as vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' All remaining functions in a file touched by a commit were labelled as non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Devign [15] used a similar data collection method by scraping vulnerability fixes directly from GitHub com- mits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A keyword approach was used to separate vul- nerability and non-vulnerability related commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The vulnerability-related commits were then filtered manually to ensure accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' All relevant functions of each commit were collected for their respective classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' D2A [16] collected source code by running a static anal- ysis tool on project versions before and after bug fixing commits of six open source repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The vulnerable class was formed from tool warnings that disappeared in the post-fix version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The non-vulnerable class consists of the remaining tool warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Each data entry indicates a function containing the original vulnerability location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We retrieved all such functions to form the D2A dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Juliet [50] contains synthetically generated examples of programs demonstrating a variety of known vulnera- ble code patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Programs are generated automatically, based on pre-defined augmentation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Each program contains a vulnerable version and non-vulnerable version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Juliet was originally created to test static and dynamic security tools, but it has also been used for training SVP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Source code data is provided in files with func- tions being annotated as vulnerable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We retrieved all functions from each annotated section of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We followed the measurement practices specified by Naka- jima and Nakatani [48] for using the ISO/IEC 25012 frame- work with respect to AI training data requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table I describes the interpretation of each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We identified the percentage of samples in a dataset that satisfy the relevant characteristics to produce an overall measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, the measurement value for each attribute lies between 0 and 1, with 1 indicating no data quality issues are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We 1http://portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='au/conf-ranks/, http://portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='au/jnl-ranks/ formally define this measurement in Equation 1, where N denotes the number of samples in a dataset and dq(i) returns 1 if a data entry i satisfies the relevant characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Further details of the measurements for each individual attribute are provided later in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Attribute = N � i=1 dq(i) N (1) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Validating Attribute Impact Finally, to validate the impact of the observed data quality issues, we investigated the performance impacts on a state-of- the-art SVP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Depending on the observed data quality issues, we either measured the performance change on a retrained model after mitigating data quality issues or altered the test setup to highlight the data quality characteristic of focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We provide further details in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For the benchmark performance, we trained a model on each dataset, without any pre-processing of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, we removed inconsistent entries for the D2A benchmark, as we were otherwise unable to produce an effective classifier for this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We ran all experiments five times using random 80:10:10 training/validation/test splits unless otherwise speci- fied, as this is a standard test setup in prior research [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We selected the LineVul SVP model [7], as it is a re- cently published model that has been shown to outperform all previous baselines for both function level and line level predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' LineVul [7] relies on CodeBERT [51] to obtain code feature representations that capture lexical and logical semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' CodeBERT is a pre-trained state-of-the-art code embedding model based on the RoBERTa architecture [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similar studies have demonstrated the effectiveness of Code- BERT for SVP [8], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' LineVul generates function-level predictions using a transformer-based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Although LineVul also has the capability to localise its predictions to the line-level after performing the function-level prediction, all of the selected datasets provide labels at the function-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, we perform prediction at the function-level granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We evaluated model performance using Recall, Precision and Matthews Correlation Coefficient (MCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We opted to use MCC as an overall indicator of performance, as its use has been recommended for similar tasks [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' MCC values range between -1 and 1, with 1 being the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' DATA QUALITY ANALYSIS Table III displays the attribute values for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Accuracy Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Accuracy defines the correctness of the data points that comprise a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This largely relates to the semantic label correctness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=', whether or not data points labelled as vulnerable or non-vulnerable genuinely align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' It has previously been observed that non-vulnerable labels are unreliable in real-world datasets as there is no ground truth label source for this class [10], [13], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' No oracle can reliably ensure the security and absence of exploits in a given code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, non-vulnerable labels are usually collected simply through the absence of a vulnerable label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Thus, our analysis was constrained to the vulnerable label source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We focused our investigation on label correctness of data points labelled as vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We determined if a label is correct via manual analysis with respect to each dataset’s labelling mechanism: whether a vulnerability accurately represents the vulnerability report or static analysis tool warning that it was derived from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In this sense, we did not verify whether a vulnerability was actually exploitable, but rather whether a code snippet is functionally relevant to the reported vulnerability of each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The following steps were taken to assess the label correctness of each entry: 1) We first extracted information relating to the vulnerability and fixing commit of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' All datasets provided a git fixing commit ID except Juliet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Big-Vul also provided CVE-IDs and D2A contained the static analysis tool trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 2) We read the fixing commit description and other available information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=', the vulnerability description from NVD for Big-Vul and the static tool trace for D2A) to gain an understanding of the vulnerability and the fixing commit changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 3) We then examined the changed lines in the fixing commit for the relevant function, as well as the entire function’s code to understand the context of the changed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Based on this code comprehension, we made an assess- ment as to whether the changed lines were functionally relevant to the information from the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 4) If we did not interpret them as functionally relevant, we examined all the fixing commit changes to identify where the root changes were to understand why the flagged function was not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 5) Afterwards, the authors discussed the labels that were in disagreement and reached a consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To facilitate our manual review, we examined 70 random samples of each dataset (90% confidence level +/- 10% [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Two of the authors of this paper conducted this manual analysis independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' each of them had two to five years of software security-related experience gained in academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The two raters achieved a Cohen Kappa value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='627 [55], which implies moderate to strong agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our findings revealed that label inaccuracy occurred within the real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We obtained accuracy values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='8 (Devign), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='543 (Big-Vul), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='286 (D2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found no TABLE III MEASURED VALUE OF EACH ATTRIBUTE FOR EACH DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Attribute Dataset Big-Vul Devign D2A Juliet Accuracy* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='286 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='000 Uniqueness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='163 Consistency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='750 Completeness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='981 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='000 Currentness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='844 Based on a sample of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' inaccuracies within the synthetic Juliet dataset, as the vulner- able cases are crafted specifically for the label rather than collected post-hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Real-world labelling works by tracing a vulnerability identifier (usually a vulnerability fix or warning) to the original code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The two authors who conducted the manual labelling noted their reasoning behind a label being correct or incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We conducted a thematic analysis [56] of the label reasoning to identify the causes of dataset inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table IV displays the proportion of each theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Irrelevant code changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The real-world datasets largely assume that code touched by a vulnerability fix is vulner- able code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, a vulnerability fixing commit may not necessarily provide a patch alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Non-functional changes, such as style changes, refactoring and code migration can confuse the data labelling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, this example fixing line2 simply converts a constant value to the equivalent macro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similarly, tan- gled commits can implement other irrelevant changes in parallel [57], which will be misinterpreted as vulnerable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Cleanup changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Vulnerability fixes can sometimes be large and disparate due to the complexity of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Tertiary changes can be made in a commit to help better facilitate a vulnerability fix, such as adding, deleting or altering variables, functions or parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For example, in this fixing commit3 example a vulnerability occurs for when read_only is set as True rather than a protected memory object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The cleanup change converts False read_only values to a nullptr, simply to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These are functional changes that relate to the vulnerability fix, so we do not consider them as irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Nonetheless, they do not indicate the location of the underlying exploitable code, and hence produce false positive labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We call these cleanup changes, although they have also been referred to as casualty changes by Sejfia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Inaccurate vulnerability fix identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' If the la- belling mechanism fails to identify a vulnerability fix, the subsequent code snippet will naturally not be a vulnera- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Datasets like Big-Vul that trace vulnerability fixes from external vulnerability reports can introduce errors into this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, we found the majority of vulnerability reports for the Chromium project to be improperly traced as this repository is not naturally hosted via GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, datasets that attempt to identify vulnerability fixes directly from commit history (Devign 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='com/FFmpeg/FFmpeg/commit/8b2fce0d3f5a56c40c28899c9237210ca8f9cf75 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='com/chromium/chromium/commit/673ce95d481ea9368c4d4d43ac756ba1d6d9e608 TABLE IV TYPES OF LABEL INACCURACY IN REAL-WORLD DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Dataset Irrelevant Cleanup Inaccurate Big-Vul 25% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='1% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='9% Devign 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='9% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='4% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='7% D2A 0 0 100% and D2A) can also be rife with errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Researchers usually attempt to identify these commits through inac- curate and unreliable keyword matching methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Lastly, D2A uses additional help from static analysis tools to identify vulnerability fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These tools produce many false positive vulnerability warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Mainly, we observed tangled commits to cause problems for current real-world data labelling heuristics [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Current datasets assume vulnerable code to be all code touched in a vulnerability fix, but commits are messy in practice [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similarly, vague, generic or unclear commit messages can make vulnerability fix identification difficult [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In contrast, correct vulnerability labels typically stem from simple, focused and well-defined vulnerability fixing commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Additionally, the datasets included samples for which we found it difficult to verify or agree upon the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This often occurred when the location of a vulnerability falls in a grey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, should the caller of a vulnerable code snippet also be labelled as such?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Herbold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [60] encountered similar problems in their investigation of tangled commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Alternatively, the label source may not contain enough information in the bug report to properly trace it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We tentatively labeled these ambiguous cases as correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, the software security domain should work towards clear definitions that prevent such ambiguous cases, to help with ensuring label correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Devign did not exhibit as many issues, as it is the only dataset for which the creators attempted to perform manual validation of the fixing commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, this accuracy as- surance comes at the cost of data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Devign is the smallest of the datasets, due to the strenuous efforts of manual validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Nonetheless, Devign still exhibits some inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The majority of the errors came from irrelevant changes, such as refactoring or code migration, which may imply the original authors did not check for such things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The accuracy for Big-Vul was lower, as many of the vulnerability fixing commits used during data extraction for this dataset were large, tangled or noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Most errors arose from inaccuracies in tracing the fixing commits, particularly for the Chromium project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 36% of the vulnerable entries in Big-Vul are from the Chromium project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Over two-thirds of the D2A labels were inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found that this was primarily due to the static analysis tool warnings being unreliable, as well as the vulnerability commit identifier being inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The majority of commits flagged by the D2A data extractor were not actually vulnerability fixes, as the context of the security-specific words was often misinterpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, not all commits that contained the word “memory” were necessarily fixing unsafe memory operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The majority of static tool warnings were also false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Static analysis tools often output an indication of the reliability of a warning, based on how confident the tool is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For example, a confident integer overflow warning would know the integer data type and variable values, whereas an unreliable report may know neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Over 97% of the static analysis warnings included in D2A are from the lowest reliability warning class, making them often inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, as the static analysis tools attempt to infer the location of the vulnerability directly, there were no false positives caused by irrelevant or cleanup code changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To evaluate the impact of inaccurate labels, we retrained each model using our manually-validated samples of each dataset as a separate holdout test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We measured model performance when using the original labels in comparison with the manually-corrected labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We could not measure MCC as the test set had no samples that were originally labelled as non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The precision decreased by 29%, 50% and 80% for Devign, Big-Vul and D2A, respectively, which we confirmed to be significant using a Mann-Whitney U test [61] (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This was because incorrect vulnerable labels caused the models to infer incorrect patterns for this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The models were taught vulnerable patterns that were actually non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, in terms of model evaluation, what were previously considered true positives became false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Correspondingly, we found that the model recall was not significantly affected (using a Mann-Whitney U test [61]) as we only uncovered label inaccuracy for the vulnerable class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' the number of false negatives was unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These impacts are still significant however, as they can lead to high false positive rates in models which would greatly increase inspection efforts during practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Accuracy is limited for some real-world datasets due to their reliance on noisy and hard-to-identify vulnerability fixing commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Accuracy issues cause SVP models to infer the wrong patterns between classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Uniqueness Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Uniqueness is not necessarily an intrinsic data property, as a real-world data distribution may contain dupli- cated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, code duplication has been demon- strated to have adverse effects on trained models [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Dupli- cates can introduce bias in a model towards certain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Inflated performance values can result when duplication occurs between the training and test sets [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, ensuring uniqueness of samples within a dataset helps models generalise towards a true data distribution [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consequently, we treat it as an inherent attribute and decided to investigate the impacts that a lack of uniqueness would have for SVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similar or identical code fragments are defined as code clones [63], of which there are four main types [64]: 1) Type-1: Identical code fragments, except for differences in white-space, layout and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 2) Type-2: Identical code fragments, except for differences in identifier names and literal values, in addition to Type- 1 clone differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 3) Type-3: Syntactically similar code fragments that differ at the statement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The fragments have statements added, modified and/or removed with respect to each other, in addition to Type-1 and Type-2 clone differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 4) Type-4: Syntactically dissimilar code fragments that im- plement the same functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We followed standard practices and considered type-3 code clones as duplicates [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Even functionally similar code fragments will include duplicated patterns and tokens that can adversely affect the model performance and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, for software vulnerability datasets, slight functional changes can form the difference between a vulnerable and non-vulnerable label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A typical vulnerability fix only alters a few lines of code [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' It is important that a model is able to capture these slight functional differences across prediction classes to avoid excessive false positive or false negative rates [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, we only considered duplicates with the same labels (vulnerable or non-vulnerable) as code clones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' An entry is not unique if it is a code clone of any other entry of the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To identify code clones, we reused the code duplicate detector tool produced by Allamanis [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We lowered the minimum token count of a sample to five, as functions are smaller than the files for which this tool was originally built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This tool outputs clusters of duplicates, as there can be more than one duplicate per function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed code duplication to occur within all the datasets, but less frequently for the Big-Vul and Devign datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We obtained a uniqueness value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='830 (Big-Vul), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='899 (Devign), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='021 (D2A), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='163 (Juliet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We manually examined a sample of 30 random duplicate clusters for each dataset (74 functions for Big-Vul, 79 functions for Devign, 210 functions for Juliet, 2288 functions for D2A) to understand why duplicate code entries are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Using thematic analysis [56], we observed three main causes of code duplication in real-world datasets: Updated code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' All real-world datasets collect code from multiple versions of the same code repository in order to maximise the number of vulnerabilities observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Across the versions, subtle functional or non-functional updates to the code introduce predominantly duplicated code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For vulnerable cases, these types of duplicates can imply that the code update either failed to fix the vulnerability or introduced a new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similar function sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A code file may contain a suite of simple modular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These functions are often identical in terms of variable names, logic, and layout but have slight functional differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For example, two functions may be implemented to start and stop a process respectively, or a set of functions may each perform a unique mathematical operation on a data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Renamed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Identical functions may be dupli- cated and renamed for use in different files and contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We illustrate these causes in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These factors are inherent in source code datasets due to both the spatial and temporal repetitiveness of code in software repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found duplication to be especially significant for D2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Each unique function in the dataset had an average of 57 duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This is because D2A produces label information at the line level, which is then abstracted to the function scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The same function can be included multiple times if unique lines are flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, the D2A labelling process introduces many additional exact duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Over 94% of the TABLE V PERFORMANCE IMPACT OF UNIQUENESS ISSUES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Dataset With Duplication Without duplication Change Precision Recall MCC Precision Recall MCC (MCC) Big-Vul 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='0% ↓ Devign 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='244 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='9% ↓ D2A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='630 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='141 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='7% ↓ Juliet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='799 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='814 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='4% ↓ D2A dataset were type-1 code clones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, two of the six repositories that comprise D2A are forks of each other (FFmpeg and Libav), which led to further duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The lack of uniqueness for D2A questions the claim of the dataset’s size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' there is limited information at the function level for this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We also found a large number of duplicates in Juliet, due to the subtlety in the variance of the test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' New test cases are produced by making slight changes to the control flow logic, internal function calls, or literal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, the non- vulnerable fixed statements can exhibit exact duplication due to having a constant corrected implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For SVP, duplicates can appear in the training set, test set, or across these two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Figure 3 illustrates these duplicate types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In-train duplicates may produce model biases [62], but it is hard to measure these aspects via model performance [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We focus our analysis on the impact of uniqueness for model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We split each dataset into a training, validation and test set, as specified in Section III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We then compared the evaluation performance of the model when cross-set duplicates to the test set were either removed or kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Allamanis [49] found cross-set duplication to be the most significant type in software engineering research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table V displays the performance change for SVP models when we removed the identified duplicate entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed that cross-set duplication is a significant factor for some datasets as the overall evaluation results (MCC) decreased for Devign, D2A and Juliet, which we confirmed to be significant using a Mann-Whitney U test [61] (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The model trained with Big-Vul data was not significantly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This implies that a lack of uniqueness may not always be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Duplicates can allow for data leakage in the evaluation setup [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' the models can trivially classify samples in the test set that are also duplicated in the training set, inflating the true performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed that duplication had a larger neg- ative influence on recall rather than precision for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' + func view_obj2() + func view_obj1() + func add_fileY_var() File Y + func do_operation_X() + func add_fileX_var() File X COMMIT 1 + func do_operation_X() File X COMMIT 2 = Updated Function = Renamed Functions = Similar Function Sets Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' An example of the three main code duplicate causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Train Set Test Set Cross-Set Duplicates In-Train Duplicates In-Test Duplicates Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Types of duplicates for ML models, adapted from Allamanis [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The removal of cross-set duplicates removed trivial samples from the test set, primarily lowering true positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This had a larger impact on recall, due to the higher ratio of false negatives in comparison to false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Recall significantly decreased for Devign (7% decrease), D2A (92% decrease) and Juliet (15% decrease) (confirmed using a Mann-Whitney U test [61]), whereas precision actually even increased after duplicate removal for Big-Vul and Juliet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, the overall performance (MCC) still decreased for each dataset other than Big-Vul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Uniqueness issues are present within all datasets due to the repetitive and incremental nature of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Duplicate code snippets can potentially inflate overall evaluation perfor- mance due to data leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency denotes that data entries should not provide conflicting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For software vulnerability datasets, this simply implies that similar code snippets should not have conflicting labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' A piece of code cannot be both vulnerable and non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Inconsistency can arise in software vulnerability data however, due to the multiple data streams that are used to construct a dataset [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency is understandably important for model training and construction, as conflicting labels confuse any AI-based model that is attempting to distinguish between two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency is related to the uniqueness attribute as we again examined duplicated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, consistency mea- sures duplicated entries with conflicting labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As slight functional changes can form the functional difference between a vulnerable and non-vulnerable code snippet, we only con- sidered type-1 code clones (exact matches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' An entry is consistent if it does not have any duplicates with conflicting labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed high consis- tency values for Big-Vul (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='999) and Devign (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='991), but lower values for D2A (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='531) and Juliet (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We manually examined a random sample of 30 inconsistent clusters to determine reasons for inconsistent vulnerability labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found that the causes of inconsistent labels were fairly unique to each data collection approach, which we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For Big-Vul, inconsistent labels were produced by latent vulnerabilities that existed within the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The la- belling heuristic of this dataset assumes that all functions in the files of a commit that were not explicitly touched are non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, these functions can actually contain vulnerabilities unknown to developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These vulnerabilities can be reported and then collected at a later date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Figure 4 illustrates this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Although the number of inconsistent cases is relatively small, these are only the latent vulnera- bilities we know about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In reality, complete knowledge of the latent vulnerabilities is unobtainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Croft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [13] observed at least twice as many latent vulnerabilities as known vulnerabilities in their dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In the Devign dataset, inconsistencies occurred due to simultaneous code branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The vulnerability fixing commit may only be identified in one branch, leaving the same commits in other branches to be treated as non-vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This primarily occurs due to merging commits on branches, as merged commits can contain vulnerability fixes but are not described as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Like the inconsistent labels of Big-Vul, this implies there are incorrect labels for the non-vulnerable class, as the other branch commits are improperly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The static analysis tools that inferred the labels of the D2A dataset produce an excessive number of warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' All functions are scanned over every analysed commit during the D2A data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, the same function can receive the same warning from the static analysis tool over different commits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' If one of the commits edits the flagged lines whereas the others do not, then inconsistent labels will be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found this occurred commonly in practice, as demonstrated by the relatively low consistency value of this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The Juliet test cases can include tertiary functions that perform unsafe operations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=', writing data to a buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Test cases are set up like this to help test the ability of vulnerability scanning tools to track data flow across functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Although these tertiary functions are vulnerable as they lack the necessary security checks, an exploit will only occur when specific values are passed to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As a result, duplicate copies of these functions are contained in both the vulnerable and non-vulnerable annotated sections of this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' From the analysis, we observe that inconsistent samples primarily point to inaccuracies within the data collection processes for the non-vulnerable class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This is due to a lack of proper label sources or checks for this class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' it is formed from the absence of vulnerability labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Commit 1 Commit 2 Commit 3 Commit i Commit i+1 Vulnerability Introducing Commit for func b() Vulnerability Fixing Commit for func a() Vulnerability Fixing Commit for func b() + func a() func b() Changes Fixed Unchanged + func b() Changes Fixed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' An example of inconsistency introduced from latent vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Function b is vulnerable in commit 3 until commit i+1, but it is only recorded as such for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' TABLE VI PERFORMANCE IMPACT OF CONSISTENCY ISSUES, WITH COMPARISON TO ORIGINAL DATA SETUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Dataset All inconsistent (original) Consistent test set Consistent train & test set Precision Recall MCC Precision Recall MCC Precision Recall MCC Big-Vul 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='919 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='774 (-) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='835 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='915 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='775 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='833 (↑) Devign 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='668 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='500 (↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='311 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='653 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='502 (↓) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='289 (↑) D2A 0 0 0 0 (-) 0 (-) 0 (-) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='948 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='599 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='748 (↑) Juliet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='998 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='985 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='987 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='999 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='999 (↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='999 (↑) Impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Like uniqueness, inconsistency can appear within the training set, test set, or across these two sets, as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Training set inconsistency would affect the patterns learnt by the model, whereas test set inconsistency would affect model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We considered both of these aspects in our impact analysis experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We removed inconsis- tency via entries from the non-vulnerable class of inconsistent clusters, as our manual analysis found these non-vulnerable entries to be incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Using the experimental setup described in Section III-D, we considered three scenarios: the original case when all inconsistent examples are retained, a consistent test set in which all within-test and cross-set inconsistencies are removed but the training set remains inconsistent, and an entirely consistent dataset in which all inconsistent entries are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We trained and evaluated a model for each setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table VI displays the performance impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed inconsistency to potentially have an effect on model evaluation as MCC performance increased when using consistent test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This is because a model will naturally make the same prediction for identical inputs, producing wrong predictions for a portion of the inconsistent entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, inconsistent samples hinder performance as the lack of distinguished labels either prevent the models from inferring important patterns or causes them to bias toward an incorrect class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' In the case of D2A, inconsistency was so prevalent that the model fails to make any correct predictions unless training with a consistent training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed the model would default predictions to the most prevalent label of an inconsistent cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' which is the non-vulnerable class in the case of D2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Using a Mann- Whitney U test [61] (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='05), we confirmed that removing inconsistency issues significantly improved performance for the most afflicted datasets (D2A and Juliet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We observed that increased consistency has a larger positive influence on precision in comparison to recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Recall actually even decreased when using consistent datasets for Devign (although the overall performance still increased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This is likely because inconsistent clusters more often produce false positives, due to the larger number of non-vulnerable samples in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Performance impacts were relatively small for Big-Vul and Devign, due to the relatively small number of affected entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We were unable to confirm whether the performance changes using these datasets were statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, we expect that these inconsistencies actually point to larger problems in the non-vulnerable classes of these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' There is likely to be a much larger number of latent vulnerabilities or misclassified fixing commits, but we only observe a low TABLE VII FREQUENCY FOR TYPES OF MISSING VALUES IN DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Dataset Truncation Empty Declaration Total Start End Both Big-Vul 32,973 133 140 0 0 33,246 Devign 814 265 9 0 0 1,088 D2A 0 0 0 10,824 13,300 24,124 number via inconsistent labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Both Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [33] and Croft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [13] found mislabelled latent vulnerabilities to impact downstream SVP models significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data collection processes must be improved to ensure consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency issues arise due to a lack of label indicators or checks for non-vulnerable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst measured values are small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' they may be an indicator of more significant prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consistency can be a significant issue that prevents the model from learning necessary patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Completeness Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Completeness can either refer to the complete- ness of information within a dataset, or to the values of indi- vidual data entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As the former requires external reference information, we focus on the latter as it is an inherent property of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For vulnerability datasets, source code can be missing information if the values do not contain all the code of the original function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To detect missing information, we automatically checked for incomplete code snippets by analysing the C/C++ function syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found that some code entries were missing or cut off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Overall, we observed completeness values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='824, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='944, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='981, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='0 for Big-Vul, Devign, D2A and Juliet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These relatively high values imply that completeness is less frequently problematic than the other data quality attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Missing information was only present in three of the four analyzed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Table VII displays the frequency of the truncation types present in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We have excluded Juliet because none of its entries contained missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found truncation at the start of functions to occur predominantly in the Big-Vul dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Return types of func- tion definitions were truncated when they were defined over multiple lines, as function parsers commonly start on the line containing the function name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We also found a few functions in the Big-Vul and Devign dataset to be cut off prematurely, missing functional lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We were unable to determine the exact cause for this truncation as we did not have access to the scripts used to produce the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We hypothesise that complexities within the source code confuse the lexicograph- ical parser being used to extract them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, many of the early truncated samples contained additional curly brackets (}) within literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' D2A was resilient to truncation but it contains empty miss- ing values, for which no code was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These occurred when the static analysis tools flagged lines in a code file outside of any containing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, D2A contains 13,300 single line function declarations that do not contain any functional source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' To see the impact of missing information on SVP models, we set aside a common test set for each dataset containing no incomplete entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We then split the remaining entries of each dataset into equal-sized halves to produce two training sets: one containing incomplete data values and the other without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The MCC performance marginally increased for the complete training sets on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, we were unable to confirm any performance change for MCC, precision or recall as significant using a Mann-Whitney U test [61] (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst the amount of information truncated can be of arbitrary complexity, it appears to be a relatively small part of the overall functions and occurs relatively infrequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, we still advise practitioners to ensure the completeness of software vulnerability data in future, as more severe issues may produce larger impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Completeness issues can arise during data collection, but these issues are easily solvable and do not have a high impact as they cause relatively little missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness Rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness aims to ensure that datasets have homogeneous temporal characteristics to their application con- texts [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This is known in the machine learning domain as concept drift [67]: a scenario in which the relationship between the input data and target variable changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' It is important for vulnerability datasets to stay up to date as vulnerabilities and source code have an evolving nature [68], [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We denote the date of an entry in a vulnerability dataset as the date that the vulnerability was reported via the dataset’s labelling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness does not relate to the synthetically created Juliet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness pertains to an entire dataset rather than individual data points, so we selected a standard non- contextual method for concept drift detection [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We used the Jensen-Shannon divergence metric [70] to represent current- ness, as it measures the statistical distance of the original and current data in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The formula for this metric is reported in [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For simplicity, we denoted the original and current data as the oldest and newest half of the dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We represented the distribution of the vulnerability data through a Bag-Of-Tokens set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We tokenised all source code in a set using a lexicographical parser and then normalised the values based on the total frequency to obtain a probability distribution of the occurrences of each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Training Validation Test 1 Test 2 Test 3 Test 4 Test 5 Timestep: 0 1 2 3 4 5 6 7 8 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness impact experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' As the Jensen-Shannon divergence metric measures dissim- ilarity, we compute currentness as one minus this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We obtained currentness values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='761 (Big-Vul), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='811 (Devign) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='844 (D2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' These values are relatively high for this attribute and are unlikely to indicate concept drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We used a similar experimental setup to McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' [69] to determine whether vulnerability data is a moving target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We sorted all entries by date and then split each dataset into ten equal partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The four earliest partitions were used to train an SVP model, the fifth partition was used for tuning, and the remaining five were used as individual test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Figure 5 displays the experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, using a Kendall rank correlation test [71] (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content='05), we observed no significant decrease in model performance for MCC, precision or recall as the time between the training and test set increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Currentness issues were not observed for software vulner- ability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' They exhibit good temporal distributions of data as they are collected over a long time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' DISCUSSION Software vulnerability datasets are particularly sensitive to data quality challenges due to the difficulties of data prepa- ration [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Most existing SVP studies focus on advances in modelling but often overshadow data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Consequently, our systematic analysis of inherent data quality attributes has revealed critical data issues afflicting the current state of soft- ware vulnerability datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data preprocessing for software vulnerability data is currently cursory or inconsistent [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' There is a lack of methods to guide data cleaning efforts for SVP research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We present the following lessons learned from our quality assessment of existing datasets: Be wary of reusing existing datasets without first check- ing the data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Uniqueness is poor for software vulnerability data, so avoid using evaluation setups that lead to significant duplication across the training and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Inconsistently labelled data points should be removed, based on the causes of such inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Source code entries with missing or incomplete informa- tion should be removed or amended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Issues in uniqueness, consistency, and completeness can be detected with rule-based syntactic filters, as we have done in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, we can theoretically solve these issues through exclusion of noisy samples that do not satisfy the quality attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' However, it may not be that easy in practice as software vulnerability data is very scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' SVP requires large datasets [16], so removing noisy samples may make datasets insufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Figure 6 displays the ratio of clean samples that we can automatically detect for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, we manually observed the data accuracy issues to be severe, but there is no existing method to automat- ically detect such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data inaccuracy could potentially decrease the number of clean entries by a further 20-71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Hence, we need to solve the underlying causes of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' From the findings we have obtained, we summarise the causes of the current major data quality issues below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We provide some directions for researchers to investigate, to help with overcoming these challenges: Automatic data collection often leads to data inaccu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found the major cause of incorrectly labelled vulnerabilities to stem from inaccuracies in vulnerability fix identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Either incorrect commits or line changes were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Substantial work has been conducted for ML-based models to identify correct vulnerability patches [12], [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Semantic filters or heuristics for correct vul- nerability fixing lines is currently lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Source code duplication may make datasets lack diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' An underlying problem for data collection is a lack of sample diversity and uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Whilst we are constrained in the vulnerability samples we can collect, we have a selection choice for the non-vulnerable class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Thus, we suggest the need for development of data collection heuristics that can obtain more diverse non- vulnerable code samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similarly, there is a need for better synthetic data generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Bug seeding has shown promising results in this regard [17], but this technique still relies on the data quality of the real-world bugs from which the technique infers the seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Unknown vulnerabilities can introduce label inconsis- tency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Label inconsistency problems arose from underly- ing problems for the non-vulnerable class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Big-Vul sam- ples contained undetected vulnerabilities, and both De- vign and D2A contained undocumented vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This is particularly problematic as we lack a label source for non-vulnerable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Semi-supervised semantic filters have shown promise for reducing noise in non-vulnerable labels [13], [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Synthetic datasets need clearly defined usage guidelines when used as training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' THREATS TO VALIDITY Construct Validity: Our interpreted data quality analysis may not perfectly represent the target attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We have formed our analysis using standard practices from relevant domains [48] and existing knowledge of software vulnerability data practices [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Requirements elicitation using domain Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Ratio of clean to unclean samples in a dataset that can be automatically detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' experts would help improve these claims in future [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' The need for manual analysis of some attributes is also a potential limitation, as it may contain bias or inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We used two independent raters to minimize such impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We used CORE rankings as a criteria for our dataset selection, even though CORE journal rankings have become deprecated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We consider these ranking still sufficient, as they were only deprecated two months prior to the date of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Internal Validity: The outcomes of our impact analysis experiments may be affected by confounding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We analysed each data attribute individually, so other data quality issues were present during each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' More work is required to examine data quality attributes cumulatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' External Validity: We have constrained our analysis to four state-of-the-art datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Measurements are also limited to datasets that contain appropriate metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' For instance, we were unable to investigate the ReVeal dataset [24] due to this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Similarly, we performed impact analysis using a single SVP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' This model has been demonstrated to be state-of- the-art [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, we considered inherent data quality attributes, so the issues remain, regardless of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' CONCLUSION We have systematically examined five data quality attributes for four state-of-the-art software vulnerability datasets, to help improve the validity and trustworthiness of downstream data- driven tasks that rely on this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Our findings revealed that some software vulnerability datasets are prone to data quality issues, particularly in terms of data accuracy, unique- ness, and consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We found 20-71% of vulnerability labels were inaccurate in real-world datasets, which altered performance up to 65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, 0-47% of the labels were inconsistent, which hindered model training completely in the most extreme circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Data quality requires ongoing consideration and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' We advise future researchers and practitioners to consider data quality in more effective detail through the means that we have provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Furthermore, we advocate the importance of data quality and the need to overcome the quality issues that we have observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Lastly, we urge the need for additional investigation into system-dependent data quality attributes to help achieve specific operational needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' DATA AVAILABILITY We have made our data and analysis scripts available via a reproduction package [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' ACKNOWLEDGMENT This work has been supported by the Cyber Security Coop- erative Research Centre Limited whose activities are partially funded by the Australian Government’s Cooperative Research Centre Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE5T4oBgHgl3EQfJQ54/content/2301.05456v1.pdf'} +page_content=' Big-Vul Devign D2A Juliet Clean UncleanREFERENCES [1] G.' metadata={'source': 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+Lue Fan, Yuxue Yang, Feng Wang, Naiyan Wang, and Zhaoxiang Zhang +Abstract—As the perception range of LiDAR expands, LiDAR-based 3D object detection contributes ever-increasingly to the long-range +perception in autonomous driving. Mainstream 3D object detectors often build dense feature maps, where the cost is quadratic to the +perception range, making them hardly scale up to the long-range settings. To enable efficient long-range detection, we first propose a fully +sparse object detector termed FSD. FSD is built upon the general sparse voxel encoder and a novel sparse instance recognition (SIR) +module. SIR groups the points into instances and applies highly-efficient instance-wise feature extraction. The instance-wise grouping +sidesteps the issue of the center feature missing, which hinders the design of the fully sparse architecture. To further enjoy the benefit of +fully sparse characteristic, we leverage temporal information to remove data redundancy and propose a super sparse detector named +FSD++. FSD++ first generates residual points, which indicate the point changes between consecutive frames. The residual points, along +with a few previous foreground points, form the super sparse input data, greatly reducing data redundancy and computational overhead. +We comprehensively analyze our method on the large-scale Waymo Open Dataset, and state-of-the-art performance is reported. To +showcase the superiority of our method in long-range detection, we also conduct experiments on Argoverse 2 Dataset, where the +perception range (200m) is much larger than Waymo Open Dataset (75m). Code is open-sourced at https://github.com/tusen-ai/SST. +Index Terms—3D object detection, LiDAR, autonomous driving, sparse, Waymo Open Dataset, instance segmentation, temporal fusion, +point clustering. +! +1 +INTRODUCTION +A +UTONOMOUS driving systems are eager for efficient +long-range perception, especially in high-speed sce- +narios. Current LiDAR-based 3D object detectors usually +convert sparse features into dense feature maps for further +feature extraction and prediction, which we name as dense +detectors. Dense detectors perform well on current pop- +ular benchmarks [1], [2], [3], where the perception range +is relatively short (less than 75 meters). However, it is +impractical to scale the dense detectors to the long-range +setting (more than 200 meters, Fig. 1). In such settings, the +computational and spatial complexity on dense feature maps +is quadratic to the perception range. Fortunately, the sparsity +of LiDAR point clouds also increases as the perception range +extends (see Fig. 1), and the calculation on the unoccupied +area is essentially unnecessary. Given the inherent sparsity, +an essential solution for efficient long-range detection is +to remove the dense feature maps and make the network +architectures fully sparse. +However, removing the dense feature map is non-trivial +since it plays an indispensable role in current designs. +Commonly adopted sparse voxel encoders [5], [6], [7] only +extract the features on the non-empty voxels. Without dense +feature maps, the object centers are usually empty, especially +for large objects. We name this issue as “Center Feature +Missing (CFM)” (Fig. 2). CFM significantly weakens the +representation power of the center voxels, even making +the center feature empty in some extreme cases like super +large vehicles. However, almost all popular voxel or pillar +based detectors [5], [6], [8], [9], [10] adopt center-based +• +Lue Fan and Yuxue Yang and Zhaoxiang Zhang are with Center for +Research on Intelligent Perception and Computing (CRIPAC), National +Laboratory of Pattern Recognition (NLPR), Institute of Automation, +Chinese Academy of Sciences (CASIA), Beijing 100190, China. E-mail: +{fanlue2019, yangyuxue2023, zhaoxiang.zhang}@ia.ac.cn. +• +Feng Wang and Naiyan Wang are with TuSimple, Beijing 100020, China. +E-mail: {feng.wff, winsty}@gmail.com. +200 𝑚 +Fig. 1. Short-range point clouds (red, from KITTI [2]) v.s. long-range +point clouds (blue, from Argoverse 2 [4]). The radius of the red circle is +75 meters. The sparsity quickly increases as the range extends. +assignment and rely on the center feature since it is an +ideal representation of the whole object. So they have to +first convert sparse voxels to dense feature maps in Bird’s +Eye View after the sparse voxel encoder. Then they resolve +the CFM issue by applying convolutions on the dense feature +maps to diffuse features to instance centers, which we name +as feature diffusion (Fig. 2). +To properly eliminate the dense feature map, we inves- +tigate the purely point-based detectors because they are +naturally fully sparse. However, two drawbacks limit the +usage of point-based methods. (1) The time-consuming +arXiv:2301.02562v1 [cs.CV] 5 Jan 2023 + +. +3JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +Fig. 2. Illustration of center feature missing and feature diffusion on +dense feature maps from Bird’s Eye View. The empty instance center (red +dot) is filled by the features diffused from occupied voxels (with LiDAR +points), after several convolutions. +neighborhood query [11] is the long-standing difficulty +to apply it to large-scale point cloud (more than 100K +points). (2) To reduce the computational overhead, point- +based methods aggressively downsample the whole scene +to a fixed number of points. The aggressive downsampling +leads to inevitable information loss and insufficient recall of +foreground objects [12], [13], especially for small ones. As a +result, very few purely point-based detectors have reached +state-of-the-art performance in the recent benchmarks with +large-scale point clouds. +In this paper, we first propose Fully Sparse Detector +(FSD) to sidestep the issue of center feature missing. FSD +is built upon a general sparse voxel encoder +[5], [6], [7] +for voxel/point feature extraction. Then FSD groups the +points into an instance, and further extract the instance- +level feature and predict a single bounding box from the +integrated instance feature, via a novel Sparse Instance +Recognition (SIR) module. In this way, predictions are made +from the whole instance feature instead of the weak or +missed center feature. As a point-based module, SIR has +several desired properties: (1) Unlike previous point-based +modules, SIR simply treats instances as groups, and does not +apply the time-consuming neighborhood query for further +grouping. (2) Similar to dynamic voxelization [14], SIR +leverages dynamic broadcast/pooling for tensor manipulation +to avoid point sampling or padding. (3) Since the group in +SIR covers the whole instance, it builds a sufficient receptive +field regardless of the physical size of the instance. +To unleash the full potential of FSD, we further utilize +temporal information and propose a Super Sparse 3D Ob- +ject Detector, named FSD++. FSD++ is inspired by human vi- +sual behavior: human is sensitive to and focuses on dynamic +parts of the physical world. In particular, FSD++ utilizes ego- +motion to remove the static parts containing heavy temporal +redundancy, while only retaining the informative dynamic +parts. We name the detected dynamic parts as residual points +since the process is similar to applying the difference between +frames. In this way, we create a super sparse point cloud +consisting of residual points and a small number of past +foreground points from history predictions. FSD++ then +takes the super sparse point cloud as input, achieving a very +efficient detection framework with temporal fusion. We owe +the credit of the high efficiency to the synergy of the fully +sparse characteristic and the super sparse input. We list our +contributions as follows. +• We introduce the concept of Fully Sparse Detector (FSD), +which is the essential solution for efficient long-range +LiDAR detection. We further propose Sparse Instance +Recognition (SIR) to sidestep the issue of Center Feature +Missing (CFM) in sparse feature maps. Combining SIR +with general sparse voxel encoders, we develop an +efficient and effective FSD implementation. +• Based on FSD, we further present the FSD++ framework, +which aggregates a super sparse point cloud from multi- +frames as input, yet removing the temporal redundancy +of point clouds. The proposed framework uncovers the +untapped potential of sparse architecture. We hope our +efforts attract the attention of the community to fully +sparse architecture. +• FSD achieves state-of-the-art performance on the com- +petitive Waymo Open Dataset. Besides, we further apply +our method to the recently released Argoverse 2 dataset +to demonstrate the superiority of FSD in long-range +detection, where FSD is much more efficient than its +dense counterparts. FSD++ achieves comparable per- +formance with mainstream state-of-the-art multi-frame +detectors with minimal additional overhead compared +with single-frame input. +2 +RELATED WORK +In reviewing the evolution of LiDAR-based 3D object de- +tectors, the previous methods could be categorized into +three types by their spatial sparsity: dense detectors, sparse +detectors, and semi-dense detectors. Below, we provide a +brief revisit of previous arts according to spatial sparsity. +2.1 +Voxel-based Dense Detectors +Pioneering work 3DFCN [15] and VoxelNet [16] use dense +convolution for voxel feature extraction. They bring con- +volutional neural networks to the field of LiDAR-based +3D object detection and achieve competitive results at the +time. However, it is inefficient to apply dense convolution +to 3D voxel representation. MV3D [17], PIXOR [18], and +PointPillars [19] adopt 2D dense convolution in Bird’s Eye +View (BEV) feature maps achieving significant efficiency +improvement. We refer to such detectors as dense detectors +since they convert the sparse point cloud into dense feature +maps. +2.2 +Point-based Sparse Detectors +Since PointNet [20] and PointNet++ [11] shed light on the +deep learning for 3D point sets, a series of point-based +detectors have emerged. These purely point-based detectors +are born to be fully sparse. PointRCNN [21] is the pioneering +work of this line of work. 3DSSD [12] accelerates the point- +based method by removing the feature propagation layer +and refinement module. VoteNet [22] first makes a center +voting and then generates proposals from the voted center +achieving better accuracy. Albeit many methods [12], [13], +[23] have tried to accelerate the point-based method, the + +. +. +. +. +.JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +time-consuming point sampling and neighborhood query +are still unaffordable in large-scale point clouds (more than +100k points per scene). So current benchmarks [1], [3] with +large-scale point clouds are still dominated by voxel-based +dense/semi-dense detectors [10], [24], [25]. +2.3 +Semi-dense Detectors +Different from dense detectors, semi-dense detectors incor- +porate both sparse features and dense features. SECOND [5] +employs sparse convolution to extract the sparse voxel +features in 3D space, which then are converted to dense +feature maps in BEV to enlarge the receptive field and +integrate with 2D detection head [26], [27], [28]. Based on +SECOND-style semi-dense detectors, a series of work [29], +[30], [31] made further improvements on the single-stage +paradigm. And other methods attach a second stage for +fine-grained feature extraction and proposal refinement [7], +[8], [10], [32], achieving superior performance. Although +semi-dense detectors become dominating in academia and +industry, related research has stagnated here because the +semi-dense detector cannot be trivially lifted to be fully +sparse as we discussed in Sec. 1. +3 +FSD: FULLY SPARSE 3D OBJECT DETECTION +3.1 +Overall Architecture +Following the motivation of instances as groups, we have +four steps to build the fully sparse detector (FSD): 1) We +first utilize a sparse voxel encoder [5], [6], [7] to extract +voxel features and casts votes for object centers(Sec. 3.2). +2) Instance Point Grouping groups foreground points into +instances based on the voting results (Sec. 3.2). 3) Given the +grouping results, Sparse Instance Recognition (SIR) module +extracts instance/point features and generates proposals +(Sec. 3.3). 4) The proposals are utilized to correct the point +grouping and refine the proposals iteratively (Sec. 3.4). +3.2 +Instance Point Grouping +Classification and Voting +We first extract voxel features +from the point cloud with a sparse voxel encoder, such +as sparse attention blocks in SST [6] or sparse convolution +encoder. Then we build point features by concatenating voxel +features and the offsets from points to their corresponding +voxel centers. These point features are passed into two heads +for foreground classification and center voting. The voting is +similar to VoteNet [22], where the model predicts the offsets +from foreground points to corresponding object centers. L1 +loss [27] and Focal Loss [33] are adopted as voting loss Lvote +and semantic classification loss Lsem. +Connected Components Labeling (CCL) +To group points +into instances, we regard all the predicted centers (red dots +in Fig. 3) as vertices in a graph. Two vertices are connected +if their distance is smaller than a certain threshold. Then +a connected component in this graph can be viewed as an +instance, and all points voted to this connected component +share a group ID. Unlike the ball query in VoteNet, our +CCL-based grouping avoids fragmented instances in most +cases. Although there are many elaborately designed instance +grouping methods [34], [35], [36], we opt for the simple CCL +because it is adequate in our design and can be implemented +by the efficient Union-Find algorithm [37] in parallel. +3.3 +Sparse Instance Recognition +3.3.1 +Preliminaries: Dynamic Broadcast/Pooling +Given N points belong to M groups, we define their cor- +responding group ID array as I in the shape of [N, ] and +their feature array as F in the shape of [N,C], where C is +the feature dimensions. F (i) is the feature array of points +belonging to the i-th group. Dynamic pooling aggregates +each F (i) into one group feature gi of shape [C, ]. Thus we +have gi = p(F (i)), where p is a symmetrical pooling function. +The dynamic pooling on all group features G of shape [M,C] +is formulated as G = p(F, I). The dynamic broadcast can be +viewed as the inverse operation to dynamic pooling, which +broadcasts gi to all the points in the i-th group. Since the +broadcasting is essentially an indexing operation, we use +the indexing notation [ ] to denote it as G[I], which is in the +shape of [N,C]. Dynamic broadcast/pooling is very efficient +because it can be implemented with high parallelism on +modern devices and well fits the sparse data with dynamic +size. We provide an efficient implementation and runtime +evaluation in Appendix A. +The prerequisite of dynamic broadcast/pooling is that +each point uniquely belongs to a group, i.e. groups should +not overlap with each other. Thanks to the fact that there is +no overlap among instances in the real 3D world, the groups +do not overlap with each other naturally. +3.3.2 +Formulation of Sparse Instance Recognition +After grouping points into instances in Sec. 3.2, we can +directly extract instance features via some basic point- +based networks like PointNet, DGCNN, etc. There are three +elements to define a basic point-based module: group center, +pair-wise feature and group feature aggregation. +Group center +The group center is the representative point +of a group. For example, in the ball query, it is the local +origin of the sphere. In SIR, the group center is defined as +the centroid of all voted centers in a group. +Pair-wise feature +defines the way to pair group center and +neighbor points input for group-aware neighbor point feature +extraction. SIR adopts two kinds of pair-wise features: 1) the +relative coordinate between the group center and each point, +2) the concatenation of the group feature and each point fea- +ture. Taking feature concatenation as an example and using +the notations in 3.3.1, the pair-wise feature can be denoted +as CAT(F, G[I]), where CAT is channel concatenation. +Group feature aggregation +In a group, a pooling function +is used to aggregate neighbor features. SIR applies dynamic +pooling to aggregate feature array F. Following the notations +in 3.3.1, we have G = p(F, I), where G is the aggregated +group features. +Integration +Combining the three basic elements, we could +build many variants of point-based operators, such as Point- +Net [20], DGCNN [38], Meta-Kernel [39], etc. Fig. 4 illustrates +the basic idea of how to build an instance-level point operator +with dynamic broadcast/pooling. In our design, we adopt +the formulation of VFE [16] as the basic structure of SIR +layers, which is basically a two-layer PointNet. In the l-th +layer of SIR module, given the input point-wise feature array + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +Input Point Cloud +Sparse Voxel Feature +Extractor +Point-wise Classification & +Center Voting +Not Connected +Connected +SIR Module +Rule out outliers +Add missing +points +SIR2 Module +Prediction 1 +Prediction 2 +Predict +proposal +Predict +proposal +Instance 1 +Instance 2 +Group Correction +Instance Point Grouping +via CCL +Instance-wise feature +extraction and prediction +Instance-wise feature +extraction and prediction +Corrected instance 1 +Corrected instance 2 +Fig. 3. Overall architecture of FSD. For simplicity, we only use two instances to illustrate the pipeline. Red dots are the voted centers from each +LiDAR point (blue dots). The SIR module and the SIR2 module all contain 3 SIR layers. +Group 1 +Group 2 +𝑁! × 3 +𝑁" × 3 +2 × 3 +𝑁 × 3 +2 × 𝐶 +𝑁 × 𝐶 +N × 𝐶 +N × 𝐶 +𝑁! + 𝑁" = 𝑁 +Grouping results +Group +centers +Broadcasted +group centers +𝑁 × 3 +Input point +coordinates +Group +features +Point-wise +subtraction +Dynamic +Pooling +Dynamic +Broadcast +Input point +features +Broadcasted +group features +Input point +features +Pair-wise +feature extraction +Pair +Instance 1 +Instance 2 +Output +point feature +Fig. 4. Illustration of building instance-level point operators with dynamic broadcast/pooling. Best viewed in color. Left: calculating center-to-neighbor +offsets given raw point clouds. Right: updating point features. Note that the operation is parallel among all instances. +Fl, point coordinates array X, the voted center X′ and group +ID array I, the output of l-th layer can be formulated as: +F ′ +l = LinNormAct +� +CAT +�Fl, X − pavg(X′, I)[I] +�� , +(1) +Fl+1 = LinNormAct (CAT (F ′ +l , pmax(F ′ +l , I)[I])) , +(2) +where LinNormAct is a fully-connected layer followed by +a normalization layer [40] and an activation function [41]. +The pavg and the pmax are average-pooling and max-pooling +function, respectively. The output Fl+1 can be further used +as the input of the next SIR layer, so our SIR module is a +stack of a couple of basic SIR layers. +3.3.3 +Sparse Prediction +With the formulation in Eqn. 1 and Eqn. 2, SIR extracts +features of all instances dynamically in parallel. And then +SIR makes sparse prediction for all groups. In contrast to +two-stage sparse prediction, our proposals (i.e., groups) +do not overlap with each other. Unlike one-stage dense +prediction, we only generate a single prediction for a group, +which significantly reduces the cost of prediction head. It +is noteworthy that the fully sparse architecture may face a +severe imbalance problem: short-range objects contain much +more points than long-range objects. Some methods [39], +[42] use hand-crafted normalization factors to mitigate the +imbalance. Instead, SIR avoids the imbalance because it +only generates a single prediction for a group regardless +of the number of points in the group. In most cases, a group +corresponds to only a single ground truth box. +Specifically, for each SIR layer, there is a Gl = pmax(F ′ +l , I) +in Eqn. 2, which can be viewed as the group features. We +concatenate all Gl from each SIR layer in channel dimension +and use the concatenated group features to predict bounding +boxes and class labels via MLPs. All the groups whose +centers fall into ground-truth boxes are positive samples. +For positive samples, the regression branch predicts the +offsets from group centers to ground-truth centers and object +sizes and orientations. L1 loss [27] and Focal Loss [33] are +adopted as regression loss Lreg and classification loss Lcls, +respectively. +3.4 +Group Correction +There is inevitable incorrect grouping in the Instance Point +Grouping module. For example, some foreground points +may be missed, or some groups may be contaminated +by background clutter. So we leverage the bounding box +proposals from SIR to correct the grouping. The points inside +a proposal belong to a corrected group regardless of their +previous group IDs. Since a few points may fall into multiple +proposals, we simply make copies for these points along +with their features and assign different copies to difference +proposals. After correction, we apply an additional SIR to +these new groups. To distinguish it from the first SIR module, +we denote the additional SIR module as SIR2. +SIR2 predicts box residual from the proposal to its +corresponding ground-truth box, following many two-stage +detectors. To make SIR2 aware of the size and location of a +proposal, we adopt the offsets from inside points to proposal +boundaries as extra point features following [43]. The regres- +sion loss is denoted as Lres = L1(∆res, � +∆res), where ∆res is +the ground-truth residual and � +∆res is the predicted residual. +Following previous methods [7], [8], the 3D Intersection over +Union (IoU) between the proposal and ground-truth serves as +the soft classification label in SIR2. Specifically, the soft label +q is defined as q = min(1, max(0, 2IoU −0.5)), where IoU is + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +5 +the area of Intersection over Union (IoU) between proposals +and corresponding ground-truths. Then cross entropy loss is +adopted to train the classification branch, denoted as Liou. +Taking all the loss functions in grouping (Sec. 3.2) and sparse +prediction into account, we have +Ltotal = Lsem + Lvote + Lreg + Lcls + Lres + Liou, +(3) +where we omit the weight of each term for simplicity. +3.5 +Discussion +The center voting in FSD is inspired by VoteNet [22], while +FSD has two essential differences from VoteNet. +• After voting, VoteNet simply aggregates features around +the voted centers without further feature extraction. FSD +goes beyond this and builds a highly efficient SIR module +taking advantage of dynamic broadcast/pooling, allowing +for further instance-level feature extraction. Thus, FSD +extracts more powerful instance features, which is experi- +mentally demonstrated in Sec. 5.5. +• VoteNet is a typical point-based method. As we discussed +in Sec. 1, it aggressively downsamples the whole scene to +a fixed number of points for efficiency, causing inevitable +information loss. Instead, the dynamic characteristic and +efficiency of SIR enable fine-grained point feature ex- +traction from any number of input points without any +downsampling. In Sec. 5.5, we showcase the efficiency of +our design in processing large-scale point clouds and the +benefits of fine-grained point representation. +4 +FSD++: FSD WITH SUPER SPARSE INPUT +It is well known that aggregating multiple frames as an +input benefits performance. However, naive aggregation +could result in a denser point cloud, which slows down +the algorithm significantly, especially in the architecture +with sparse operations. This motivates us to pursue more +sparse input data by removing temporal redundancy from +the original point cloud stream. Thanks to the fully sparse +characteristic, the fully sparse model could greatly benefit +from the increase of sparsity after redundancy removal. Thus +a natural question arises: How can we remove the redundancy +while retaining the informative parts in advance? The similarities +between consecutive point cloud frames offer us a potential +solution to this question. +In particular, the spatial distribution of points varies con- +tinuously and smoothly in a sequence. We name the points +that change between consecutive frames as residual points. +The residual points are informative since they represent new +observations in a time step. Combining the residual points +and history predictions, detectors have sufficient knowledge +to infer about current objects. In this paradigm, the residual +points and previous foreground points together form a super +sparse point cloud. FSD could directly take them as input for +much more efficient object detection. +4.1 +Residual Points Probing +LiDAR sensors capture plenty of newly observed foreground +points at each time step. These points can be attributed to +two main sources: (1) objects moving to new positions; (2) +occluded regions becoming visible. These newly observed +points are referred to as residual points. Residual points are +critical to locate moving objects and detect recently emerged +objects. As we mentioned before, the residual points could +be detected from the changes of point spatial distribution. +The residual point detection algorithm must fulfill two +key requirements. (1) The algorithm is supposed to be +robust to tiny disturbances of points, which might be caused +by sensing noise or tiny ego-motion estimation error. It +is unexpected that such point disturbances are detected +as residual points. (2) The algorithm should be highly +efficient to handle millions of points from multiple frames. In +particular, each frame in WOD contains up to 200,000 points. +Several straightforward solutions meet the first require- +ment, i.e. ball query or voxelization into dense occupancy +maps. A point can be viewed as a residual point if no +previous points fall into its neighborhood defined by the +ball query radii. The residual points can also be detected +by the simple difference between the two dense occupancy +maps. Although proper ball query radii or voxel sizes bring +robustness to point disturbances, these solutions still come +with either high computational complexity (O(N 2)) or a +huge memory footprint. +To fulfill both of the demands outlined above, we resort +to hashing and design an algorithm shown in Algo. 1, named +Residual Points Probing (RPP). RPP consists of two steps. (1) +It first quantizes the point coordinates into integers. The +granularity of quantization controls the robustness to point +disturbances. (2) For each point, RPP verifies if it is a residual +point by hash probing. Specifically, RPP first builds a hash +table from previous quantized points. The key set of the hash +table is denoted as K ⊂ Z3, which is the quantized integer +coordinates. And value set of the hash table is denoted +as V = {1, 0}, where 1 indicates the slot is occupied and +0 indicates the slot is unoccupied. RPP then uses current +quantized coordinates to probe the hash table. If a current +point hits an unoccupied slot, it is treated as a residual point. +Here is a hidden assumption in RPP that we assume two +points are the same if they occupy the same voxel after +quantization. We adopt the well-known open addressing for +probing and the double hashing as the hash function to reduce +hash collisions. +Algorithm 1: Efficient Residual Points Probing +Input: current points Pcur, previous points Ppre, +voxel size s, load factor α, hash function h +Output: Residual points of current frame ∆Pcur +�Pcur ← Quantize(Pcur, s); +�Ppre ← Quantize(Ppre, s); +Initialize empty hash table T of length | �Ppre|/α; +Initialize empty residual point set ∆Pcur; +foreach �pi in �Ppre do +sloti ← Probe(T, h(�pi)); +if sloti is not occupied then +sloti ← occupied flag; +foreach �pi in �Pcur do +sloti ← Probe(T, h(�pi)); +if sloti is not occupied then +Add pi to ∆Pcur; +return ∆Pcur + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +6 +✘ +✓ +✘ +✘ +✘ +✘ +✘ +✘ +✘ +✘ +✘ +✘ +✘ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +(a) Single- +frame points +(b) Multi- +frame points +(c) Residual +points with +change +blindness +(d) Residual +points w/o +change +blindness +𝑇! +𝑇" +𝑇# +𝑇$ +𝑇% +Fig. 5. Illustration of temporal point manipulations. Different colors indicate points from different time steps. Gray points are the sampled skeleton +points. Red cross  means the observation is too weak to be recognized as foreground objects (not really, just for illustration). Green check mark  +means the observation is strong enough. (a) Points from a single frame are too weak. (b) After multi-frame point aggregation, the detector generates +true positives from time step T1. (c) Single-frame residual points suffer from change blindness. Since the detector cannot generate a true positive in +T0 for skeleton point sampling, we still only have the weak residual points in T1. In this way, the detector outputs false negatives all the time. (d) +Detector makes the right prediction in T1 with residual points from two frames (max age is 2). So in the T2, the detector could leverage the prediction +in T1 for skeleton point sampling, alleviating the change blindness. +Residual Point Probing is efficient in terms of both +memory and speed. For instance, we assume there are N +unique quantized coordinates in a point cloud clip. If we +expect the collision rate less than α, the length of the hash +table should be N/α. Empirically, N is around 500,000 in a +5-frame point cloud in WOD. A slot state can be represented +as a single bit, and let α equal to 0.1. We have that the +memory cost of this hash table is around 0.6MB. Moreover, +the probing of each point is independent with each other, +allowing for high parallelism in GPU. +Formally, we denote the RPP process as follows: +∆Pt = Pt − +B +� +i=1 +Pt−i, +(4) +where ∆Pt is the detected residual points in time step t +and Pt is the raw points in time step t. The notation “X − +Y ” means removing the intersection of X and Y from X, +equivalent to X \(X ∩Y ). And the union means point cloud +concatenation. B is the number of previous frames used in +RPP, which we term as base frames. +4.2 +Skeleton Point Sampling +Since residual points contain only new observations of the +current time step, detectors require additional information +from previous frames for sufficient input. To incorporate +this historical data, we use previously predicted boxes to +crop previous foreground points, while discarding others +outside of the boxes. The cropped points are placed into +the current frame after ego-motion compensation. However, +the foreground points from multiple previous frames are +still essentially redundant. Especially, the quite many points +on short-range objects from multiple frames could lead to +unnecessary overhead. +To reduce the redundancy from multi-frame foreground +points, we further sample within these cropped points. +Intuitively, we expect the sampled points contain the minimal +information models need to make proper predictions. In this +sense, we refer to such a minimal subset of cropped points +as skeleton points, because they depict the basic structure +or “skeleton” of objects. Specifically, we try three kinds of +sampling methods: random sampling, farthest points sampling, +and voxel sampling. All the sampling methods are applied +inside the previously predicted bounding boxes. For random +sampling and farthest points sampling, we adopt a prede- +fined maximal point threshold NT . We sample NT points +inside the bounding boxes which contain points more than +NT . For voxel sampling, we adopt dynamic voxelization [14] +to voxelize points. All the points falling into a voxel are +reduced to a single point by average pooling. +4.3 +Treatment to Change Blindness +Theoretically, by combining the skeleton points and residual +points, a model is able to make predictions in current frames. +However, a phenomenon known as “change blindness” +can hinder performance. Change blindness refers to the +human visual system’s tendency to overlook progressive +small changes in a scene, even if the aggregated changes of +multiple time steps are significant. A similar issue can occur +in our case. Thinking of a vehicle nearly entering into the +sensing range of LiDARs in time step t, only a small part of +the vehicle can be observed. The detector is very likely to +recognize it as background, so RPP will remove these points +in time step t + 1 and only keep a small number of new +points of the vehicle as residual points. In this way, if the +vehicle appears slowly, the detector might never recognize it. +Fig. 5 demonstrates the change blindness. + +4 +2 +0 +-2 +-4 +-2 +0 +2 +4 +6 +84 +2 +. +. +8 +. +. +. +0 +！ +. +. +1 +. +. +-2 +-4 +-2 +0 +2 +4 +84 +2 +. +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +-2 +-4 +-2 +0 +4 +84 +2 +0 +-2 +-4 +-2 +0 +4 +84 +2 +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +. +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +. +. +. +. +. +. +-2 +-4 +-2 +0 +2 +4 +84 +2 +. +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +06881 +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +. += +0 +. +-2 +-4 +-2 +0 +2 +4 +84 +2 +0 +-2 +-4 +-2 +0 +2 +4 +84 +2 +! +. +0 +. +. +. +. +. +8 +. +-2 +-4 +-2 +0 +2 +4 +84 +2 +8 +. +. +. +0 +！ +. +. +. +. +-2 +-4 +-2 +0 +2 +4 +0JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +7 +To remedy the change blindness, we introduce a max age +M for residual points. In other words, the detector takes +residual points from at most M steps as input. Formally, the +detector takes �M−1 +i=0 ∆Pt−i as accumulated residual points +for input in time step t. +4.4 +Integrated Super Sparse Input +The input point clouds consist of two parts: previous skeleton +points and residual points from multiple time steps. Formally, +for an N-frame FSD++ detector, we have the final input +points in time step t as follows: +P in +t += +� N +� +i=1 +P s +t−i +� +∪ +�M−1 +� +i=0 +∆Pt−i +� +, +(5) +where P s +t is the skeleton points at time step t. P in +t +is much +more sparse than the raw point clouds and directly sent into +the FSD detector. Fig. 7 shows examples of P s, ∆P and P in. +4.5 +Training and Inference Pipeline +The training and inference pipeline for FSD++ differs from +the standard approach due to its use of history predictions +and temporal information. Fig. 6 summarizes the overall +pipeline. +4.5.1 +Training +To utilize history predictions, the input point cloud stream +must be arranged in the temporal order. However, the +ordered input stream affects the model training due to a +lack of data shuffling. Another problem is that the history +predictions are not reliable in the early training stages. +Considering these two issues, we use a well-trained FSD +detector to generate offline predictions of the entire training +set. During training, for every sample, we load it along with +its previous offline predictions to sample skeleton points. +Ground truth boxes seem to be an alternative to the offline +predictions. However, the distribution gap between ground +truth used in training and predicted boxes used in inference +is considerable. Thus we adopt the offline predictions instead +of ground-truth boxes. +4.5.2 +Inference +In the online inference phase, the input point cloud stream is +naturally in the temporal order. For a point cloud sequence, +the predictions of its first frame are from the well-trained +FSD detector. These predictions are regarded as the “previous +predictions” of the first frame, which are called the seed +predictions of a sequence. We maintain several queues to +cache some historical data that could be used more than +once. For example, in a N-frame FSD++ pipeline, the raw +points and skeleton points of time step t could be reused +from time step t + 1 to t + N − 1. +5 +EXPERIMENTS +5.1 +Setup +5.1.1 +Dataset +Waymo Open Dataset (WOD) In our experiments, we use +WOD [1] as the primary dataset to evaluate the performance +of our proposed method. WOD is the most trustworthy +benchmark for LiDAR-based 3D object detection. With 1150 +sequences and more than 200,000 frames, WOD is currently +the largest dataset of its kind. Among them, 798 sequences +are used for training, 202 for validation, and 150 for testing. +The detection range in WOD is 75 meters (cover area of +150m × 150m). +Argoverse 2 (AV2) We further conduct long-range experi- +ments on the recently released Argoverse 2 dataset [4] to +demonstrate the superiority of FSD in long-range detection. +AV2 has a similar scale to WOD, and it contains 1000 +sequences in total, 700 for training, 150 for validation, and +150 for testing. In addition to average precision (AP), AV2 +adopts a composite score as an evaluation metric, which takes +both AP and localization errors into account. The perception +range in AV2 is 200 meters (cover area of 400m × 400m), +which is much larger than WOD. Such a large perception +range leads to a huge memory footprint for dense detectors. +5.1.2 +Model Configuration +To demonstrate the generality of SIR, we build two FSD vari- +ants. FSDsst adopts the emerging single stride sparse trans- +former [6] as the sparse voxel feature extractor. FSDspconv is +built upon sparse convolution based U-Net in PartA2 [7]. In +the experiments of FSD, we use FSDsst in the experiments +unless otherwise specified. In the experiments of FSD++, we +use FSDspconv as our detector since the highly optimized +engineering of SpConv makes it more efficient than the SST +backbone with multi-frame input. +5.1.3 +Implementation Details +Our implementation is based on popular MMDetec- +tion3D(v0.15) [44]. In FSDsst, we use 4 sparse regional +attention blocks [6] as our voxel feature extractor. The SIR +module and SIR2 module consist of 3 and 6 SIR layers, +respectively. A SIR layer is defined by Eqn. 1 and Eqn. 2. Our +SST-based model converges much faster than SST, so we train +our models for 6 epochs for ablation study, instead of the 2× +schedule (24 epochs) in SST. For FSDspconv, in addition to the +6-epoch schedule, we adopt a longer schedule (12 epochs) +for better performance. Different from the default setting in +MMDetection3D, we decrease the number of pasted instances +in the CopyPaste augmentation. In FSD, some scarce classes +like cyclist prone to be over-fitted with too many pasted +instances. All experiments in Argoverse 2 dataset adopt a +12-epoch schedule. The models for performance analysis +(Sec. 5.3 ∼ Sec. 5.6) are trained on 8 RTX 2080Ti GPUs with +batch-size 2. And the models in Table. 1 are trained on 8 RTX +3090 GPUs with batch-size 2. More details can be found in +our released code. +5.2 +Main Results of FSD and FSD++ +We first compare FSD with state-of-the-art detectors and +our baseline in Table 1 and Table 2. In the validation split, +FSD/FSD++ achieves state-of-the-art average performance +(L2 mAPH) in single-frame/multi-frame settings, respec- +tively. In test split, FSD achieves the best performance on all +classes among all single-frame detectors. Meanwhile, FSD++ +with 7-frame input surpasses all detectors with up to 100- +frame input, in terms of average metric. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +8 +𝑃!"# +𝐵!"# +∗ +𝑃!"% +𝐵!"% +∗ +𝑃!"% +Skeleton +Sampling +𝑃!"# +& +𝑃!"% +& +Residual Point +Probing +𝑃! +𝑃!"# +Δ𝑃! +FSD +Skeleton +Sampling +𝐵! +𝑃!"% +𝐵!"% +𝑃!"# +& +𝑃!"% +& +Residual Point +Probing +Δ𝑃! +FSD +Skeleton +Sampling +𝐵! +𝑃!"% +𝑃!"# +𝑃! +𝑃 +Point clouds +loaded from disk +𝐵∗ +Offline predicted boxes +loaded from disk +Data loaded from +memory +Data to be cached +in memory +Points concatenation +Training data flow +Inference data flow +Fig. 6. The overall architecture of FSD++. In training, we adopt offline predictions to approximate history predictions. During inference, the detector +uses previous online predictions for skeleton point sampling. When the max age is larger than one, there will be some other ∆Pt−i from time step +t − i. For simplicity, we only present ∆Pt here. +(a) Moving cars with static ego-vehicle +(b) Moving cars with moving ego-vehicle +(c) Static cars and moving pedestrian +Fig. 7. Examples of super sparse input point clouds. Residual points are colored in red. Previous foreground points are in blue. Gray points will not be +sent into the detector. (a) Moving cars cause apparent residual points. And some occluded points in the ground plane become visible due to car +movement. (b) The ego-vehicle is moving, which causes some points in the ground plane to be detected as residual points. (c) Residual points are +detected on the moving pedestrian instead of the static cars. +It is also noteworthy that FSD and FSD++ are much more +efficient than most of the previous arts, especially in the +multi-frame setting and long-range setting. We elaborate this +in Sec. 5.4 and Sec. 5.7. +5.3 +Study of Treatments to Center Feature Missing +5.3.1 +Quantitative Experiments +In what follows, we conduct experiments on WOD to inves- +tigate the issue of Center Feature Missing (CFM). We first +develop several models with different characteristics. Note +that all the following models adopt the same voxelization +resolution, so they face the same degree of CFM at the +beginning. +• FSDplain: After the sparse voxel encoder, FSDplain directly +predicts the box from each voxel. The voxels inside ground- +truth boxes are assigned as positive. Although FSDplain uses +the most straightforward solution for CFM, it suffers from +the large variance of regression targets and low-quality +predictions from informative voxels. +• SSTcenter: It replaces the anchor-based head in SST with +CenterHead [9], [28]. Based on the sparse voxel encoder, +SSTcenter converts sparse voxels into dense feature maps +and applies several convolutions to diffuse features to the +empty object centers as in Fig. 2. Then it makes predictions +from the diffused center feature. +• FSDnogc: It removes the group correction and SIR2 module +in FSD. +• CenterPoint-PP: It does not resort to any sparse voxel +encoders. Instead, it applies multiple dense convolutions +soon after voxelization for feature diffusion, greatly elimi- +nating CFM. It is also equipped with CenterHead to avoid +large variance of regression targets. +There is usually a quite large unoccupied area around +the centers of large vehicles. Thus the performance of large +vehicles is an appropriate indicator that reveals the effect of +CFM. So we build a customized evaluation tool, which breaks +down the object length following the COCO evaluation [59]. +Then we use it to evaluate the performance of vehicles with +different lengths. Table 3 shows the results, and we list our +findings as follows. +• Comparing FSDplain with SSTcenter, they share the same +attention-based sparse voxel encoder. However, the trend +is totally opposite w.r.t vehicle size. With feature diffusion, + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +9 +Methods +#. +frames +mAP/mAPH +L2 +Vehicle 3D AP/APH +Pedestrian 3D AP/APH +Cyclist 3D AP/APH +L1 +L2 +L1 +L2 +L1 +L2 +SECOND [5] +1 +61.0/57.2 +72.3/71.7 +63.9/63.3 +68.7/58.2 +60.7/51.3 +60.6/59.3 +58.3/57.0 +MVF [14] +1 +-/- +62.9/- +-/- +65.3/- +-/- +-/- +-/- +AFDet [45] +1 +-/- +63.7/- +-/- +-/- +-/- +-/- +-/- +Pillar-OD [46] +1 +-/- +69.8/- +-/- +72.5/- +-/- +-/- +-/- +RangeDet [39] +1 +65.0/63.2 +72.9/72.3 +64.0/63.6 +75.9/71.9 +67.6/63.9 +65.7/64.4 +63.3/62.1 +PointPillars [19] +1 +62.8/57.8 +72.1/71.5 +63.6/63.1 +70.6/56.7 +62.8/50.3 +64.4/62.3 +61.9/59.9 +Voxel RCNN [32] +1 +-/- +75.6/- +66.6/- +-/- +-/- +-/- +-/- +RCD [42] +1 +-/- +69.0/68.5 +-/- +-/- +-/- +-/- +-/- +VoTr-TSD [47] +1 +-/- +74.9/74.3 +65.9/65.3 +-/- +-/- +-/- +-/- +LiDAR-RCNN [43] +1 +65.8/61.3 +76.0/75.5 +68.3/67.9 +71.2/58.7 +63.1/51.7 +68.6/66.9 +66.1/64.4 +Pyramid RCNN [48] +1 +-/- +76.3/75.7 +67.2/66.7 +-/- +-/- +-/- +-/- +Voxel-to-Point [49] +1 +-/- +77.2/- +69.8/- +-/- +-/- +-/- +-/- +3D-MAN [50] +16 +-/- +74.5/74.0 +67.6/67.1 +71.7/67.7 +62.6/59.0 +-/- +-/- +M3DETR [51] +1 +61.8/58.7 +75.7/75.1 +66.6/66.0 +65.0/56.4 +56.0/48.4 +65.4/64.2 +62.7/61.5 +Part-A2-Net [7] +1 +66.9/63.8 +77.1/76.5 +68.5/68.0 +75.2/66.9 +66.2/58.6 +68.6/67.4 +66.1/64.9 +CenterPoint-Pillar [9] +1 +-/- +76.1/75.5 +68.0/67.5 +76.1/65.1 +68.1/57.9 +-/- +-/- +CenterPoint-Voxel [9] +1 +69.8/67.6 +76.6/76.0 +68.9/68.4 +79.0/73.4 +71.0/65.8 +72.1/71.0 +69.5/68.5 +IA-SSD [13] +1 +62.3/58.1 +70.5/69.7 +61.6/61.0 +69.4/58.5 +60.3/50.7 +67.7/65.3 +65.0/62.7 +PV-RCNN [8] +1 +66.8/63.3 +77.5/76.9 +69.0/68.4 +75.0/65.6 +66.0/57.6 +67.8/66.4 +65.4/64.0 +RSN [52] +1 +-/- +75.1/74.6 +66.0/65.5 +77.8/72.7 +68.3/63.7 +-/- +-/- +SST TS [6] +1 +-/- +76.2/75.8 +68.0/67.6 +81.4/74.0 +72.8/65.9 +-/- +-/- +SST [6] +1 +67.8/64.6 +74.2/73.8 +65.5/65.1 +78.7/69.6 +70.0/61.7 +70.7/69.6 +68.0/66.9 +AFDetV2 [24] +1 +71.0/68.8 +77.6/77.1 +69.7/69.2 +80.2/74.6 +72.2/67.0 +73.7/72.7 +71.0/70.1 +PillarNet-34 [53] +1 +71.0/68.5 +79.1/78.6 +70.9 / 70.5 +80.6/74.0 +72.3/66.2 +72.3/71.2 +69.7/68.7 +PV-RCNN++ [10] +1 +68.4/64.9 +78.8/78.2 +70.3/69.7 +76.7/67.2 +68.5/59.7 +69.0/67.6 +66.5/65.2 +PV-RCNN++(center) [10] +1 +71.7/69.5 +79.3 / 78.8 +70.6/70.2 +81.3/76.3 +73.2/68.0 +73.7/72.7 +71.2/70.2 +CenterFormer [54] +8 +75.1/73.7 +78.8/78.3 +74.3/73.8 +82.1/79.3 +77.8/75.0 +75.2/74.4 +73.2/72.3 +INT [55] +10 +-/73.6 +-/- +-/73.3 +-/- +-/71.9 +-/- +-/75.6 +MPPNet [56] +16 +75.6/74.9 +82.7 / 82.3 +75.4 / 75.0 +84.7/82.3 +77.4/75.1 +77.3/76.7 +75.1/74.5 +FSDspconv (ours) +1 +71.9/69.7 +77.8/77.3 +68.9/68.5 +81.9/76.4 +73.2/68.0 +76.5/75.2 +73.8/72.5 +FSDsst (ours) +1 +71.5/69.2 +76.8/76.3 +67.9/67.5 +81.3/75.3 +72.5/67.0 +77.2/76.0 +74.4/73.2 +FSDspconv (ours)† +1 +72.9 / 70.8 +79.2/78.8 +70.5/70.1 +82.6 / 77.3 +73.9 / 69.1 +77.1 / 76.0 +74.4 / 73.3 +FSD++ (ours)† +7 +76.8 / 75.5 +81.4/80.9 +73.3/72.9 +85.1 / 82.2 +78.2 / 75.4 +81.2 / 80.3 +78.9 / 78.1 +TABLE 1 +Performances on the Waymo Open Dataset validation split. All reported results are from single model without any test-time augmentations. †: Longer +schedule (12 epochs). We mark the best single-frame results and multi-frame results with gray boxes and cyan boxes, respectively. +SSTcenter attains much worse performance than FSDplain +on large vehicles. It suggests feature diffusion is a sub- +optimal solution for CFM in the case of large objects. For +those large objects, the features may not be diffused to +the centers or the diffused features are too weak to make +accurate predictions. +• However, FSDplain obtains the worst performance among +all detectors on vehicles with normal sizes. Note that the +CFM issue is minor for the normal-size vehicles. So, in this +case, the center-based assignment in SSTcenter shows its +superiority to the assignment in FSDplain. It suggests the +solution for CFM in FSDplain is also sub-optimal, even if it +achieves better performance in large objects. +• Comparing FSDnogc with SSTcenter, they share the same +sparse voxel encoder while FSDnogc replaces the dense +part in SSTcenter with SIR. The huge improvements of +FSDnogc on large vehicles fairly reveal that SIR effectively +resolves CFM and is better than feature diffusion. +• CenterPoint-PP suffers much less from CFM because it +leverages dense feature maps from the very beginning of +the network. It is also equipped with advanced center- +based assignment. Even so, FSDnogc and FSD still outper- +form CenterPoint-PP, especially on large vehicles. +5.3.2 +Qualitative Analysis +In addition to the quantitative experiments, we demonstrate +the qualitative effect of CFM and our treatment, shown in +Fig. 8. Fig. 8 showcases the voted centers of FSD and the +Fig. 8. An intuitive illustration of the center feature missing. Left: Voted +centers of FSD. Right: Predicted heatmap of SSTcenter. +predicted heatmap of center-based SST. Both of them yield +high-quality predictions for vehicles of normal size, but their +predictions (votes) are usually ambiguous for large vehicles. +Center-based dense detectors make predictions from such +ambiguous heatmaps, so they are prone to make flawed +final predictions. Although the center voting of FSD on large +vehicles is also mediocre, FSD only uses the votes to obtain +point groups (i.e., instance segmentation), which does not + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +10 +Methods +#. +frames +mAP/mAPH +L2 +Vehicle 3D AP/APH +Pedestrian 3D AP/APH +Cyclist 3D AP/APH +L1 +L2 +L1 +L2 +L1 +L2 +CenterPoint [9] +1 +-/69.0 +-/- +-/71.9 +-/- +-/67.0 +-/- +-/68.2 +AFDetV2-lite [24] +1 +72.2/70.0 +80.5/80.0 +73.0/72.6 +79.8/74.3 +73.7/68.6 +72.4/71.2 +69.8/69.7 +PV-RCNN [8] +1 +71.3/68.8 +80.6/80.1 +72.8/72.4 +78.2/72.0 +71.8/66.0 +71.8/70.4 +69.1/67.8 +PV-RCNN++ [10] +1 +72.4/70.2 +81.6/81.2 +73.9/73.5 +80.4/75.0 +74.1/69.0 +71.9/70.8 +69.3/68.2 +Graph R-CNN [57] +1 +73.8/71.6 +83.6 / 83.1 +76.0 / 75.6 +81.9/76.5 +75.6/70.5 +72.5/71.3 +69.8/68.7 +AFDetV2 [24] +2 +74.6/73.1 +81.7/81.2 +74.3/73.9 +81.3/78.0 +75.5/72.4 +76.4/75.4 +74.1/73.0 +CenterPoint++ [9] +3 +74.2/72.8 +82.8/82.3 +75.5/75.1 +81.0/78.2 +75.1/72.4 +74.4/73.3 +72.0/71.0 +BEVFusion∗ [58] +3 +77.7/76.3 +85.0 / 84.6 +77.9/77.5 +84.7 / 82.0 +79.1/76.4 +78.5/77.5 +76.0/75.1 +DeepFusion∗ [25] +5 +76.9/75.5 +83.3/82.8 +76.1/75.6 +84.6/81.8 +79.2/76.4 +77.8/76.8 +75.5/74.5 +CenterFormer [54] +16 +76.9/75.6 +84.7/84.4 +78.1 / 77.7 +84.6/81.8 +79.4 / 76.6 +75.5/74.5 +73.3/72.4 +INT [55] +100 +76.6/75.2 +84.7/84.3 +78.0/77.6 +82.4/79.7 +76.6/74.0 +77.4/76.3 +75.2/74.1 +MPPNet [56] +16 +76.9/75.7 +84.3/83.9 +77.3/76.9 +84.1/81.5 +78.4/75.9 +77.1/76.4 +74.9/74.2 +FSDspconv (ours)† +1 +74.4 / 72.4 +82.7/82.3 +74.4/74.1 +82.9 / 77.9 +75.9 / 71.3 +75.6 / 74.4 +72.9 / 71.8 +FSD++ (ours)† +7 +78.4 / 77.1 +84.5/84.1 +77.1/76.7 +84.5/81.7 +79.0/76.2 +81.4 / 80.5 +79.2 / 78.3 +TABLE 2 +Performances on the Waymo Open Dataset test split. All results are in single-model setting without ensemble or test-time augmentations. ∗: +Multi-modal methods with camera information. We mark the best single-frame results and multi-frame results with gray boxes and cyan boxes, +respectively. †: 12-epoch schedule. +Vehicle length (m) +Methods +[0, 4) +[4, 8) +[8, 12) +[12, +∞) +Official∗ +CenterPoint-PP† +34.3 +69.3 +42.0 +43.6 +66.2 +FSDplain +32.2 +64.6 +41.3 +42.2 +62.3 +SSTcenter [6] +36.0 +69.4 +33.7 +30.5 +66.3 +FSDnogc +33.5 ↓ 2.5 +68.2 ↓ 1.2 +47.7 ↑ 14.0 +47.9 ↑ 17.4 +65.2 ↓ 1.1 +FSD +36.7 ↑ 0.7 +71.0 ↑ 1.6 +51.3 ↑ 17.6 +53.7 ↑ 23.2 +69.3 ↑ 3.0 +TABLE 3 +Vehicle detection with vehicle length breakdown. †: re-implemented +ourselves. ∗: official Waymo L2 overall metric. Arrows indicate the +performance changes from SSTcenter. +necessitate perfect center voting. The final predictions of FSD +are derived from the complete point groups rather than the +weak center features, thereby sidestepping the issue of CFM. +3.4 +10.4 +200 +50 +100 +150 +Perception Range (m) +Training Memory (GB) +1.0 +3.0 +5.0 +7.0 +9.0 +11.0 +13.0 +> 24 +11.8 +1.4 +3.4 +6.3 +4.9 +5.9 +5.2 +3.6 +15.0 +OOM +> 24 +5.6 +15.5 +> 24 +> 24 +1.1 +1.4 +1.9 +2.3 +FSD_sst +CenterPoint +CenterPoint-PP +SST_center +FSD_spconv +50 +100 +200 +150 +Perception Range (m) +Inference Latency (ms) +50 +100 +150 +200 +250 +300 +400 +500 +> 800 +94 +90 +64 +81 +83 +97 +434 +238 +89 +105 +164 +232 +80 +714 +208 +400 +626 +54 +61 +67 +Fig. 9. Memory footprints and inference latency in different perception +ranges. Statistics are obtained on a single 3090 GPU with batch size +1. Inference latency is evaluated by the standard benchmark script in +MMDetection3D without any test-time optimization. CenterPoint-PP and +SSTcenter are defined in Sec. 5.3. Best viewed in color. +5.4 +Long-range Detection +Several widely adopted 3D detection benchmarks [1], [2], [3] +have relatively short perception range. To unleash the poten- +tial of FSD, we conduct long-range detection experiments +on the recently released Argoverse 2 dataset (AV2). AV2 +has a perception range up to 200 meters, making it an ideal +testbed for our method. In addition, AV2 contains objects in +30 classes, exhibiting the long-tail distribution, which is also +another challenge for FSD. +5.4.1 +Main results +We first list the main results of FSD on AV2 in Table 4. +The authors of AV2 provide a baseline CenterPoint model, +but the results are mediocre. To make a fair comparison, +we re-implement a stronger CenterPoint model on the +AV2 dataset. The re-implemented CenterPoint adopts the +same training scheme with FSD, including ground-truth +sampling to alleviate the long-tail issue. FSD outperforms +CenterPoint in the average metric. It is noteworthy that FSD +significantly outperforms CenterPoint in some tiny objects +(e.g., Pedestrian, Construction Cone) as well as some objects +with extremely large sizes (e.g., Articulated Bus, School +Bus). We owe this to the virtue of instance-level fine-grained +feature extraction in SIR. +5.4.2 +Range Scaling +To demonstrate the efficiency of FSD in long-range detection, +we depict the trend of training memory and inference latency +of three detectors when the perception range increases in +Fig. 9. Fig. 9 shows that dense detectors experience a dramatic +increase in latency and memory footprint as the perception +range grows. Designed to be fully sparse, the resource needed +for FSD is roughly linear to the number of input points, so its +memory and latency only slightly increase as the perception +range extends. +5.5 +Performance Inspection of FSD +5.5.1 +Effectiveness of Components +In addition to FSDplain and FSDnogc (Sec. 5.3), we also +degrade FSD to FSDagg to gain insights into its mechanism. +In FSDagg, we aggregate grouped point features by dynamic +pooling after Instance Point Grouping, and then directly + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +11 +Methods +Average +Vehicle +Bus +Pedestrian +Stop Sign +Box Truck +Bollard +C-Barrel +Motorcyclist +MPC-Sign +Motorcycle +Bicycle +A-Bus +School Bus +Truck Cab +C-Cone +V-Trailer +Sign +Large Vehicle +Stroller +Bicyclist +Precision +CenterPoint† [9] +13.5 +61.0 +36.0 +33.0 +28.0 +26.0 +25.0 +22.5 +16.0 +16.0 +12.5 +9.5 +8.5 +7.5 +8.0 +8.0 +7.0 +6.5 +3.0 +2.0 +14 +CenterPoint∗ +22.0 +67.6 +38.9 +46.5 +16.9 +37.4 +40.1 +32.2 +28.6 +27.4 +33.4 +24.5 +8.7 +25.8 +22.6 +29.5 +22.4 +6.3 +3.9 +0.5 +20.1 +FSD +24.0 +67.1 +39.8 +57.4 +21.3 +38.3 +38.3 +38.1 +30.0 +23.6 +38.1 +25.5 +15.6 +30.0 +20.1 +38.9 +23.9 +7.9 +5.1 +5.7 +27.0 +FSD‡ +28.2 +68.1 +40.9 +59.0 +29.0 +38.5 +41.8 +42.6 +39.7 +26.2 +49.0 +38.6 +20.4 +30.5 +14.8 +41.2 +26.9 +11.9 +5.9 +13.8 +33.4 +Composite Score +CenterPoint∗ +17.6 +57.2 +32.0 +35.7 +13.2 +31.0 +28.9 +25.6 +22.2 +19.1 +28.2 +19.6 +6.8 +22.5 +17.4 +22.4 +17.2 +4.8 +3.0 +0.4 +16.7 +FSD +19.1 +56.0 +33.0 +45.7 +16.7 +31.6 +27.7 +30.4 +23.8 +16.4 +31.9 +20.5 +12.0 +25.6 +15.9 +29.2 +18.1 +6.4 +3.8 +4.5 +22.1 +FSD‡ +22.7 +57.7 +34.2 +47.5 +23.4 +31.7 +30.9 +34.4 +32.3 +18.0 +41.4 +32.0 +15.9 +26.1 +11.0 +30.7 +20.5 +9.5 +4.4 +11.5 +28.0 +TABLE 4 +Performance in Argoverse 2 validation split. †: provided by authors of AV2 dataset. ‡: Weak CopyPaste augmentation for preventing overfitting (one +instance per class). ∗: re-implemented by ourselves. C-Barrel: construction barrel. MPC-Sign: mobile pedestrian crossing sign. A-Bus: articulated +bus. C-Cone: construction cone. V-Trailer: vehicular trailer. We omit the results of dog, wheelchair and message board trailer because these +categories contain very few instances. The average results take all categories into account, including the omitted categories. We mark the categories +attaining notable improvements in bold. +make predictions from the pooled features. FSDagg is similar +to the way in VoteNet [22] as we discussed in Sec. 3.5. Thus, +FSDagg can explicitly leverage instance-level features other +than the point-level features in FSDplain, mitigating the issue +of CFM. However, FSDagg cannot take advantage of further +point feature extraction in SIR. As can be seen in Table 5, the +improvement is limited if we only apply grouping without +SIR. The combination of grouping and SIR, on the other hand, +yields significant improvements. +Grouping +SIR +Group +Correction +L2 3D APH +Vehicle +Pedestrian +Cyclist +FSDplain +62.29 +64.31 +64.49 +FSDagg +✓ +63.13 +65.13 +64.52 +FSDnogc +✓ +✓ +65.20 +67.39 +67.78 +FSD +✓ +✓ +✓ +69.30 +69.30 +69.60 +TABLE 5 +Ablation of design factors in SIR. Performances are evaluated on Waymo +validation split. +5.5.2 +Downsampling in SIR +The efficiency of SIR makes it feasible to extract fine-grained +point features without any point downsampling. This is +another notable difference between FSD and VoteNet. To +demonstrate the superiority, we apply voxelization on the +raw points before the SIR module and treat the centroids of +voxels as downsampled points. We conduct experiments on +the AV2 dataset because it contains a couple of categories +in a tiny size, which may be sensitive to downsampling. As +expected, small objects have notable performance loss when +adopting downsampling, and we list some of them in Table +6. We also evaluate the inference latency of the SIR module +on a 3090 GPU, which is highly efficient. +5.5.3 +HD Map-assisted Detection +Argoverse 2 dataset provides a highly reliable HD map, +which could be utilized as a prior to remove uninterested +regions making the scene more sparse. Thus we proceed with +experiments removing some uninterested regions to show +the advantages of FSD in more sparse scenarios. The results +are summarized in Table 7. FSD has a significantly lower +memory footprint and latency with an acceptable precision +AP +Voxel size +CC +Bollard +Bicyclist +Stop Sign +Latency (ms)† +30cm +35.4 +36.5 +24.6 +18.3 +3.5 +20cm +37.3 +37.3 +26.4 +20.0 +4.1 +10cm +38.9 +38.3 +27.0 +21.3 +4.5 +Point +39.3 +38.6 +27.1 +21.5 +6.3 +TABLE 6 +Performances with different representation granularity. †: Latency of SIR +module. +FSD +CenterPoint +Mem. +Latency(ms) +mAP +Mem. +Latency(ms) +mAP +all +5.9 +97 +24.0 +10.4 +232 +22.0 +only RoI† +3.2 ↓ 45.8% +81↓ 16.5% +23.2 +9.9↓ 4.8% +227↓ 2.2% +21.5 +w/o ground +2.3 ↓ 61.0% +74↓ 25.8% +21.0 +9.7↓ 6.7% +217↓ 6.4% +19.8 +TABLE 7 +Performance with different detection areas. †: Region of Interest is +defined by the HD map in AV2 dataset. +loss after removing the uninterested regions. On the contrary, +the efficiency improvement of CenterPoint is minor. It reveals +that FSD benefits more from the increase of data sparsity, +which is another advantage of the fully sparse architecture. +5.6 +Comprehensive Analysis of FSD++ +5.6.1 +Preliminary Settings for the Analysis of FSD++ +In this section, we conduct extensive experiments to reveal +the inner workings of FSD++. Here we first present the +setting of our baseline model in this section, which is slightly +different from the best FSD++ model in Table. 1 and Table. 2. +Unless otherwise specified, the default hyper-parameters of +all the FSD++ models in Sec. 5.6 are listed in the first column +of Table 8. +The model latency reported in this section is measured on +a single RTX 3090 GPU with a mini-batch size of 1 in float32 +precision. To ensure accuracy, we only consider the latency +of the model in all evaluations, excluding the latency of IO, +which is potentially unstable in the multi-frame setting. +It is worth noting that we have observed run-to-run +variation in the performance of cyclist class, likely due to +its low number in the dataset. As a result, we mark the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +12 +Baseline FSD++ +Best FSD++ +Multi-frame FSD +Schedule +6 epochs +12 epochs +6 epochs +SPS† +Random +Random +- +#. frames +6 +7 +6 +Max age +2 +2 +- +Backbone∗ +SpUNet-base +SpUNet-large +SpUNet-base +#. layers in SIR2 +3 +3 +3 +RPP size +(0.25, 0.25, 0.4) +(0.25, 0.25, 0.4) +- +TABLE 8 +Basic hyper-parameter choice of models adopted in this section +(Sec. 5.6) †: SPS stands for skeleton point sampling. ∗: SpUNet-large +has one more stage than SpUNet-base [7] and the number of channels +of its first is doubled. +performance of this class in gray in some experiments to +indicate that it may not be reliable. +5.6.2 +Skeleton Point Sampling +Table 9 shows the performance with different skeleton point +sampling strategies. We find that there are no significant +differences between the three strategies considered. However, +using random skeleton sampling considerably reduces the +latency of FSD++ without sacrificing performance. And +skeleton sampling consistently boosts the performance of +the cyclist class, which suggests that appropriate sampling +alleviates the overfitting for rare classes. The performance +of pedestrian is better without sampling, which reveals that +more points might be helpful for small objects. In practice, +although different sampling strategies could be adopted for +different classes, we use the random sampling for all classes +for simplicity and generality. +L2 3D AP/APH +Latency +(ms) +Mean +Vehicle +Pedestrian +Cyclist +Random +76.10/74.73 +72.20/71.74 +76.93/74.11 +79.20/78.33 +68.7 +Object FPS +75.56/74.21 +72.06/71.59 +76.94/74.14 +77.68/76.88 +72.3 +Voxel Sampling +75.76/74.40 +72.10/71.66 +76.80/74.05 +78.37/77.49 +71.4 +None +75.23/73.91 +71.93/71.50 +77.54/74.72 +76.33/75.51 +73.9 +TABLE 9 +Effectiveness of different skeleton point sampling. +5.6.3 +Different Number of Frames +FSD++ samples skeleton points from multiple previous +frames. Table 10 showcases how the number of used frames +affects its performance. There are two interesting findings. +• Performance becomes better as the number of frames +grows. In the meantime, the latency does not signif- +icantly increase. We owe the credit to residual point +probing, which removes most of the background. It +could offer even more clean residual points if more base +frames (Eqn. 4) are used. +• FSD++ outperforms FSD with the same number of +frames in vehicle and cyclist class. We also intuitively +owe it to RPP since it removes most background clutter +and eases the burden of the segmentation. The slightly +lower performance of pedestrian suggests it might be +better to retain all points for pedestrian. However, the +performance loss is acceptable since FSD++ achieves +better average performance and much lower latency. +#. frames +L2 3D AP/APH +Latency +(ms) +Mean +Vehicle +Pedestrian +Cyclist +2 +73.39/71.83 +69.54/69.12 +74.68/71.35 +75.94/75.02 +66.1 +3 +75.20/73.74 +70.95/70.52 +76.13/73.09 +78.51/77.62 +67.0 +4 +75.44/74.03 +71.67/71.21 +76.48/73.50 +78.18/77.37 +67.3 +5 +75.13/73.72 +71.50/71.04 +76.71/73.83 +77.17/76.29 +68.7 +6 +76.10/74.73 +72.19/71.74 +76.92/74.11 +79.20/78.33 +68.7 +6 (FSD)† +75.65/74.28 +71.54/71.07 +78.04/75.22 +77.37/76.54 +116.2 +TABLE 10 +Performance of FSD++ with the different number of frames. Since +FSD++ uses previous foreground points, it needs at least two frames. †: +multi-frame FSD model with simple point concatenation. Performance is +unstable in the scarce cyclist class, so we mark the numbers in gray. +5.6.4 +Drifting Analysis +It would be a major concern if FSD++ suffers from the drifting +error given its reliance on history predictions. In particular, +if the detector makes inaccurate predictions at time step t, it +is likely that the detector becomes worse at time step t + 1 +since the predictions in t + 1 rely on the predictions from t +(skeleton point sampling). +To prevent potential drifting, we insert some keyframes +at regular intervals during the inference of a sequence. +At keyframes, we use the predictions from standard FSD +for skeleton point sampling, which could be viewed as a +rectification of the potential drifting. Table. 11 shows that +FSD++ achieves competitive results without any keyframes. +And the minor gap between the first row and the last row +confirms that the drifting of FSD++ is negligible. +Gap between +key frames +L2 3D AP/APH +Mean +Vehicle +Pedestrian +Cyclist +5 +76.05/74.67 +72.23/71.77 +76.83/74.02 +79.08/78.21 +10 +76.03/74.66 +72.20/71.75 +76.88/74.07 +79.00/78.15 +20 +76.07/74.68 +72.20/71.74 +76.90/74.08 +79.11/78.23 +50 +76.10/74.72 +72.20/71.74 +76.91/74.10 +79.20/78.32 +None +76.10/74.73 +72.20/71.74 +76.93/74.11 +79.20/78.33 +TABLE 11 +The performance of different keyframe gaps. “None” means using only +initial predictions. +5.6.5 +Change Blindness Ablation +Due to the change blindness we discussed in Sec. 4.3, newly +emerged objects might be ignored by the detector. Max age +is proposed to mitigate the issue of change blindness, and +Table 12 shows its effect. We find keep residual points for +two time steps (max age 2) is enough. +Max age +L2 3D AP/APH +Latency +(ms) +Mean +Vehicle +Pedestrian +Cyclist +1 +75.17/73.82 +71.38/70.94 +76.74/73.97 +77.40/76.56 +65.3 +2 +76.10/74.73 +72.20/71.74 +76.93/74.11 +79.20/78.33 +68.7 +3 +76.14/74.74 +72.22/71.75 +77.06/74.20 +79.13/78.27 +70.2 +TABLE 12 +Different max ages of residual points. Performance is unstable in the +scarce cyclist class, so we mark the numbers in gray. +For a closer look at the issue of change blindness, we split +the objects in the original WOD validation set to emerging +objects and existing objects for further ablations. Emerging +objects mean those objects do not appear in the first frame of +a sequence, while emerging later. The results are shown in + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +13 +Table 13. The emerging objects recall of FSD++(1)1 is inferior +to FSD 6f with the same number of frames. This suggests +that change blindness is indeed an issue for FSD++. However, +prolonging the max age makes FSD++ outperform FSD 6f +in all classes, which demonstrates the proposed max age +effectively mitigates change blindness. +Method +Recall of emerging objects +Mean +Vehicle +Pedestrian +Cyclist +FSD +73.05 +66.01 +72.62 +80.53 +FSD 6f∗ +77.54 +69.74 +78.34 +84.53 +FSD++(1) +75.58 +69.18 +77.48 +83.09 +FSD++(2) +77.82 +70.56 +78.10 +84.79 +FSD++(3) +78.14 +70.60 +78.30 +85.52 +TABLE 13 +Performance for emerging objects. ∗: FSD with 6-frame concatenated +input. In the WOD validation split, the number of emerging objects count +for around 42.4%/37.1%/52.6% in all objects for vehicle / pedestrian / +cyclist, respectively. +5.6.6 +Robustness to Seed Quality +During inference of every point cloud sequence, FSD++ +needs the predictions in the initial frame as a seed to start, as +we discussed in Sec. 4.5.2. Here we figure out how the quality +of seed predictions affects the performance. Concretely, we +add two typical kinds of random noise to seed predictions, +including random box drop and random box insertion. They +are designed to simulate false negatives and false positives. +All the experiments share a trained FSD++ detector. The +modifications above are applied during inference and are not +adopted for training-time augmentation. +Table 14 shows FSD++ is robust to both two types of +noises. Particularly, for box drop, there is only marginal +performance degradation even after dropping all the initial +seed boxes. We explain this surprising phenomenon in two +aspects: (1) FSD++ is born to be robust to the dropping +of moving objects. This is because moving objects create a +considerable amount of residual points and FSD++ is capable +of making predictions from these residual points. (2) There +is still a small number of residual points in the static objects +due to the change of viewpoint. Moreover, the mechanism +of max age also helps accumulate residual points on static +objects. +In the case of box insertion, FSD++ is almost unaffected +because they can be easily identified as background in the +segmentation stage of FSD. +Noise type +L2 3D AP/APH +Mean +Vehicle +Pedestrian +Cyclist +None +76.10/74.73 +72.20/71.74 +76.93/74.11 +79.20/78.33 +Drop (10%)† +75.95/74.58 +72.11/71.66 +76.81/73.99 +78.92/78.08 +Drop (50%) +75.76/74.39 +71.86/71.41 +76.63/73.82 +78.78/77.95 +Drop (100%) +74.69/73.35 +70.47/70.02 +75.41/72.69 +78.20/77.33 +Insertion (10%) +76.00/74.62 +72.16/71.70 +76.82/73.99 +79.01/78.16 +Insertion (50%) +76.02/74.64 +72.14/71.69 +76.86/74.04 +79.05/78.19 +Insertion (100%) +75.98/74.61 +72.16/71.71 +76.84/74.02 +78.95/78.10 +TABLE 14 +Robustness to the noisy seed predictions. †: the percentage in +parentheses denotes the ratio of dropped/inserted instances. +1. The numbers in the parenthesis denote the max ages. +5.6.7 +Analysis of Residual Point Probing +Quantization size and the number of base frames are two +important hyper-parameters in RPP. Here we show how they +affect the output residual points and performance. +Quantization size makes RPP robust to small point distur- +bance. We list the results of different quantization sizes in +Table 15. It could be seen from the “residual point ratio” +that RPP with larger quantization sizes leads to less residual +points making the detector more efficient but leading to +slightly lower performance. +Quantization +size +L2 3D AP/APH +Residual +point ratio† +Latency +(ms) +Vehicle +Pedestrian +Cyclist +(0.15, 0.15, 0.4) +72.06/71.60 +77.19/74.43 +77.53/76.67 +17.4% +72.3 +(0.25, 0.25, 0.4) +72.20/71.74 +76.93/74.11 +79.20/78.33 +9.6% +68.7 +(0.35, 0.35, 0.4) +72.04/71.58 +76.88/74.03 +78.54/77.68 +6.0% +65.2 +TABLE 15 +The effectiveness of quantization size in Residual Point Probing. †: +residual point ratio means the average ratio of the residual points to the +total points in a single frame. +Base frame (in Eqn. 4) also has a considerable effect on RPP. +The more base frames are incorporated, the less residual +points could be obtained, leading to higher efficiency. More- +over, the performance is hardly affected by the increase of +base frames. +#. RPP base +frames +L2 3D AP/APH +Residual +point ratio† +Latency +(ms) +Vehicle +Pedestrian +Cyclist +3 +72.48/72.03 +77.16/74.35 +77.60/76.78 +14.4% +70.2 +4 +72.24/71.79 +77.04/74.20 +78.44/77.60 +10.8% +69.0 +5 +72.20/71.74 +76.93/74.11 +79.20/78.33 +9.6% +68.7 +6 +72.22/71.77 +77.41/74.59 +78.60/77.74 +9.4% +68.5 +TABLE 16 +The effectiveness of the number of base frames in Residual Point +Probing. †: residual point ratio means the average ratio of the residual +points to the total points in a single frame. +5.7 +Detailed Runtime Evaluation +Here we elaborate on the efficiency of each component of FSD +and FSD++. All evaluated models use SpUNet-large as the +backbone. Evaluations are conducted on a single RTX 3090 +in FP32 precision without any test-time optimizations. We +only record the single-sample forward latency of the detector +implemented with MMDetection3Dv0.15, ignoring the IO +of point clouds which is unstable in the multi-frame setting. +Fig. 10 shows the detailed results, which are average numbers +evaluated on the first ten sequences of validation split. +As can be seen from the figure, the latency of the +segmentor is greatly reduced, which consists of the sparse +voxel encode (i.e., backbone) and segmentation head. As +a results, FSD++ is as fast as the single-frame FSD, yet +achieves better performance than FSD 6f (Table 10). It is +worth emphasizing that the “others” part of latency is usually +brought by some serialized operations, such as class-wise +detection heads and class-wise NMS. This part of latency +could be greatly reduced in deployment. +6 +CONCLUSION +This paper first proposes a LiDAR-based fully sparse 3D +object detection framework, namely FSD. FSD forsakes the + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +14 +Latency Breakdowns +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +Latency (ms) +FSD_6f +FSD +FSD++ +Segmentor +Skeleton Sampling and RPP +SIR +Group Correction +SIR2 +Clustering +Others +Segmentor +Skeleton Sampling and RPP +SIR +Group Correction +SIR2 +Clustering +Others +Fig. 10. Latency breakdowns of multi-frame FSD, single-frame FSD and +FSD++. +widely adopted dense BEV feature map in previous arts, +which is the hindrance to making detectors fully sparse. +Instead, FSD consists of a general sparse voxel encoder and a +highly-efficient sparse instance recognition (SIR) module. SIR +remedies the issue of the center feature missing, which is the +essential difficulty of fully sparse architecture. FSD not only +actualizes efficient long-range (up to 200 meters) detection +on Argoverse 2 dataset, but also achieves state-of-the-art +performance on the competitive Waymo Open Dataset. +To unleash the potential of FSD, we propose lever- +aging temporal information to remove data redundancy. +The proposed Skeleton Point Sampling and Residual Point +Probing offer FSD a super sparse input point cloud, which +constitutes the FSD++ framework. FSD++ achieves state-of- +the-art single-model performance on both validation and test +split Waymo Open Dataset, and maintains high efficiency. +We hope our work lightens future research direction for +LiDAR-based point cloud recognition. +APPENDIX A +EFFICIENT DYNAMIC POOLING +The proposed SIR consists of three basic operations: MLP, +dynamic broadcasting, and dynamic pooling. MLP is highly +optimized in mainstream deep learning frameworks. Dy- +namic broadcasting is essentially an indexing operation, +which is highly parallel in modern GPUs. The efficiency +bottleneck of SIR lies in dynamic pooling. Thus we provide +an efficient implementation of dynamic pooling in this +section. +A.1 +Implementation +The dynamic pooling implemented by PyTorch is known +as scatter operation. In this implementation, each thread +manages one feature and performs simple atomic operations +for feature reduction. Intensive atomic operations in large +groups are detrimental to parallelism. +We optimize the dynamic pooling operator in three +aspects. (1) Partitioning large groups into small sub-groups +with fixed sizes to balance the workload. (2) Sorting the +Partition +…… +Atomic Operation +Group 𝑖 (small) +Group 𝑗 (large) +𝑁 point features +𝐶 +block 𝑗, part 1 +block 𝑖 +block 𝑗, part 2 +… +… +… +Loop +Reduction +Warp of +Threads +Loop +Reduction +Warp of +Threads +Fig. 11. Illustration of dynamic pooling implementation in CUDA. Best +viewed in color. We take two groups with different sizes as examples. +Dynamic pooling reduces each group to a single feature. +features with the same group ID in adjacent positions to +enhance the memory locality. Note that the group IDs of +each element remain unchanged throughout the SIR module, +so the sort is only applied once. (3) Threads in a warp are +assigned to adjacent channels for coalesced memory access +without warp divergence. +In particular, each thread corresponds to a feature dimen- +sion and reduces features in a partitioned sub-group through +loops. Besides, the calls to atomic operators are reduced from +once per thread to once per block, which contributes to high +parallelism. Fig. 11 illustrates our efficient implementation. +A.2 +Runtime Evaluation +We take dynamic max-pooling as an example and evaluate +the latency of our implementation and torch scatter in differ- +ent cases, including multiple feature dimensions, multiple +group sizes, and whether the group size is balanced. The +total number of groups in the evaluation is 100. The results +are shown in Table 17 and Table 18. With different data sizes, +our implementation achieves 2.48× to 39.58× speedup. And +we have a more significant speedup with imbalanced group +sizes. +Feature +dimension +Latency (ms) with Different Group Sizes +[100, 101) +[101, 102) +[102, 103) +[103, 104) +64 +0.06/0.16 +0.06/0.16 +0.08/1.49 +0.24/14.05 +256 +0.06/0.18 +0.06/0.33 +0.14/3.52 +0.72/20.53 +1024 +0.09/0.16 +0.10/0.85 +0.37/6.53 +4.82/65.28 +Speedup +2.48× +5.56× +20.47× +30.53× +TABLE 17 +Latency of dynamic max pooling on data with balanced group sizes. +Each item denotes the latency of ours/torch scatter in milliseconds. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +15 +Feature +dimension +Latency (ms) with Different Group Sizes +[100, 101)∗ +[101, 102)∗ +[102, 103)∗ +[103, 104)∗ +64 +0.06/0.16 +0.06/0.32 +0.09/2.60 +0.36/20.87 +256 +0.06/0.15 +0.08/0.78 +0.20/5.44 +1.19/32.75 +1024 +0.06/0.24 +0.12/1.36 +0.57/9.24 +5.91/196.4 +Speedup +3.05× +8.81× +24.01× +39.58× +TABLE 18 +Latency of dynamic max pooling on data with imbalanced group sizes. +Each item denotes the latency of ours/torch scatter in milliseconds. ∗: +The sizes of one-tenth groups are enlarged by 10× to create imbalanced +data. +REFERENCES +[1] +P. 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Zitnick, “Microsoft COCO: Common Objects in +Context,” in European conference on computer vision. +Springer, 2014. + diff --git a/rNE0T4oBgHgl3EQfrQEH/content/tmp_files/load_file.txt b/rNE0T4oBgHgl3EQfrQEH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7b72787e5e4e51ba9b6e280be3002b54a524573 --- /dev/null +++ b/rNE0T4oBgHgl3EQfrQEH/content/tmp_files/load_file.txt @@ -0,0 +1,2458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf,len=2457 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 1 Super Sparse 3D Object Detection Lue Fan, Yuxue Yang, Feng Wang, Naiyan Wang, and Zhaoxiang Zhang Abstract—As the perception range of LiDAR expands, LiDAR-based 3D object detection contributes ever-increasingly to the long-range perception in autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Mainstream 3D object detectors often build dense feature maps, where the cost is quadratic to the perception range, making them hardly scale up to the long-range settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To enable efficient long-range detection, we first propose a fully sparse object detector termed FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD is built upon the general sparse voxel encoder and a novel sparse instance recognition (SIR) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SIR groups the points into instances and applies highly-efficient instance-wise feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The instance-wise grouping sidesteps the issue of the center feature missing, which hinders the design of the fully sparse architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To further enjoy the benefit of fully sparse characteristic, we leverage temporal information to remove data redundancy and propose a super sparse detector named FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD++ first generates residual points, which indicate the point changes between consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The residual points, along with a few previous foreground points, form the super sparse input data, greatly reducing data redundancy and computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We comprehensively analyze our method on the large-scale Waymo Open Dataset, and state-of-the-art performance is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To showcase the superiority of our method in long-range detection, we also conduct experiments on Argoverse 2 Dataset, where the perception range (200m) is much larger than Waymo Open Dataset (75m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Code is open-sourced at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='com/tusen-ai/SST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Index Terms—3D object detection, LiDAR, autonomous driving, sparse, Waymo Open Dataset, instance segmentation, temporal fusion, point clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1 INTRODUCTION A UTONOMOUS driving systems are eager for efficient long-range perception, especially in high-speed sce- narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Current LiDAR-based 3D object detectors usually convert sparse features into dense feature maps for further feature extraction and prediction, which we name as dense detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Dense detectors perform well on current pop- ular benchmarks [1], [2], [3], where the perception range is relatively short (less than 75 meters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, it is impractical to scale the dense detectors to the long-range setting (more than 200 meters, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In such settings, the computational and spatial complexity on dense feature maps is quadratic to the perception range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fortunately, the sparsity of LiDAR point clouds also increases as the perception range extends (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1), and the calculation on the unoccupied area is essentially unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Given the inherent sparsity, an essential solution for efficient long-range detection is to remove the dense feature maps and make the network architectures fully sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, removing the dense feature map is non-trivial since it plays an indispensable role in current designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Commonly adopted sparse voxel encoders [5], [6], [7] only extract the features on the non-empty voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Without dense feature maps, the object centers are usually empty, especially for large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We name this issue as “Center Feature Missing (CFM)” (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' CFM significantly weakens the representation power of the center voxels, even making the center feature empty in some extreme cases like super large vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, almost all popular voxel or pillar based detectors [5], [6], [8], [9], [10] adopt center-based Lue Fan and Yuxue Yang and Zhaoxiang Zhang are with Center for Research on Intelligent Perception and Computing (CRIPAC), National Laboratory of Pattern Recognition (NLPR), Institute of Automation, Chinese Academy of Sciences (CASIA), Beijing 100190, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' E-mail: {fanlue2019, yangyuxue2023, zhaoxiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='zhang}@ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Feng Wang and Naiyan Wang are with TuSimple, Beijing 100020, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' E-mail: {feng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='wff, winsty}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 200 𝑚 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Short-range point clouds (red, from KITTI [2]) v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' long-range point clouds (blue, from Argoverse 2 [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The radius of the red circle is 75 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The sparsity quickly increases as the range extends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' assignment and rely on the center feature since it is an ideal representation of the whole object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' So they have to first convert sparse voxels to dense feature maps in Bird’s Eye View after the sparse voxel encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then they resolve the CFM issue by applying convolutions on the dense feature maps to diffuse features to instance centers, which we name as feature diffusion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To properly eliminate the dense feature map, we inves- tigate the purely point-based detectors because they are naturally fully sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, two drawbacks limit the usage of point-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (1) The time-consuming arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='02562v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='CV] 5 Jan 2023 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Illustration of center feature missing and feature diffusion on dense feature maps from Bird’s Eye View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The empty instance center (red dot) is filled by the features diffused from occupied voxels (with LiDAR points), after several convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' neighborhood query [11] is the long-standing difficulty to apply it to large-scale point cloud (more than 100K points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) To reduce the computational overhead, point- based methods aggressively downsample the whole scene to a fixed number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The aggressive downsampling leads to inevitable information loss and insufficient recall of foreground objects [12], [13], especially for small ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As a result, very few purely point-based detectors have reached state-of-the-art performance in the recent benchmarks with large-scale point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this paper, we first propose Fully Sparse Detector (FSD) to sidestep the issue of center feature missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD is built upon a general sparse voxel encoder [5], [6], [7] for voxel/point feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then FSD groups the points into an instance, and further extract the instance- level feature and predict a single bounding box from the integrated instance feature, via a novel Sparse Instance Recognition (SIR) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this way, predictions are made from the whole instance feature instead of the weak or missed center feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As a point-based module, SIR has several desired properties: (1) Unlike previous point-based modules, SIR simply treats instances as groups, and does not apply the time-consuming neighborhood query for further grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) Similar to dynamic voxelization [14], SIR leverages dynamic broadcast/pooling for tensor manipulation to avoid point sampling or padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (3) Since the group in SIR covers the whole instance, it builds a sufficient receptive field regardless of the physical size of the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To unleash the full potential of FSD, we further utilize temporal information and propose a Super Sparse 3D Ob- ject Detector, named FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD++ is inspired by human vi- sual behavior: human is sensitive to and focuses on dynamic parts of the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In particular, FSD++ utilizes ego- motion to remove the static parts containing heavy temporal redundancy, while only retaining the informative dynamic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We name the detected dynamic parts as residual points since the process is similar to applying the difference between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this way, we create a super sparse point cloud consisting of residual points and a small number of past foreground points from history predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD++ then takes the super sparse point cloud as input, achieving a very efficient detection framework with temporal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We owe the credit of the high efficiency to the synergy of the fully sparse characteristic and the super sparse input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We list our contributions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We introduce the concept of Fully Sparse Detector (FSD), which is the essential solution for efficient long-range LiDAR detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We further propose Sparse Instance Recognition (SIR) to sidestep the issue of Center Feature Missing (CFM) in sparse feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Combining SIR with general sparse voxel encoders, we develop an efficient and effective FSD implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Based on FSD, we further present the FSD++ framework, which aggregates a super sparse point cloud from multi- frames as input, yet removing the temporal redundancy of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The proposed framework uncovers the untapped potential of sparse architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We hope our efforts attract the attention of the community to fully sparse architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD achieves state-of-the-art performance on the com- petitive Waymo Open Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Besides, we further apply our method to the recently released Argoverse 2 dataset to demonstrate the superiority of FSD in long-range detection, where FSD is much more efficient than its dense counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD++ achieves comparable per- formance with mainstream state-of-the-art multi-frame detectors with minimal additional overhead compared with single-frame input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2 RELATED WORK In reviewing the evolution of LiDAR-based 3D object de- tectors, the previous methods could be categorized into three types by their spatial sparsity: dense detectors, sparse detectors, and semi-dense detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Below, we provide a brief revisit of previous arts according to spatial sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Voxel-based Dense Detectors Pioneering work 3DFCN [15] and VoxelNet [16] use dense convolution for voxel feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' They bring con- volutional neural networks to the field of LiDAR-based 3D object detection and achieve competitive results at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, it is inefficient to apply dense convolution to 3D voxel representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' MV3D [17], PIXOR [18], and PointPillars [19] adopt 2D dense convolution in Bird’s Eye View (BEV) feature maps achieving significant efficiency improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We refer to such detectors as dense detectors since they convert the sparse point cloud into dense feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Point-based Sparse Detectors Since PointNet [20] and PointNet++ [11] shed light on the deep learning for 3D point sets, a series of point-based detectors have emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' These purely point-based detectors are born to be fully sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' PointRCNN [21] is the pioneering work of this line of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3DSSD [12] accelerates the point- based method by removing the feature propagation layer and refinement module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' VoteNet [22] first makes a center voting and then generates proposals from the voted center achieving better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Albeit many methods [12], [13], [23] have tried to accelerate the point-based method, the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 3 time-consuming point sampling and neighborhood query are still unaffordable in large-scale point clouds (more than 100k points per scene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' So current benchmarks [1], [3] with large-scale point clouds are still dominated by voxel-based dense/semi-dense detectors [10], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Semi-dense Detectors Different from dense detectors, semi-dense detectors incor- porate both sparse features and dense features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SECOND [5] employs sparse convolution to extract the sparse voxel features in 3D space, which then are converted to dense feature maps in BEV to enlarge the receptive field and integrate with 2D detection head [26], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Based on SECOND-style semi-dense detectors, a series of work [29], [30], [31] made further improvements on the single-stage paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And other methods attach a second stage for fine-grained feature extraction and proposal refinement [7], [8], [10], [32], achieving superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Although semi-dense detectors become dominating in academia and industry, related research has stagnated here because the semi-dense detector cannot be trivially lifted to be fully sparse as we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3 FSD: FULLY SPARSE 3D OBJECT DETECTION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Overall Architecture Following the motivation of instances as groups, we have four steps to build the fully sparse detector (FSD): 1) We first utilize a sparse voxel encoder [5], [6], [7] to extract voxel features and casts votes for object centers(Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2) Instance Point Grouping groups foreground points into instances based on the voting results (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3) Given the grouping results, Sparse Instance Recognition (SIR) module extracts instance/point features and generates proposals (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4) The proposals are utilized to correct the point grouping and refine the proposals iteratively (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Instance Point Grouping Classification and Voting We first extract voxel features from the point cloud with a sparse voxel encoder, such as sparse attention blocks in SST [6] or sparse convolution encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then we build point features by concatenating voxel features and the offsets from points to their corresponding voxel centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' These point features are passed into two heads for foreground classification and center voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The voting is similar to VoteNet [22], where the model predicts the offsets from foreground points to corresponding object centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' L1 loss [27] and Focal Loss [33] are adopted as voting loss Lvote and semantic classification loss Lsem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Connected Components Labeling (CCL) To group points into instances, we regard all the predicted centers (red dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3) as vertices in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Two vertices are connected if their distance is smaller than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then a connected component in this graph can be viewed as an instance, and all points voted to this connected component share a group ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Unlike the ball query in VoteNet, our CCL-based grouping avoids fragmented instances in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Although there are many elaborately designed instance grouping methods [34], [35], [36], we opt for the simple CCL because it is adequate in our design and can be implemented by the efficient Union-Find algorithm [37] in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Sparse Instance Recognition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Preliminaries: Dynamic Broadcast/Pooling Given N points belong to M groups, we define their cor- responding group ID array as I in the shape of [N, ] and their feature array as F in the shape of [N,C], where C is the feature dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' F (i) is the feature array of points belonging to the i-th group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Dynamic pooling aggregates each F (i) into one group feature gi of shape [C, ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus we have gi = p(F (i)), where p is a symmetrical pooling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The dynamic pooling on all group features G of shape [M,C] is formulated as G = p(F, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The dynamic broadcast can be viewed as the inverse operation to dynamic pooling, which broadcasts gi to all the points in the i-th group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Since the broadcasting is essentially an indexing operation, we use the indexing notation [ ] to denote it as G[I], which is in the shape of [N,C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Dynamic broadcast/pooling is very efficient because it can be implemented with high parallelism on modern devices and well fits the sparse data with dynamic size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We provide an efficient implementation and runtime evaluation in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The prerequisite of dynamic broadcast/pooling is that each point uniquely belongs to a group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' groups should not overlap with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thanks to the fact that there is no overlap among instances in the real 3D world, the groups do not overlap with each other naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Formulation of Sparse Instance Recognition After grouping points into instances in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2, we can directly extract instance features via some basic point- based networks like PointNet, DGCNN, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' There are three elements to define a basic point-based module: group center, pair-wise feature and group feature aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Group center The group center is the representative point of a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For example, in the ball query, it is the local origin of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In SIR, the group center is defined as the centroid of all voted centers in a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Pair-wise feature defines the way to pair group center and neighbor points input for group-aware neighbor point feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SIR adopts two kinds of pair-wise features: 1) the relative coordinate between the group center and each point, 2) the concatenation of the group feature and each point fea- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Taking feature concatenation as an example and using the notations in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1, the pair-wise feature can be denoted as CAT(F, G[I]), where CAT is channel concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Group feature aggregation In a group, a pooling function is used to aggregate neighbor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SIR applies dynamic pooling to aggregate feature array F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Following the notations in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1, we have G = p(F, I), where G is the aggregated group features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Integration Combining the three basic elements, we could build many variants of point-based operators, such as Point- Net [20], DGCNN [38], Meta-Kernel [39], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4 illustrates the basic idea of how to build an instance-level point operator with dynamic broadcast/pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In our design, we adopt the formulation of VFE [16] as the basic structure of SIR layers, which is basically a two-layer PointNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the l-th layer of SIR module, given the input point-wise feature array JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 4 Input Point Cloud Sparse Voxel Feature Extractor Point-wise Classification & Center Voting Not Connected Connected SIR Module Rule out outliers Add missing points SIR2 Module Prediction 1 Prediction 2 Predict proposal Predict proposal Instance 1 Instance 2 Group Correction Instance Point Grouping via CCL Instance-wise feature extraction and prediction Instance-wise feature extraction and prediction Corrected instance 1 Corrected instance 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Overall architecture of FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For simplicity, we only use two instances to illustrate the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Red dots are the voted centers from each LiDAR point (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The SIR module and the SIR2 module all contain 3 SIR layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Group 1 Group 2 𝑁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' × 3 𝑁" × 3 2 × 3 𝑁 × 3 2 × 𝐶 𝑁 × 𝐶 N × 𝐶 N × 𝐶 𝑁!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' + 𝑁" = 𝑁 Grouping results Group centers Broadcasted group centers 𝑁 × 3 Input point coordinates Group features Point-wise subtraction Dynamic Pooling Dynamic Broadcast Input point features Broadcasted group features Input point features Pair-wise feature extraction Pair Instance 1 Instance 2 Output point feature Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Illustration of building instance-level point operators with dynamic broadcast/pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Left: calculating center-to-neighbor offsets given raw point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Right: updating point features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Note that the operation is parallel among all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fl, point coordinates array X, the voted center X′ and group ID array I, the output of l-th layer can be formulated as: F ′ l = LinNormAct � CAT �Fl, X − pavg(X′, I)[I] �� , (1) Fl+1 = LinNormAct (CAT (F ′ l , pmax(F ′ l , I)[I])) , (2) where LinNormAct is a fully-connected layer followed by a normalization layer [40] and an activation function [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The pavg and the pmax are average-pooling and max-pooling function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The output Fl+1 can be further used as the input of the next SIR layer, so our SIR module is a stack of a couple of basic SIR layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Sparse Prediction With the formulation in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2, SIR extracts features of all instances dynamically in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And then SIR makes sparse prediction for all groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In contrast to two-stage sparse prediction, our proposals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=', groups) do not overlap with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Unlike one-stage dense prediction, we only generate a single prediction for a group, which significantly reduces the cost of prediction head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is noteworthy that the fully sparse architecture may face a severe imbalance problem: short-range objects contain much more points than long-range objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Some methods [39], [42] use hand-crafted normalization factors to mitigate the imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Instead, SIR avoids the imbalance because it only generates a single prediction for a group regardless of the number of points in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In most cases, a group corresponds to only a single ground truth box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Specifically, for each SIR layer, there is a Gl = pmax(F ′ l , I) in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2, which can be viewed as the group features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We concatenate all Gl from each SIR layer in channel dimension and use the concatenated group features to predict bounding boxes and class labels via MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All the groups whose centers fall into ground-truth boxes are positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For positive samples, the regression branch predicts the offsets from group centers to ground-truth centers and object sizes and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' L1 loss [27] and Focal Loss [33] are adopted as regression loss Lreg and classification loss Lcls, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 Group Correction There is inevitable incorrect grouping in the Instance Point Grouping module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For example, some foreground points may be missed, or some groups may be contaminated by background clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' So we leverage the bounding box proposals from SIR to correct the grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The points inside a proposal belong to a corrected group regardless of their previous group IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Since a few points may fall into multiple proposals, we simply make copies for these points along with their features and assign different copies to difference proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' After correction, we apply an additional SIR to these new groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To distinguish it from the first SIR module, we denote the additional SIR module as SIR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SIR2 predicts box residual from the proposal to its corresponding ground-truth box, following many two-stage detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To make SIR2 aware of the size and location of a proposal, we adopt the offsets from inside points to proposal boundaries as extra point features following [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The regres- sion loss is denoted as Lres = L1(∆res, � ∆res), where ∆res is the ground-truth residual and � ∆res is the predicted residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Following previous methods [7], [8], the 3D Intersection over Union (IoU) between the proposal and ground-truth serves as the soft classification label in SIR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Specifically, the soft label q is defined as q = min(1, max(0, 2IoU −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5)), where IoU is JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 5 the area of Intersection over Union (IoU) between proposals and corresponding ground-truths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then cross entropy loss is adopted to train the classification branch, denoted as Liou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Taking all the loss functions in grouping (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2) and sparse prediction into account, we have Ltotal = Lsem + Lvote + Lreg + Lcls + Lres + Liou, (3) where we omit the weight of each term for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 Discussion The center voting in FSD is inspired by VoteNet [22], while FSD has two essential differences from VoteNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' After voting, VoteNet simply aggregates features around the voted centers without further feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD goes beyond this and builds a highly efficient SIR module taking advantage of dynamic broadcast/pooling, allowing for further instance-level feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus, FSD extracts more powerful instance features, which is experi- mentally demonstrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' VoteNet is a typical point-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1, it aggressively downsamples the whole scene to a fixed number of points for efficiency, causing inevitable information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Instead, the dynamic characteristic and efficiency of SIR enable fine-grained point feature ex- traction from any number of input points without any downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5, we showcase the efficiency of our design in processing large-scale point clouds and the benefits of fine-grained point representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4 FSD++: FSD WITH SUPER SPARSE INPUT It is well known that aggregating multiple frames as an input benefits performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, naive aggregation could result in a denser point cloud, which slows down the algorithm significantly, especially in the architecture with sparse operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' This motivates us to pursue more sparse input data by removing temporal redundancy from the original point cloud stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thanks to the fully sparse characteristic, the fully sparse model could greatly benefit from the increase of sparsity after redundancy removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus a natural question arises: How can we remove the redundancy while retaining the informative parts in advance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The similarities between consecutive point cloud frames offer us a potential solution to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In particular, the spatial distribution of points varies con- tinuously and smoothly in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We name the points that change between consecutive frames as residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The residual points are informative since they represent new observations in a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Combining the residual points and history predictions, detectors have sufficient knowledge to infer about current objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this paradigm, the residual points and previous foreground points together form a super sparse point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD could directly take them as input for much more efficient object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Residual Points Probing LiDAR sensors capture plenty of newly observed foreground points at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' These points can be attributed to two main sources: (1) objects moving to new positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) occluded regions becoming visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' These newly observed points are referred to as residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Residual points are critical to locate moving objects and detect recently emerged objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As we mentioned before, the residual points could be detected from the changes of point spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The residual point detection algorithm must fulfill two key requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (1) The algorithm is supposed to be robust to tiny disturbances of points, which might be caused by sensing noise or tiny ego-motion estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is unexpected that such point disturbances are detected as residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) The algorithm should be highly efficient to handle millions of points from multiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In particular, each frame in WOD contains up to 200,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Several straightforward solutions meet the first require- ment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ball query or voxelization into dense occupancy maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A point can be viewed as a residual point if no previous points fall into its neighborhood defined by the ball query radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The residual points can also be detected by the simple difference between the two dense occupancy maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Although proper ball query radii or voxel sizes bring robustness to point disturbances, these solutions still come with either high computational complexity (O(N 2)) or a huge memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To fulfill both of the demands outlined above, we resort to hashing and design an algorithm shown in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1, named Residual Points Probing (RPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' RPP consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (1) It first quantizes the point coordinates into integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The granularity of quantization controls the robustness to point disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) For each point, RPP verifies if it is a residual point by hash probing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Specifically, RPP first builds a hash table from previous quantized points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The key set of the hash table is denoted as K ⊂ Z3, which is the quantized integer coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And value set of the hash table is denoted as V = {1, 0}, where 1 indicates the slot is occupied and 0 indicates the slot is unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' RPP then uses current quantized coordinates to probe the hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' If a current point hits an unoccupied slot, it is treated as a residual point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Here is a hidden assumption in RPP that we assume two points are the same if they occupy the same voxel after quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We adopt the well-known open addressing for probing and the double hashing as the hash function to reduce hash collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Algorithm 1: Efficient Residual Points Probing Input: current points Pcur, previous points Ppre, voxel size s, load factor α, hash function h Output: Residual points of current frame ∆Pcur �Pcur ← Quantize(Pcur, s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' �Ppre ← Quantize(Ppre, s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Initialize empty hash table T of length | �Ppre|/α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Initialize empty residual point set ∆Pcur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' foreach �pi in �Ppre do sloti ← Probe(T, h(�pi));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' if sloti is not occupied then sloti ← occupied flag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' foreach �pi in �Pcur do sloti ← Probe(T, h(�pi));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' if sloti is not occupied then Add pi to ∆Pcur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' return ∆Pcur JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 6 ✘ ✓ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✓ ✓ ✓ ✓ ✓ ✓ ✓ (a) Single- frame points (b) Multi- frame points (c) Residual points with change blindness (d) Residual points w/o change blindness 𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 𝑇" 𝑇# 𝑇$ 𝑇% Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Illustration of temporal point manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Different colors indicate points from different time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Gray points are the sampled skeleton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Red cross \x17 means the observation is too weak to be recognized as foreground objects (not really, just for illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Green check mark \x13 means the observation is strong enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (a) Points from a single frame are too weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (b) After multi-frame point aggregation, the detector generates true positives from time step T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (c) Single-frame residual points suffer from change blindness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Since the detector cannot generate a true positive in T0 for skeleton point sampling, we still only have the weak residual points in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this way, the detector outputs false negatives all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (d) Detector makes the right prediction in T1 with residual points from two frames (max age is 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' So in the T2, the detector could leverage the prediction in T1 for skeleton point sampling, alleviating the change blindness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Residual Point Probing is efficient in terms of both memory and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For instance, we assume there are N unique quantized coordinates in a point cloud clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' If we expect the collision rate less than α, the length of the hash table should be N/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Empirically, N is around 500,000 in a 5-frame point cloud in WOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A slot state can be represented as a single bit, and let α equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We have that the memory cost of this hash table is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Moreover, the probing of each point is independent with each other, allowing for high parallelism in GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Formally, we denote the RPP process as follows: ∆Pt = Pt − B � i=1 Pt−i, (4) where ∆Pt is the detected residual points in time step t and Pt is the raw points in time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The notation “X − Y ” means removing the intersection of X and Y from X, equivalent to X \\(X ∩Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And the union means point cloud concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' B is the number of previous frames used in RPP, which we term as base frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Skeleton Point Sampling Since residual points contain only new observations of the current time step, detectors require additional information from previous frames for sufficient input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To incorporate this historical data, we use previously predicted boxes to crop previous foreground points, while discarding others outside of the boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The cropped points are placed into the current frame after ego-motion compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, the foreground points from multiple previous frames are still essentially redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Especially, the quite many points on short-range objects from multiple frames could lead to unnecessary overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To reduce the redundancy from multi-frame foreground points, we further sample within these cropped points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Intuitively, we expect the sampled points contain the minimal information models need to make proper predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this sense, we refer to such a minimal subset of cropped points as skeleton points, because they depict the basic structure or “skeleton” of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Specifically, we try three kinds of sampling methods: random sampling, farthest points sampling, and voxel sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All the sampling methods are applied inside the previously predicted bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For random sampling and farthest points sampling, we adopt a prede- fined maximal point threshold NT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We sample NT points inside the bounding boxes which contain points more than NT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For voxel sampling, we adopt dynamic voxelization [14] to voxelize points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All the points falling into a voxel are reduced to a single point by average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Treatment to Change Blindness Theoretically, by combining the skeleton points and residual points, a model is able to make predictions in current frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, a phenomenon known as “change blindness” can hinder performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Change blindness refers to the human visual system’s tendency to overlook progressive small changes in a scene, even if the aggregated changes of multiple time steps are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A similar issue can occur in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thinking of a vehicle nearly entering into the sensing range of LiDARs in time step t, only a small part of the vehicle can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The detector is very likely to recognize it as background, so RPP will remove these points in time step t + 1 and only keep a small number of new points of the vehicle as residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this way, if the vehicle appears slowly, the detector might never recognize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5 demonstrates the change blindness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4 2 0 2 4 2 0 2 4 6 84 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': 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+page_content=' 2 4 2 0 2 4 84 2 0 2 4 2 0 2 4 84 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2 4 2 0 2 4 84 2 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 0 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2 4 2 0 2 4 0JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 7 To remedy the change blindness, we introduce a max age M for residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In other words, the detector takes residual points from at most M steps as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Formally, the detector takes �M−1 i=0 ∆Pt−i as accumulated residual points for input in time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 Integrated Super Sparse Input The input point clouds consist of two parts: previous skeleton points and residual points from multiple time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Formally, for an N-frame FSD++ detector, we have the final input points in time step t as follows: P in t = � N � i=1 P s t−i � ∪ �M−1 � i=0 ∆Pt−i � , (5) where P s t is the skeleton points at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' P in t is much more sparse than the raw point clouds and directly sent into the FSD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 7 shows examples of P s, ∆P and P in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 Training and Inference Pipeline The training and inference pipeline for FSD++ differs from the standard approach due to its use of history predictions and temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 6 summarizes the overall pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Training To utilize history predictions, the input point cloud stream must be arranged in the temporal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, the ordered input stream affects the model training due to a lack of data shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Another problem is that the history predictions are not reliable in the early training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Considering these two issues, we use a well-trained FSD detector to generate offline predictions of the entire training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' During training, for every sample, we load it along with its previous offline predictions to sample skeleton points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Ground truth boxes seem to be an alternative to the offline predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, the distribution gap between ground truth used in training and predicted boxes used in inference is considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus we adopt the offline predictions instead of ground-truth boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Inference In the online inference phase, the input point cloud stream is naturally in the temporal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For a point cloud sequence, the predictions of its first frame are from the well-trained FSD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' These predictions are regarded as the “previous predictions” of the first frame, which are called the seed predictions of a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We maintain several queues to cache some historical data that could be used more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For example, in a N-frame FSD++ pipeline, the raw points and skeleton points of time step t could be reused from time step t + 1 to t + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5 EXPERIMENTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Dataset Waymo Open Dataset (WOD) In our experiments, we use WOD [1] as the primary dataset to evaluate the performance of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' WOD is the most trustworthy benchmark for LiDAR-based 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' With 1150 sequences and more than 200,000 frames, WOD is currently the largest dataset of its kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Among them, 798 sequences are used for training, 202 for validation, and 150 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The detection range in WOD is 75 meters (cover area of 150m × 150m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Argoverse 2 (AV2) We further conduct long-range experi- ments on the recently released Argoverse 2 dataset [4] to demonstrate the superiority of FSD in long-range detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' AV2 has a similar scale to WOD, and it contains 1000 sequences in total, 700 for training, 150 for validation, and 150 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In addition to average precision (AP), AV2 adopts a composite score as an evaluation metric, which takes both AP and localization errors into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The perception range in AV2 is 200 meters (cover area of 400m × 400m), which is much larger than WOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Such a large perception range leads to a huge memory footprint for dense detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Model Configuration To demonstrate the generality of SIR, we build two FSD vari- ants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSDsst adopts the emerging single stride sparse trans- former [6] as the sparse voxel feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSDspconv is built upon sparse convolution based U-Net in PartA2 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the experiments of FSD, we use FSDsst in the experiments unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the experiments of FSD++, we use FSDspconv as our detector since the highly optimized engineering of SpConv makes it more efficient than the SST backbone with multi-frame input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Implementation Details Our implementation is based on popular MMDetec- tion3D(v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='15) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In FSDsst, we use 4 sparse regional attention blocks [6] as our voxel feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The SIR module and SIR2 module consist of 3 and 6 SIR layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A SIR layer is defined by Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Our SST-based model converges much faster than SST, so we train our models for 6 epochs for ablation study, instead of the 2× schedule (24 epochs) in SST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For FSDspconv, in addition to the 6-epoch schedule, we adopt a longer schedule (12 epochs) for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Different from the default setting in MMDetection3D, we decrease the number of pasted instances in the CopyPaste augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In FSD, some scarce classes like cyclist prone to be over-fitted with too many pasted instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All experiments in Argoverse 2 dataset adopt a 12-epoch schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The models for performance analysis (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 ∼ Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6) are trained on 8 RTX 2080Ti GPUs with batch-size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And the models in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1 are trained on 8 RTX 3090 GPUs with batch-size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' More details can be found in our released code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Main Results of FSD and FSD++ We first compare FSD with state-of-the-art detectors and our baseline in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the validation split, FSD/FSD++ achieves state-of-the-art average performance (L2 mAPH) in single-frame/multi-frame settings, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In test split, FSD achieves the best performance on all classes among all single-frame detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Meanwhile, FSD++ with 7-frame input surpasses all detectors with up to 100- frame input, in terms of average metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 8 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "# 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "# ∗ 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% ∗ 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% Skeleton Sampling 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "# & 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% & Residual Point Probing 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "# Δ𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD Skeleton Sampling 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "# & 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% & Residual Point Probing Δ𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD Skeleton Sampling 𝐵!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "% 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' "# 𝑃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 𝑃 Point clouds loaded from disk 𝐵∗ Offline predicted boxes loaded from disk Data loaded from memory Data to be cached in memory Points concatenation Training data flow Inference data flow Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The overall architecture of FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In training, we adopt offline predictions to approximate history predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' During inference, the detector uses previous online predictions for skeleton point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' When the max age is larger than one, there will be some other ∆Pt−i from time step t − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For simplicity, we only present ∆Pt here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (a) Moving cars with static ego-vehicle (b) Moving cars with moving ego-vehicle (c) Static cars and moving pedestrian Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Examples of super sparse input point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Residual points are colored in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Previous foreground points are in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Gray points will not be sent into the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (a) Moving cars cause apparent residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And some occluded points in the ground plane become visible due to car movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (b) The ego-vehicle is moving, which causes some points in the ground plane to be detected as residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (c) Residual points are detected on the moving pedestrian instead of the static cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is also noteworthy that FSD and FSD++ are much more efficient than most of the previous arts, especially in the multi-frame setting and long-range setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We elaborate this in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Study of Treatments to Center Feature Missing 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Quantitative Experiments In what follows, we conduct experiments on WOD to inves- tigate the issue of Center Feature Missing (CFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We first develop several models with different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Note that all the following models adopt the same voxelization resolution, so they face the same degree of CFM at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSDplain: After the sparse voxel encoder, FSDplain directly predicts the box from each voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The voxels inside ground- truth boxes are assigned as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Although FSDplain uses the most straightforward solution for CFM, it suffers from the large variance of regression targets and low-quality predictions from informative voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SSTcenter: It replaces the anchor-based head in SST with CenterHead [9], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Based on the sparse voxel encoder, SSTcenter converts sparse voxels into dense feature maps and applies several convolutions to diffuse features to the empty object centers as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then it makes predictions from the diffused center feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSDnogc: It removes the group correction and SIR2 module in FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' CenterPoint-PP: It does not resort to any sparse voxel encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Instead, it applies multiple dense convolutions soon after voxelization for feature diffusion, greatly elimi- nating CFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is also equipped with CenterHead to avoid large variance of regression targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' There is usually a quite large unoccupied area around the centers of large vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus the performance of large vehicles is an appropriate indicator that reveals the effect of CFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' So we build a customized evaluation tool, which breaks down the object length following the COCO evaluation [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Then we use it to evaluate the performance of vehicles with different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Table 3 shows the results, and we list our findings as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Comparing FSDplain with SSTcenter, they share the same attention-based sparse voxel encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, the trend is totally opposite w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='t vehicle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' With feature diffusion, JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 9 Methods #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' frames mAP/mAPH L2 Vehicle 3D AP/APH 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 FSDspconv (ours)† 1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 / 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 / 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 / 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 / 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 FSD++ (ours)† 7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 / 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3/72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 / 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 / 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 / 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 / 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 TABLE 1 Performances on the Waymo Open Dataset validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All reported results are from single model without any test-time augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: Longer schedule (12 epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We mark the best single-frame results and multi-frame results with gray boxes and cyan boxes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SSTcenter attains much worse performance than FSDplain on large vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It suggests feature diffusion is a sub- optimal solution for CFM in the case of large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For those large objects, the features may not be diffused to the centers or the diffused features are too weak to make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, FSDplain obtains the worst performance among all detectors on vehicles with normal sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Note that the CFM issue is minor for the normal-size vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' So, in this case, the center-based assignment in SSTcenter shows its superiority to the assignment in FSDplain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It suggests the solution for CFM in FSDplain is also sub-optimal, even if it achieves better performance in large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Comparing FSDnogc with SSTcenter, they share the same sparse voxel encoder while FSDnogc replaces the dense part in SSTcenter with SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The huge improvements of FSDnogc on large vehicles fairly reveal that SIR effectively resolves CFM and is better than feature diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' CenterPoint-PP suffers much less from CFM because it leverages dense feature maps from the very beginning of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is also equipped with advanced center- based assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Even so, FSDnogc and FSD still outper- form CenterPoint-PP, especially on large vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Qualitative Analysis In addition to the quantitative experiments, we demonstrate the qualitative effect of CFM and our treatment, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8 showcases the voted centers of FSD and the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' An intuitive illustration of the center feature missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Left: Voted centers of FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Right: Predicted heatmap of SSTcenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' predicted heatmap of center-based SST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Both of them yield high-quality predictions for vehicles of normal size, but their predictions (votes) are usually ambiguous for large vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Center-based dense detectors make predictions from such ambiguous heatmaps, so they are prone to make flawed final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Although the center voting of FSD on large vehicles is also mediocre, FSD only uses the votes to obtain point groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=', instance segmentation), which does not JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 10 Methods #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' frames mAP/mAPH L2 Vehicle 3D AP/APH Pedestrian 3D AP/APH Cyclist 3D AP/APH L1 L2 L1 L2 L1 L2 CenterPoint [9] 1 /69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 /- /71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 /- /67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 /- /68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 AFDetV2-lite [24] 1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0/72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 79.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 / 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 / 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 FSD++ (ours)† 7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 / 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5/84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5/81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 / 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 / 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 TABLE 2 Performances on the Waymo Open Dataset test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All results are in single-model setting without ensemble or test-time augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ∗: Multi-modal methods with camera information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We mark the best single-frame results and multi-frame results with gray boxes and cyan boxes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: 12-epoch schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Vehicle length (m) Methods [0, 4) [4, 8) [8, 12) [12, +∞) Official∗ CenterPoint-PP† 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 FSDplain 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 SSTcenter [6] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 FSDnogc 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 ↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 ↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 ↑ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 ↑ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 ↓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 FSD 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 ↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 ↑ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 ↑ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 ↑ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 TABLE 3 Vehicle detection with vehicle length breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: re-implemented ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ∗: official Waymo L2 overall metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Arrows indicate the performance changes from SSTcenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' necessitate perfect center voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The final predictions of FSD are derived from the complete point groups rather than the weak center features, thereby sidestepping the issue of CFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 200 50 100 150 Perception Range (m) Training Memory (GB) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 > 24 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 OOM > 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 > 24 > 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 FSD_sst CenterPoint CenterPoint-PP SST_center FSD_spconv 50 100 200 150 Perception Range (m) Inference Latency (ms) 50 100 150 200 250 300 400 500 > 800 94 90 64 81 83 97 434 238 89 105 164 232 80 714 208 400 626 54 61 67 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Memory footprints and inference latency in different perception ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Statistics are obtained on a single 3090 GPU with batch size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Inference latency is evaluated by the standard benchmark script in MMDetection3D without any test-time optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' CenterPoint-PP and SSTcenter are defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 Long-range Detection Several widely adopted 3D detection benchmarks [1], [2], [3] have relatively short perception range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To unleash the poten- tial of FSD, we conduct long-range detection experiments on the recently released Argoverse 2 dataset (AV2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' AV2 has a perception range up to 200 meters, making it an ideal testbed for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In addition, AV2 contains objects in 30 classes, exhibiting the long-tail distribution, which is also another challenge for FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Main results We first list the main results of FSD on AV2 in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The authors of AV2 provide a baseline CenterPoint model, but the results are mediocre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To make a fair comparison, we re-implement a stronger CenterPoint model on the AV2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The re-implemented CenterPoint adopts the same training scheme with FSD, including ground-truth sampling to alleviate the long-tail issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD outperforms CenterPoint in the average metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is noteworthy that FSD significantly outperforms CenterPoint in some tiny objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=', Pedestrian, Construction Cone) as well as some objects with extremely large sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=', Articulated Bus, School Bus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We owe this to the virtue of instance-level fine-grained feature extraction in SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Range Scaling To demonstrate the efficiency of FSD in long-range detection, we depict the trend of training memory and inference latency of three detectors when the perception range increases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 9 shows that dense detectors experience a dramatic increase in latency and memory footprint as the perception range grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Designed to be fully sparse, the resource needed for FSD is roughly linear to the number of input points, so its memory and latency only slightly increase as the perception range extends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 Performance Inspection of FSD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Effectiveness of Components In addition to FSDplain and FSDnogc (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3), we also degrade FSD to FSDagg to gain insights into its mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In FSDagg, we aggregate grouped point features by dynamic pooling after Instance Point Grouping, and then directly JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 11 Methods Average Vehicle Bus Pedestrian Stop Sign Box Truck Bollard C-Barrel Motorcyclist MPC-Sign Motorcycle Bicycle A-Bus School Bus Truck Cab C-Cone V-Trailer Sign Large Vehicle Stroller Bicyclist Precision CenterPoint† [9] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 14 CenterPoint∗ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 46.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 FSD 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 57.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 FSD‡ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 59.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 FSD 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 45.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 FSD‡ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 TABLE 4 Performance in Argoverse 2 validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: provided by authors of AV2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ‡: Weak CopyPaste augmentation for preventing overfitting (one instance per class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ∗: re-implemented by ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' C-Barrel: construction barrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' MPC-Sign: mobile pedestrian crossing sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A-Bus: articulated bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' C-Cone: construction cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' V-Trailer: vehicular trailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We omit the results of dog, wheelchair and message board trailer because these categories contain very few instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The average results take all categories into account, including the omitted categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We mark the categories attaining notable improvements in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' make predictions from the pooled features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSDagg is similar to the way in VoteNet [22] as we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus, FSDagg can explicitly leverage instance-level features other than the point-level features in FSDplain, mitigating the issue of CFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, FSDagg cannot take advantage of further point feature extraction in SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As can be seen in Table 5, the improvement is limited if we only apply grouping without SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The combination of grouping and SIR, on the other hand, yields significant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Grouping SIR Group Correction L2 3D APH Vehicle Pedestrian Cyclist FSDplain 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='29 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='31 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='49 FSDagg ✓ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='13 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='13 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='52 FSDnogc ✓ ✓ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='39 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='78 FSD ✓ ✓ ✓ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='30 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='30 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='60 TABLE 5 Ablation of design factors in SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Performances are evaluated on Waymo validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Downsampling in SIR The efficiency of SIR makes it feasible to extract fine-grained point features without any point downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' This is another notable difference between FSD and VoteNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To demonstrate the superiority, we apply voxelization on the raw points before the SIR module and treat the centroids of voxels as downsampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We conduct experiments on the AV2 dataset because it contains a couple of categories in a tiny size, which may be sensitive to downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As expected, small objects have notable performance loss when adopting downsampling, and we list some of them in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We also evaluate the inference latency of the SIR module on a 3090 GPU, which is highly efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 HD Map-assisted Detection Argoverse 2 dataset provides a highly reliable HD map, which could be utilized as a prior to remove uninterested regions making the scene more sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus we proceed with experiments removing some uninterested regions to show the advantages of FSD in more sparse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The results are summarized in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD has a significantly lower memory footprint and latency with an acceptable precision AP Voxel size CC Bollard Bicyclist Stop Sign Latency (ms)† 30cm 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 20cm 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 10cm 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 Point 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 TABLE 6 Performances with different representation granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: Latency of SIR module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD CenterPoint Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Latency(ms) mAP Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Latency(ms) mAP all 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 97 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 232 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 only RoI† 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 ↓ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8% 81↓ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9↓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8% 227↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 w/o ground 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 ↓ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0% 74↓ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7↓ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7% 217↓ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='8 TABLE 7 Performance with different detection areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: Region of Interest is defined by the HD map in AV2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' loss after removing the uninterested regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' On the contrary, the efficiency improvement of CenterPoint is minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It reveals that FSD benefits more from the increase of data sparsity, which is another advantage of the fully sparse architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 Comprehensive Analysis of FSD++ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Preliminary Settings for the Analysis of FSD++ In this section, we conduct extensive experiments to reveal the inner workings of FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Here we first present the setting of our baseline model in this section, which is slightly different from the best FSD++ model in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1 and Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Unless otherwise specified, the default hyper-parameters of all the FSD++ models in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 are listed in the first column of Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The model latency reported in this section is measured on a single RTX 3090 GPU with a mini-batch size of 1 in float32 precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To ensure accuracy, we only consider the latency of the model in all evaluations, excluding the latency of IO, which is potentially unstable in the multi-frame setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is worth noting that we have observed run-to-run variation in the performance of cyclist class, likely due to its low number in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As a result, we mark the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 12 Baseline FSD++ Best FSD++ Multi-frame FSD Schedule 6 epochs 12 epochs 6 epochs SPS† Random Random #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' frames 6 7 6 Max age 2 2 Backbone∗ SpUNet-base SpUNet-large SpUNet-base #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' layers in SIR2 3 3 3 RPP size (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4) TABLE 8 Basic hyper-parameter choice of models adopted in this section (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6) †: SPS stands for skeleton point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ∗: SpUNet-large has one more stage than SpUNet-base [7] and the number of channels of its first is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' performance of this class in gray in some experiments to indicate that it may not be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Skeleton Point Sampling Table 9 shows the performance with different skeleton point sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We find that there are no significant differences between the three strategies considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, using random skeleton sampling considerably reduces the latency of FSD++ without sacrificing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And skeleton sampling consistently boosts the performance of the cyclist class, which suggests that appropriate sampling alleviates the overfitting for rare classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The performance of pedestrian is better without sampling, which reveals that more points might be helpful for small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In practice, although different sampling strategies could be adopted for different classes, we use the random sampling for all classes for simplicity and generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' L2 3D AP/APH Latency (ms) Mean Vehicle Pedestrian Cyclist Random 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='73 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='93/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 Object FPS 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='56/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='21 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='59 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='94/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='14 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='68/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='88 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Voxel Sampling 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='76/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='40 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='66 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='80/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='05 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='37/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='49 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 None 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='23/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='91 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='93/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='50 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='54/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='72 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33/75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='51 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='9 TABLE 9 Effectiveness of different skeleton point sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 Different Number of Frames FSD++ samples skeleton points from multiple previous frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Table 10 showcases how the number of used frames affects its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' There are two interesting findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Performance becomes better as the number of frames grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the meantime, the latency does not signif- icantly increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We owe the credit to residual point probing, which removes most of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It could offer even more clean residual points if more base frames (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD++ outperforms FSD with the same number of frames in vehicle and cyclist class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We also intuitively owe it to RPP since it removes most background clutter and eases the burden of the segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The slightly lower performance of pedestrian suggests it might be better to retain all points for pedestrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, the performance loss is acceptable since FSD++ achieves better average performance and much lower latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' frames L2 3D AP/APH Latency (ms) Mean Vehicle Pedestrian Cyclist 2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='39/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='83 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='54/69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='12 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='68/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='35 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='94/75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='02 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='95/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='52 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='13/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='09 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='51/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='62 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0 4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='44/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='03 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='67/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='21 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='48/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='50 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='18/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='37 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='13/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='72 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='50/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='04 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='71/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='83 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='17/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='29 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 6 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='73 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='19/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='92/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 6 (FSD)† 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='65/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='28 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='54/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='07 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='04/75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='22 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='37/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='54 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 TABLE 10 Performance of FSD++ with the different number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Since FSD++ uses previous foreground points, it needs at least two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: multi-frame FSD model with simple point concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Performance is unstable in the scarce cyclist class, so we mark the numbers in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 Drifting Analysis It would be a major concern if FSD++ suffers from the drifting error given its reliance on history predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In particular, if the detector makes inaccurate predictions at time step t, it is likely that the detector becomes worse at time step t + 1 since the predictions in t + 1 rely on the predictions from t (skeleton point sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To prevent potential drifting, we insert some keyframes at regular intervals during the inference of a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' At keyframes, we use the predictions from standard FSD for skeleton point sampling, which could be viewed as a rectification of the potential drifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 11 shows that FSD++ achieves competitive results without any keyframes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And the minor gap between the first row and the last row confirms that the drifting of FSD++ is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Gap between key frames L2 3D AP/APH Mean Vehicle Pedestrian Cyclist 5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='05/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='67 72.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='66 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='75 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='88/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='07 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='00/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='15 20 76.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='23 50 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='72 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='91/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='32 None 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='73 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='93/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 TABLE 11 The performance of different keyframe gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' “None” means using only initial predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 Change Blindness Ablation Due to the change blindness we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3, newly emerged objects might be ignored by the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Max age is proposed to mitigate the issue of change blindness, and Table 12 shows its effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We find keep residual points for two time steps (max age 2) is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Max age L2 3D AP/APH Latency (ms) Mean Vehicle Pedestrian Cyclist 1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='17/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='82 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='38/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='94 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='97 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='40/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='56 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='73 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='93/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='14/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='22/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='75 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='13/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='27 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 TABLE 12 Different max ages of residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Performance is unstable in the scarce cyclist class, so we mark the numbers in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' For a closer look at the issue of change blindness, we split the objects in the original WOD validation set to emerging objects and existing objects for further ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Emerging objects mean those objects do not appear in the first frame of a sequence, while emerging later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The results are shown in JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 13 Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The emerging objects recall of FSD++(1)1 is inferior to FSD 6f with the same number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' This suggests that change blindness is indeed an issue for FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' However, prolonging the max age makes FSD++ outperform FSD 6f in all classes, which demonstrates the proposed max age effectively mitigates change blindness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Method Recall of emerging objects Mean Vehicle Pedestrian Cyclist FSD 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='05 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='01 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='62 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='53 FSD 6f∗ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='54 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='34 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='53 FSD++(1) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='58 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='18 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='48 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='09 FSD++(2) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='82 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='56 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='79 FSD++(3) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='14 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='60 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='30 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='52 TABLE 13 Performance for emerging objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ∗: FSD with 6-frame concatenated input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the WOD validation split, the number of emerging objects count for around 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4%/37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1%/52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6% in all objects for vehicle / pedestrian / cyclist, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6 Robustness to Seed Quality During inference of every point cloud sequence, FSD++ needs the predictions in the initial frame as a seed to start, as we discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Here we figure out how the quality of seed predictions affects the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Concretely, we add two typical kinds of random noise to seed predictions, including random box drop and random box insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' They are designed to simulate false negatives and false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All the experiments share a trained FSD++ detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The modifications above are applied during inference and are not adopted for training-time augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Table 14 shows FSD++ is robust to both two types of noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Particularly, for box drop, there is only marginal performance degradation even after dropping all the initial seed boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We explain this surprising phenomenon in two aspects: (1) FSD++ is born to be robust to the dropping of moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' This is because moving objects create a considerable amount of residual points and FSD++ is capable of making predictions from these residual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) There is still a small number of residual points in the static objects due to the change of viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Moreover, the mechanism of max age also helps accumulate residual points on static objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In the case of box insertion, FSD++ is almost unaffected because they can be easily identified as background in the segmentation stage of FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Noise type L2 3D AP/APH Mean Vehicle Pedestrian Cyclist None 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='73 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='93/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 Drop (10%)† 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='95/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='58 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='66 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='81/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='99 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='92/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='08 Drop (50%) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='76/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='39 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='86/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='41 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='63/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='82 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='78/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='95 Drop (100%) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='69/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='35 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='47/70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='02 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='41/72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='69 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 Insertion (10%) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='00/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='62 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='16/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='70 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='82/73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='99 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='01/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='16 Insertion (50%) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='02/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='64 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='14/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='69 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='86/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='04 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='05/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='19 Insertion (100%) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='98/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='61 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='16/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='71 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='84/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='02 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='95/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10 TABLE 14 Robustness to the noisy seed predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: the percentage in parentheses denotes the ratio of dropped/inserted instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The numbers in the parenthesis denote the max ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 Analysis of Residual Point Probing Quantization size and the number of base frames are two important hyper-parameters in RPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Here we show how they affect the output residual points and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Quantization size makes RPP robust to small point distur- bance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We list the results of different quantization sizes in Table 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It could be seen from the “residual point ratio” that RPP with larger quantization sizes leads to less residual points making the detector more efficient but leading to slightly lower performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Quantization size L2 3D AP/APH Residual point ratio† Latency (ms) Vehicle Pedestrian Cyclist (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='60 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='19/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='43 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='53/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='67 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='93/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='04/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='58 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='88/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='03 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='54/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='0% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 TABLE 15 The effectiveness of quantization size in Residual Point Probing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: residual point ratio means the average ratio of the residual points to the total points in a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Base frame (in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 4) also has a considerable effect on RPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The more base frames are incorporated, the less residual points could be obtained, leading to higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' More- over, the performance is hardly affected by the increase of base frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' #.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' RPP base frames L2 3D AP/APH Residual point ratio† Latency (ms) Vehicle Pedestrian Cyclist 3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='48/72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='03 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='16/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='35 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='60/76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='78 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='24/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='79 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='04/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20 78.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='33 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='6% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='22/71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='77 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='41/74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='59 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='60/77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='74 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='5 TABLE 16 The effectiveness of the number of base frames in Residual Point Probing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' †: residual point ratio means the average ratio of the residual points to the total points in a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='7 Detailed Runtime Evaluation Here we elaborate on the efficiency of each component of FSD and FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' All evaluated models use SpUNet-large as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Evaluations are conducted on a single RTX 3090 in FP32 precision without any test-time optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We only record the single-sample forward latency of the detector implemented with MMDetection3Dv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='15, ignoring the IO of point clouds which is unstable in the multi-frame setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 10 shows the detailed results, which are average numbers evaluated on the first ten sequences of validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As can be seen from the figure, the latency of the segmentor is greatly reduced, which consists of the sparse voxel encode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=', backbone) and segmentation head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' As a results, FSD++ is as fast as the single-frame FSD, yet achieves better performance than FSD 6f (Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' It is worth emphasizing that the “others” part of latency is usually brought by some serialized operations, such as class-wise detection heads and class-wise NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' This part of latency could be greatly reduced in deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 6 CONCLUSION This paper first proposes a LiDAR-based fully sparse 3D object detection framework, namely FSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD forsakes the JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 14 Latency Breakdowns 100 80 60 40 20 0 100 80 60 40 20 0 Latency (ms) FSD_6f FSD FSD++ Segmentor Skeleton Sampling and RPP SIR Group Correction SIR2 Clustering Others Segmentor Skeleton Sampling and RPP SIR Group Correction SIR2 Clustering Others Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Latency breakdowns of multi-frame FSD, single-frame FSD and FSD++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' widely adopted dense BEV feature map in previous arts, which is the hindrance to making detectors fully sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Instead, FSD consists of a general sparse voxel encoder and a highly-efficient sparse instance recognition (SIR) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' SIR remedies the issue of the center feature missing, which is the essential difficulty of fully sparse architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD not only actualizes efficient long-range (up to 200 meters) detection on Argoverse 2 dataset, but also achieves state-of-the-art performance on the competitive Waymo Open Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' To unleash the potential of FSD, we propose lever- aging temporal information to remove data redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The proposed Skeleton Point Sampling and Residual Point Probing offer FSD a super sparse input point cloud, which constitutes the FSD++ framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' FSD++ achieves state-of- the-art single-model performance on both validation and test split Waymo Open Dataset, and maintains high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We hope our work lightens future research direction for LiDAR-based point cloud recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' APPENDIX A EFFICIENT DYNAMIC POOLING The proposed SIR consists of three basic operations: MLP, dynamic broadcasting, and dynamic pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' MLP is highly optimized in mainstream deep learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Dy- namic broadcasting is essentially an indexing operation, which is highly parallel in modern GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The efficiency bottleneck of SIR lies in dynamic pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Thus we provide an efficient implementation of dynamic pooling in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='1 Implementation The dynamic pooling implemented by PyTorch is known as scatter operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In this implementation, each thread manages one feature and performs simple atomic operations for feature reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Intensive atomic operations in large groups are detrimental to parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We optimize the dynamic pooling operator in three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (1) Partitioning large groups into small sub-groups with fixed sizes to balance the workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (2) Sorting the Partition ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Atomic Operation Group 𝑖 (small) Group 𝑗 (large) 𝑁 point features 𝐶 block 𝑗, part 1 block 𝑖 block 𝑗, part 2 … … … Loop Reduction Warp of Threads Loop Reduction Warp of Threads Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Illustration of dynamic pooling implementation in CUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' We take two groups with different sizes as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Dynamic pooling reduces each group to a single feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' features with the same group ID in adjacent positions to enhance the memory locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Note that the group IDs of each element remain unchanged throughout the SIR module, so the sort is only applied once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' (3) Threads in a warp are assigned to adjacent channels for coalesced memory access without warp divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' In particular, each thread corresponds to a feature dimen- sion and reduces features in a partitioned sub-group through loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Besides, the calls to atomic operators are reduced from once per thread to once per block, which contributes to high parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 11 illustrates our efficient implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='2 Runtime Evaluation We take dynamic max-pooling as an example and evaluate the latency of our implementation and torch scatter in differ- ent cases, including multiple feature dimensions, multiple group sizes, and whether the group size is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The total number of groups in the evaluation is 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' The results are shown in Table 17 and Table 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' With different data sizes, our implementation achieves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='48× to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='58× speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' And we have a more significant speedup with imbalanced group sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Feature dimension Latency (ms) with Different Group Sizes [100, 101) [101, 102) [102, 103) [103, 104) 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='53 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='09/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='10/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='37/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='82/65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='28 Speedup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='48× 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='56× 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='47× 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='53× TABLE 17 Latency of dynamic max pooling on data with balanced group sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Each item denotes the latency of ours/torch scatter in milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' 8, AUGUST 2015 15 Feature dimension Latency (ms) with Different Group Sizes [100, 101)∗ [101, 102)∗ [102, 103)∗ [103, 104)∗ 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='08/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='20/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='19/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='75 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='06/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='12/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='57/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='91/196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='4 Speedup 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='05× 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='81× 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='01× 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content='58× TABLE 18 Latency of dynamic max pooling on data with imbalanced group sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Each item denotes the latency of ours/torch scatter in milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' ∗: The sizes of one-tenth groups are enlarged by 10× to create imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Kretzschmar, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE0T4oBgHgl3EQfrQEH/content/2301.02562v1.pdf'} +page_content=' Dotiwalla, A.' 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/dev/null +++ b/rtFJT4oBgHgl3EQfbiyy/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:698a4fa5223d7e2850592abffea90e3306e4b4c66185326f43b886821266b166 +size 171099 diff --git a/sdE2T4oBgHgl3EQf1gg8/content/tmp_files/2301.04151v1.pdf.txt b/sdE2T4oBgHgl3EQf1gg8/content/tmp_files/2301.04151v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae6a9e919ccaaf3ca62e2857d022f2e646c9aab6 --- /dev/null +++ b/sdE2T4oBgHgl3EQf1gg8/content/tmp_files/2301.04151v1.pdf.txt @@ -0,0 +1,5698 @@ +Prepared for submission to JHEP +Blowup Equations for Little Strings +Hee-Cheol Kim,a,b Minsung Kim,a Yuji Sugimotoa +aDepartment of Physics, POSTECH, Pohang 790-784, Korea +bAsia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea +Abstract: We propose blowup equations for 6d little string theories which generalize +Nakajima-Yoshioka’s blowup equations for the 4d/5d instanton partition functions on +Omega background. We find that unlike the blowup equations for standard SQFTs, +we need to sum over auxiliary magnetic fluxes on the blown-up P1 for a non-dynamical +2-form gauge field which plays a role in canceling the mixed anomalies of the gauge +symmetries. We demonstrate with explicit examples that the blowup equations, when +combined with the modular properties, can be solved in order to determine the elliptic +genera of little strings. +arXiv:2301.04151v1 [hep-th] 10 Jan 2023 + +Contents +1 +Introduction +1 +2 +Blowup equations and Modular bootstrap +4 +2.1 +Blowup equations for 6d SCFTs +5 +2.2 +Blowup equations for LSTs +9 +2.3 +Bootstrapping LSTs +14 +3 +Examples +18 +3.1 +ˆA1 LSTs +18 +3.1.1 +IIA picture +19 +3.1.2 +IIB picture +25 +3.2 +Heterotic LSTs +29 +3.2.1 +E8 × E8 picture +29 +3.2.2 +SO(32) picture +37 +3.3 +SU(3) + 1sym + 1Λ2 +43 +4 +Conclusion +49 +A Elliptic functions +50 +A.1 Modular forms +50 +A.2 Jacobi forms +51 +B Derivation of elliptic genera +55 +B.1 Elliptic genus of E8 × E8 heterotic LST +56 +B.2 Elliptic genus of SO(32) heterotic LST +59 +B.3 Elliptic genus of SU(3) + 1sym + 1Λ2 +61 +1 +Introduction +Since the introduction of supersymmetric quantum field theories (SQFTs), numerous +advancements have been made. From a classification perspective, five-dimensional +supersymmetric field theories are classified by examining the consistency of physics on +the Coulomb branch of moduli space [1–4], utilizing geometric descriptions [3, 5–10], +and analyzing the RG-flows of 6d superconformal field theories (SCFTs) on a circle +[11–17]. Six-dimensional supersymmetric field theories are classified based on the +types of non-compact bases and the methods of gluing them together in F-theory +– 1 – + +compactified on non-compact elliptically fibered Calabi-Yau threefolds [18–21]. The +classification of 6d little string theories (LSTs) is also discussed in [21, 22] which will +be explained right after. +There have been significant quantitative studies conducted on higher dimensional +SQFTs. For instance, it is possible to calculate the supersymmetric partition functions +of these theories on Ω-deformed R4 using various methods. This partition function is +a type of Witten index that counts BPS states on the Coulomb branch. For theories +with classical gauge groups, it can be computed using supersymmetric localization +based on ADHM constructions of the instanton moduli space [23, 24] or using the +topological vertex formalism introduced in [25–27]. We can also calculate the partition +function in the presence of the codimension two or four defects in these theories +[28–44]. While the SUSY localization and topological vertex formalism are effective +methods for qualitatively studying higher dimensional SQFTs, one disadvantage of +these methods is that we need to know ADHM constructions for instanton moduli +space of gauge theories or brane web descriptions of these SQFTs in Type IIB string +theory. +An alternative approach to calculating the instanton partition functions is by +utilizing the blowup equation, which was initially developed for the study of Donaldson +invariants in mathematics. In [45], a systematic method for formulating and solving +the blowup equations to obtain Nekrasov’s instanton partition functions of 4d N = 2 +SU(N) gauge theories was proposed, and this method was subsequently generalized +to 5d N = 1 SU(N) gauge theories in [46, 47]. More recently, several extensions +have been made to compute various observables in higher dimensional field theories: +the instanton partition functions of 5d SUSY gauge theories with generic gauge +groups and matter representations were computed in [48–50], the blowup equations +for refined topological strings on certain local Calabi-Yau 3-folds were formulated in +[50, 51], and the elliptic genera of self-dual strings in 6d SCFTs on tensor branch +were calculated using elliptic blowup equations in [50, 52–54]. Also, as another +extension, the blowup equations for 5d and 6d supersymmetric field theories with +codimension four defects were proposed in [55]. One of the key benefits of the blowup +approach is its capability to systematically calculate the partition functions, including +non-perturbative contributions, through the use of the effective prepotential and +consistent magnetic fluxes on the blowup background which can be systematically +obtained for any 5d or 6d supersymmetric field theory [50, 51]. While we now have a +large amount of examples for the blowup formalism, it remains to be verified whether +it can be applied to little string theories. +Little string theories were originally introduced as worldvolume theories of NS5- +branes in the gravity decoupling limit [56–60]. Depending on the space in which +the NS5-branes reside, there are N = (2, 0), (1, 1), and (1, 0) LSTs in 6 dimensions. +The decoupling limit is achieved by taking the string coupling constant to zero +while maintaining an intrinsic string scaling finite. The resulting theory becomes a +– 2 – + +non-local theory without gravity and it has some stringy properties such as T-duality. +In this sense, this theory is an intermediate theory between local quantum field +theories and the usual string theories, so a deep understanding of the LSTs may +help us understand both subjects. Additionally, since NS5-branes are known to be +one of the most challenging and interesting non-perturbative objects to study, a +better understanding of these objects is desired. Beside this, there are also various +motivations for studying LSTs, including the investigation of discrete light-cone +quantization and holography in a linear dilation background, etc. For a brief overview +of LSTs in these contexts, we recommend readers to see [61, 62]. +A systematic construction of the LSTs is proposed based on the geometric phases +of F-theory in [22]. This construction allows us to classify LSTs according to the +types of base curves and the manner in which they are connected, generalizing the +geometric classification of 6d SCFTs. In quantitative studies, the partition functions +of A-type LSTs engineered by NS5-branes on A-type singularities have been obtained +using the localization method applied to the worldvolume theories of 2d instantonic +strings [63]. Similarly, the elliptic genera of LSTs on some of D-type singularities have +been computed based on the localization method [64]. Also, in [65], the elliptic genera +of some LSTs were calculated using T-dualities and the modular ansatz, which is +based on the modular properties of the elliptic genera. However, these computations +have only been carried out in a few simple cases, as the localization method requires +ADHM-like constructions of little string worldvolume theories that are currently +unavailable in most cases. The modular ansatz method also has some limitations, +including a rapid increase in the number of unknown coefficients as the string number +increases and the need for precise knowledge of T-duality for LSTs. As such, it is +important to find alternative methods for calculating the partition functions of LSTs. +In this paper, we propose a systematic method for constructing blowup equations +for LSTs and provide examples of its application in the explicit calculation of partition +functions. The blowup equations can be formulated by using two key ingredients: the +effective prepotential evaluated on the Ω-background, which can be obtained from +the effective cubic and mixed Chern-Simons terms on the tensor branch, and a set of +magnetic fluxes on the blown-up P1. We will explain how to obtain these ingredients +for arbitrary LSTs. +It turns out that the blowup equations for LSTs are rather different from those +for 5d/6d SQFTs. Interestingly, the blowup equations for LSTs involve summation +over magnetic fluxes for an auxiliary gauge field as well as those for the dynamical +gauge fields. This auxiliary gauge field is a non-dynamical 2-form field used to +cancel mixed anomalies of gauge symmetries. We will explain the precise role of this +auxiliary gauge field in the next section. However, we note that the summation over +auxiliary magnetic fluxes is not convergent in terms of Kähler parameters. This is +essentially due to the absence of a quadratic kinetic term for the auxiliary gauge field. +Despite this, we show that the partition functions of the LSTs calculated from other +– 3 – + +methods satisfy the blowup equations that we have proposed when certain upper and +lower bounds are placed on the power of Kähler parameters coupled to the auxiliary +magnetic fluxes. Furthermore, we will show that elliptic genera of the LSTs can be +calculated by solving the blowup equations when combined with the modular ansatz. +We illustrate our approach using ˆA1 type IIA and IIB LSTs, E8 × E8 and SO(32) +Heterotic LSTs as rank-1 LSTs, and the SU(3) gauge theory with a symmetric and +an antisymmetric hypermultiplets as a rank-2 LST1. +The rest of this paper is organized as follows. In Section 2, we provide a review of +the blowup equations and modular bootstrap approach for 6d SCFTs, and present the +proposed blowup equations for LSTs. In Section 3, we demonstrate how our proposal +works with several examples. In Section 4, we summarize our results and discuss some +future directions. In Appendix A, we collect some facts about the elliptic functions +used in the main context. In Appendix B, we present the computations of the elliptic +genera of some LSTs using ADHM constructions of instantonic strings. +2 +Blowup equations and Modular bootstrap +In this section, we propose the blowup equations for the partition functions of 6d +little string theories on T 2 × R4. We begin by reviewing the formulation of blowup +equations for 6d SCFTs and then extend this approach to construct the blowup +equations for 6d LSTs. We also describe how to calculate the elliptic genera of strings +in LSTs using their modular properties and by solving the blowup equations. +The elliptic genera of strings in 6d LSTs on a torus T 2 times Ω-deformed R4, +which is a Witten index, is defined as +Zk(τ, φ, m; ϵ1,2) = TrRR +� +(−1)Fe2πi(τHL−¯τHR)e2πiϵ1(J1+JR)e2πiϵ2(J2+JR)e2πiφ·Πe2πim·F� +, +(2.1) +where τ is the complex structure of the torus, HL and HR are left-moving and right- +moving Hamiltonians in the 2d worldsheet, J1 and J2 are Cartan generators of the +SO(4) Lorentz rotation on R4, JR is the Cartan for SU(2)R R-symmetry, ϵ1 and ϵ2 +are the Ω-deformation parameters, Π and F are gauge and flavor charges, and φ and +m collectively denote chemical potentials for gauge and flavor symmetries, repectively. +The supercharge Q and its conjugate Q† commute with J1 + JR and J2 + JR, and +the right-moving Hamiltonian is given by 2HR = {Q, Q†}. This elliptic genus counts +BPS spectrum in the Ramond sector annihilated by Q and Q†, so it is independent +of ¯τ, in the 2d worldsheet SCFT living on strings with tensor charge denoted by k. +1Here we use a word “rank” as a number of dynamical parameters which is different from [22]. +– 4 – + +2.1 +Blowup equations for 6d SCFTs +Let us review the blowup equations for the 6d SCFT which are functional equations +for the full partition function defined by +Z(τ, ϕ, φ, m; ϵ1,2) = e−2πiEZpert × +� +k +e−kϕZk , +(2.2) +with Zk=0 = 1 where Zpert is the perturbative contribution of the partition function +on T 2 × R4, E is called the effective prepotential and ϕ here denotes the tension of +the self-dual strings with charge k parameterized by the scalar vacuum expectation +values in the tensor multiplets. We note that the partition function (2.2) can be +written as a factorized expression of +Z = e−2πiEZGV = e−2πiE PE +� � +jl,jr,d +(−1)2(jl+jr)N d +jl,jr +√p1p2χjl(ϵ−)χjr(ϵ+) +(1 − p1)(1 − p2) +e2πid·m +� +, +(2.3) +where ZGV is the refined Gopakumar-Vafa (GV) invariants [66, 67] counting the BPS +degeneracies N d +jl,jr on Ω-background. Here, PE[f(µ)] = exp +��∞ +n=1 +1 +nf(µn) +� +is the +Plethystic exponential, m collectively denotes chemical potentials (ϕ, φ, m) for tensor, +gauge and flavor symmetries, d is the electric charge of the BPS state with Lorentz +spin (jl, jr) = ( J1−J2 +2 +, J1+J2 +2 +), χj is the SU(2) character of spin j representation, +ϵ± = ϵ1±ϵ2 +2 +and p1,2 = e2πiϵ1,2. +The effective prepotential E in the prefactor arises from the classical action and +the regularization factors in the path integral. We can compute it by evaluating +the low energy effective action on the Ω-background in the presence of non-trivial +background gauge fields. The low energy effective action of 6d SCFTs on a circle +which was computed in [14, 50, 68–70] consists of the classical (or tree-level) and the +1-loop contributions. The tree-level action for a 6d SCFT is given by +Stree = +� � +−1 +2ΩαβGα ∧ ∗Gβ − ΩαβBα ∧ X4β + · · · +� +(2.4) +where · · · stands for the SUSY completions, Ωαβ = Σα · Σβ is the intersection form +of the tensor multiplets corresponding to the intersection pairing of compact cycles +Σα in the base surface of the associated elliptic Calabi-Yau threefold, and Gα is +the 3-form field strength for a self-dual 2-form tensor field Bα. The second term is +the Green-Schwarz term which is required to cancel the 1-loop gauge anomalies via +Green-Schwarz-Sagnotti mechanism [71]. The Green-Schwarz term contributes to the +6d anomaly 8-form as +IGS = −1 +2ΩαβX4αX4β , +(2.5) +– 5 – + +where the 4-form X4α, which appears in the Bianchi identity dGα = X4α, is given by +X4α = −1 +4aαp1(T6) + 1 +4 +� +a +ba,α Tr F 2 +a + cαc2(R) . +(2.6) +Here, p1(T6) and c2(R) are the first Pontryagin class of the 6d spacetime tangent +bundle and the second Chern class of SU(2)R R-symmetry bundle, respectively, and +Fa is the field strength of the a-th symmetry group including all gauge and flavor +symmetries. The coefficients aα, ba,α and cα are determined from anomaly cancellation +conditions. +The classical action provides non-trivial contributions to the effective prepotential. +The characteristic classes in the Green-Schwarz terms can be replaced by the Ω- +deformation parameters as +p1(T6) �→ −(ϵ2 +1 + ϵ2 +2) , +c2(R) �→ ϵ2 ++ , +(2.7) +with ϵ± = ϵ1±ϵ2 +2 +. Using this, the tree-level effective prepotential on the Ω-deformed +background is evaluated as +Etree = +1 +ϵ1ϵ2 +� +−τ +2Ωαβφα,0φβ,0 − Ωαβφα,0 +�aβ +4 (ϵ2 +1 + ϵ2 +2) + ba,β +2 Ka,ijφa,iφa,j + cβϵ2 ++ +�� +, +(2.8) +where φα,0 = iϕα/2π denotes the scalar VEV in the tensor multiplet, Ka,ij is the +Killing form of the a-th symmetry group, and φa,i are the holonomies for gauge and +flavor symmetries. +There are also 1-loop contributions to the effective prepotential E which can be +calculated as follows. The 6d SCFT compactified on a circle leads to a 5d Kaluza-Klein +(KK) theory. The low energy theory of this 5d KK theory on a Coulomb branch +is characterized by topological Chern-Simons couplings which involve contributions +from the Kaluza-Klein momentum states along the circle as well as zero momentum +states. The 1-loop Chern-Simons terms at low energy can be written as +S1−loop = +� � Cijk +24π2Ai ∧ Fj ∧ Fk − 1 +48CG +i Ai ∧ p1(T6) + 1 +2CR +i Ai ∧ c2(R) +� +, +(2.9) +where the first term is the cubic Chern-Simons term for the gauge and the flavor +symmetries, the second and third terms are the mixed gauge-gravity and gauge- +SU(2)R R-symmetry Chern-Simons terms, respectively. +The cubic Chern-Simons terms are determined by the cubic prepotential given +by [3, 68–70] +F1−loop = 1 +12 +� +n∈Z +� +�� +e∈R +|nτ + e · φ|3 − +� +f +� +w∈wf +|nτ + w · φ + mf|3 +� +�, +(2.10) +– 6 – + +where R is the set of root vectors of the 6d gauge group, wf and mf are a set of +weights and mass parameters, respectively, of the f-th charged hypermultiplet. The +summation over all integer KK charges n can be performed by using the zeta function +regularization2. The mixed Chern-Simons coefficients CG +i and CR +i can be computed +in a similar manner. At this time, the contributions of the positive KK charge states +and negative KK charge states cancel each other, and so we find +CG +i = −∂i +� +�� +e∈R +|e · φ| − +� +f +� +w∈wf +|w · φ + mf| +� +� , +CR +i = 1 +2∂i +� +e∈R +|e · φ| . +(2.11) +The 1-loop contribution to the effective prepotential is comprised of the collection of +the Chern-Simons contributions. +E1−loop = +1 +ϵ1ϵ2 +� +F1−loop + ϵ2 +1 + ϵ2 +2 +48 +CG +i φi + ϵ2 ++ +2 CR +i φi +� +. +(2.12) +The full effective prepotential of a 6d SCFT on a torus times Ω-deformed R4 is +then given by the sum of the classical and the 1-loop contributions: +E = Etree + E1−loop . +(2.13) +This explains how to compute the effective prepotentials E for arbitrary 6d SCFTs. +We remark that when the 6d SCFT is compactified on a circle with automorphism +twists, the intersection form Ωαβ, the Killing form Ka,ij, and the gauge and flavor +algebra appearing in the effective prepotential should be replaced by those of the +twisted theory. See [50] for explicit calculations of E for many interesting 6d SCFTs +on T 2 × R4 with/without twists. +Let us now explain how to formulate the blowup equations for 6d SCFTs. Consider +a 6d SCFT on a blowup geometry ˆC2 obtained by replacing the origin of the C2 by a +2-sphere P1. The partition function on this ˆC2 background, which we will call ˆZ, is +factorized under the supersymmetric localization as a product of two contributions +coming from the north and south pole of the P1 at the origin [45, 46]. It turns out that +the partition function is independent of the volume of the P1, and thus blowing down +the P1 results in a smooth transition from ˆZ to the ordinary partition function Z on +C2 without the P1 at the origin. More precisely, as indicated in [50], the partition +function ˆZ defined on ˆC2 is related after the blowdown transition to the ordinary +partition function Z on C2 by replacing (−1)F in (2.1) and (2.2) by (−1)2JR. This +replacement of the fermion number operator can be implemented by shifting the +Ω-deformation parameter ϵ1 for the angular momentum to ϵ1 + 1. Thus one finds +ˆZ(φ, m, ϵ1, ϵ2) = e−2πiE(φ,m,ϵ1,ϵ2) ˆZGV(φ, m, ϵ1, ϵ2) , +ˆZGV(φ, m, ϵ1, ϵ2) = ZGV(φ, m, ϵ1 + 1, ϵ2) . +(2.14) +2For a 6d SCFT with twist, we can have fractional KK-momentum states. See [50]. +– 7 – + +Note that ϵ1 in the prefactor E remains the same because shifting ϵ1 in the GV- +invariant does not affect the regularization factor. +Now, by identifying the blowup partition function ˆZ on ˆC2, which takes a +factorized expression under localization, with the partition function Z on the ordinary +C2 background, we can find a functional equation so-called a blowup equation as +follows [45–47] (See also [49–55, 72, 73] for various generalizations) : +Λ(m, ϵ1, ϵ2) ˆZ(φ, m, ϵ1, ϵ2) = +� +⃗n +(−1)|⃗n| ˆZ(N)(⃗n, ⃗B) ˆZ(S)(⃗n, ⃗B) , +(2.15) +where |⃗n| = � +i ni denotes the sum of magnetic fluxes ni for the dynamical tensors +and gauge symmetry groups on P1, ⃗B denotes the background magnetic fluxes for +the global symmetries, and Λ is a constant prefactor independent of the dynamical +Kähler parameters φ. Here, ˆZ(N) and ˆZ(S) are localized partition functions near the +north and south poles of the P1. They can be obtained, since the local geometries +can be approximated as C2, from the ordinary partition function Z by shifting the +chemical potentials as +ˆZ(N)(⃗n, ⃗B) = ˆZ(φi + ϵ1ni, mj + ϵ1Bj, ϵ1, ϵ2 − ϵ1) , +ˆZ(S)(⃗n, ⃗B) = ˆZ(φi + ϵ2ni, mj + ϵ2Bj, ϵ1 − ϵ2, ϵ2) . +(2.16) +Here φi collectively denotes the scalar VEVs in the tensor and gauge multiplets. +The prefactor Λ can be zero. In this case, the blowup equation is called vanishing +blowup equation. For example, when the 6d SCFT contains a half hypermultiplet +that does not form a full hypermultiplet, the theory admits only vanishing blowup +equations. We will not discuss vanishing blowup equations in this paper. +We cannot turn on arbitrary magenetic fluxes (⃗n, ⃗B) on the P1, but they must +be correctly quantized. The proper quantization conditions for the maginetic fluxes +are [51] +(⃗n, ⃗B) · e is integral/half-integral ⇔ 2(jl + jr) is odd/even, +(2.17) +for all BPS particles of the gauge and flavor charge e and spin (jl, jr). Among (⃗n, ⃗B) +satisfying the quantization conditions, a set of special magnetic fluxes called consistent +magnetic fluxes can give a blowup equation (2.15) which the partition function Z +obeys. We refer the reader to [50] for a detailed discussion on the process of identifying +the consistent magnetic fluxes and solving the blowup equations to calculate the BPS +spectra of 5d and 6d SQFTs. +It is more convenient to express the blowup equation (2.15) in terms of the +GV-invariant as follows: +Λ(m, ϵ1, ϵ2) ˆZGV(φ, m, ϵ1, ϵ2) = +� +⃗n +(−1)|⃗n|e−2πiV ˆZ(N) +GV (⃗n, ⃗B) ˆZ(S) +GV(⃗n, ⃗B), +(2.18) +– 8 – + +where +V = E(φi, mj, ϵ1, ϵ2) − E(φi + ϵ1ni, mj + ϵ1Bj, ϵ1, ϵ2 − ϵ1) +− E(φi + ϵ2ni, mj + ϵ2Bj, ϵ1 − ϵ2, ϵ2) . +(2.19) +For a 6d SCFT, we can split the GV-invariant part as +ZGV = Zpert × Zstr = Zpert × +� +⃗k +e−2πiΩαβkαφβ,0Z⃗k . +(2.20) +Here, Zpert is the 1-loop perturbative contributions from the tensor, vector and +hypermultiplets. +Zstr is the self-dual string contributions which are given by a +summation over the elliptic genera Z⃗k of the worldsheet SCFTs on self-dual strings +with tensor charge ⃗k ≡ (k1, k2, · · · , kN) where kα ∈ Z≥0 for all α. The explicit form +of the 1-loop contributions is given by +Zpert = PE [Itensor + Ivector + Ihyper] , +(2.21) +where the single-letter contributions Itensor, Ivector, Ihyper of tensor, vector and hyper- +multiplet are +Itensor = − +p1 + p2 +(1 − p1)(1 − p2) +1 +1 − q , +(2.22) +Ivector = − +1 + p1p2 +(1 − p1)(1 − p2) +1 +2 +� +n∈Z +� +ρ∈R +e2πi|nτ+ρ·φ| , +(2.23) +Ihyper = +√p1p2 +(1 − p1)(1 − p2) +� +n∈Z +� +f +� +w∈wf +e2πi|nτ+w·φ+mf| , +(2.24) +where q = e2πiτ. +For a given 6d SCFT, we can systematically compute the effective prepotential +E and find the consistent magnetic fluxes (⃗n, ⃗B), and thus formulate the blowup +equations as in (2.18). We then expand the blowup equations in terms of the Kähler +parameters e2πid·m and solve them iteratively to calculate the BPS degeneracies of +N d +jl,jr of the 6d theory. +2.2 +Blowup equations for LSTs +We will now extend the blowup formalism for 6d SCFTs to 6d LSTs. Little string +theories are characterized by a collection of 2-form tensor fields whose intersection +pairing, represented by Ωαβ, is negative semi-definite and has a single null direction. +This means that there exists a unit vector ℓα in the string charge lattice such that +Ωαβℓβ = 0. As a result, the tensor field corresponding to the null direction ℓα in the +string charge lattice is non-dynamical. We will call the strings with tensor charges +propotional to ℓα as the full winding strings. The tension of these full winding strings, +– 9 – + +represented by T ∼ M 2 +string, is always finite, and it defines the intrinsic scale of the +LST. The full winding strings in LSTs are therefore distinguised from the self-dual +strings in 6d SCFTs which have tensionless limit. +We are interested in the elliptic genera of 2d worldsheet SCFTs of LSTs on tensor +branch. We can write the contributions from the dynamical strings to the partition +function as a collection of the elliptic genera of the strings as follows: +Zstr = +� +⃗k +vk1 +1 · · · vkN +N Z⃗k , +(2.25) +where +vα ≡ e−2πiΩαβφβ,0 , +vN ≡ e2πi(w−ΩNβφβ,0) , +(2.26) +with α = 1, · · · , N − 1 and β = 1, · · · , N. Here, the scalar VEVs of the N − 1 tensor +multiplets φβ,0 and the little string tension w ∼ T play the role of chemical potentials +for the string charges. The full winding string states are represented by the fugacity +e2πiw, but are independent of other tensor scalar VEVs φβ,0. +There is a natural limit w → i∞ while keeping φα,0 finite. In this limit the full +winding string states are truncated and the LST is reduced to a 6d SCFT with N − 1 +tensor multiplets. From this point of view, the LST can be considered as an affine +extension of the 6d SCFT by attaching an affine tensor node to the tensor quiver +diagram. This leads to the intersection form Ωαβ with N tensor nodes of the LST +[22]. The partition function (2.25) under this 6d SCFT limit becomes that of the +self-dual strings in the 6d SCFT and it satisfies the blowup equation discussed in the +previous subsection. +We now proceed to construct the blowup equations for the partition function of +the LSTs and use them to compute their elliptic genera. One of the distinguished +features of the LSTs from 6d SCFTs is that the mixed gauge-global anomalies in +LSTs are not completely canceled by the standard Green-Schwarz mechanism. The +anomaly 8-form for the mixed anomalies should take a factorized form as +Imixed +8 += Y4 ∧ X4,0 , +(2.27) +where the first factor Y4 is a 4-form given in terms of the second Chern classes for +the dynamical gauge fields +Y4 = 1 +4 +N +� +α=1 +ℓαTrF 2 +Gα , +(2.28) +and the second factor X4,0 is a 4-form independent of the dynamical gauge field which +can be written as +X4,0 = −1 +4a0p1(T6) + 1 +4 +� +a +ba,0 Tr F 2 +a + c0c2(R) . +(2.29) +– 10 – + +Here, FGα and Fa are the field strength for the gauge group Gα and that for the a-th +flavor group respectively. We normalize the instanton number as kα = 1 +4TrF 2 +Gα ∈ Z +when integrated over a 4-manifold and it parametrizes the α-th direction in the +string charge lattice. The coefficents a0, ba,0, and c0 are fixed by the 1-loop and the +Green-Schwarz anomaly calculations. When the theory has a F-theory construction +on an elliptic Calabi-Yau threefold, we can identify them as +a0 = K · ΣLST , +ba,0 = ΣFa · ΣLST , +c0 = ℓαh∨ +α , +(2.30) +where K is the canonical class of the base B in the CY 3-fold, ΣLST is the curve class +associated to the little string scale satisfying Ω · ΣLST = 0, ΣFa is the curve class +supporting the 7-brane with a-th flavor symmetry, h∨ +α is the dual Coxeter number +of the group Gα, and the dot · between two curve classes stands for the intersection +number of the curves. +The non-vanishing mixed anomalies are inconsistent with the dynamical gauge +symmetries in the presence of background gauge fields for the global symmetries. +Therefore, there must be a regularization scheme that cancels these mixed gauge +anomalies while preserving the dynamical gauge symmetries, even in the presence of +non-trivial background fields for the global symmetries. +There are some choices of regularization scheme. For instance, in [74], the mixed +gauge anomalies were canceled by adding Green-Schwarz counterterms involving a 2- +form background gauge field coupled to the 2-form instanton currents J ∼ ⋆TrFG∧FG. +Here, the 2-form background gauge field transforms under the background global +symmetry transformation and also under the local Lorentz transformation. This +results in a continuous 2-group global symmetry. +In this paper we will introduce another counterterm which leads to the consis- +tent blowup equations for LSTs as we will explain below. We shall introduce the +counterterm defined as +∆S = − +� +B0 ∧ X4,0 , +(2.31) +with a 2-form gauge field B0 which transforms under the dynamical gauge transfor- +mation parametrized by ΛG as +B0 → B0 + 1 +4ℓα TrΛGαFGα . +(2.32) +This modifies the Bianchi identity for the 3-form field strength H0 = dB0 as +dH0 = Y4 . +(2.33) +Then the gauge variation of the counterterm cancels the gauge anomalies arising +from Imixed +8 +in the presence of the background fields for the global symmetries. Let +us emphasize that the 2-form field B0 here is not a fixed background field since it +– 11 – + +transforms non-trivially under the dynamical gauge transformation. We need to +integrate this field in the path integral although it has no kinetic term in the action. +This 2-form field can be considered as a kind of Lagrange multiplier introduced to +cancel the mixed gauge-global anomalies. +We are now ready to formulate the blowup equations for the LSTs. We first +need to prepare the effective prepotential on the Ω-background. The tree-level action +for a LST is almost the same as that of 6d SCFTs, but now there are additional +contributions from the gauge kinetic terms coupled to the little string tension w and +the counterterm (2.31). We propose that the tree-level effective prepotential for a +LST is +ELST +tree = ESCFT +tree ++ E(0) +tree , +(2.34) +E(0) +tree = +1 +ϵ1ϵ2 +�w +2 ℓαKα,ijφα,iφα,j − φ0,0 +�a0 +4 (ϵ2 +1 + ϵ2 +2) + ba,0 +2 Ka,ijma,ima,j + c0ϵ2 ++ +�� +. +Here, we introduced an auxiliary scalar VEV φ0,0 to take into account the magnetic +flux of the 2-form B0 in (2.31) on the blowup background, which will be explained in +more detail below. The first term with w in E(0) +tree is the gauge kinetic terms evaluated +on the Ω-background and the second term with φ0,0 is the contribution from the +counterterm (2.31). The 1-loop contributions to the effective prepotential E1−loop and +to the GV-invariant Zpert can be calculated in the same way as those for 6d SCFTs +presented in (2.12) and in (2.21) respectively in the preivous subsection. +Now we claim that the partition function Z = e−2πiE × ZGV of a little string +theory satisfies the blowup equation +Λ(m; ϵ1, ϵ2) ˆZ(φ, m; ϵ1, ϵ2) = +� +⃗n +(−1)|⃗n| ˆZ(N)(⃗n, ⃗B) ˆZ(S)(⃗n, ⃗B) +⇔ Λ(m, ϵ1, ϵ2) ˆZGV(φ, m, ϵ1, ϵ2) = +� +⃗n +(−1)|⃗n|e−2πiV ˆZ(N) +GV (⃗n, ⃗B) ˆZ(S) +GV(⃗n, ⃗B) , +(2.35) +with a set of consistent magnetic fluxes ⃗n, ⃗B satisfying the quantization in (2.17). ˆZ +is again the partition function with the ϵ1 shift given in (2.14), and ˆZ(N) and ˆZ(S) +are the local partition functions near the north pole and south pole of the P1 defined +by (2.16). +There are a few remarks for the blowup equations for LSTs. Firstly, the magnetic +fluxes ⃗n on the blownup P1 in the blowup equation involve not only the magnetic +fluxes for the dynamical tensor and gauge symmetry groups, but also the magnetic +flux for the 2-form gauge field B0 that is added to cancel the mixed gauge-global +anomalies. As explained above, the 2-form field B0 behaves like a Lagrange multiplier +and we should sum over its magnetic fluxes on the blowup background. Otherwise, +it will not be possible to activate background fields for the symmetries that have +mixed anomalies with gauge symmetries. In the blowup equation, turning on a +– 12 – + +flux n0,0 for B0 is implemented by a shift of the auxiliary scalar field in the form +φ0,0 → φ0,0 +n0,0ϵ1,2 with n0,0 ∈ Z. The summation of these auxiliary magnetic fluxes +is crucial to construct a consistent blowup equation for LSTs that have mixed gauge +anomalies. We note that the auxiliary field φ0,0 only serves the purpose of activating +the fluxes n0,0ϵ1,2 and ultimately disappears in the blowup equation through the use +of the combination V = E(N) + E(S) − E. +Secondly, the sum over the auxiliary magnetic flux n0,0 on P1 in the blowup +equation is not convergent. It turns out that the n0,0 dependent terms appear only in +the exponent V in the blowup equation and they are all linear in n0,0. Namely, the +right side of the blowup equation contains a sum over n0,0 of the form +� +n0,0∈Z +e−n0,0f(m;ϵ1,2)+··· × · · · , +(2.36) +with a function f(m; ϵ1,2) independent of the dynamical Kähler parameters φ. This +sum is obviously divergent, so it seems that the blowup equation is not well-defined. +Nevertheless, we assert that the blowup equation of a LST is still valid in the +following sense. As we will demonstrate explicitly with examples in the next section, +the LST partition functions satisfy the blowup equations if we first expand them in +terms of Kähler parameters and then sum over the auxiliary magnetic fluxes n0,0. +Surprisingly, if one sums up the fluxes |n0,0| ≤ nmax, one finds that every order in +the Kähler parameter expansion is exactly canceled, leaving a few terms coming +from the maximum flux |n0,0| = nmax. These remaining terms are also canceled +iteratively by new terms appearing when the maximum flux is increased such as +nmax → nmax + 1 → nmax + 2 → nmax + 3, and so on. Hence, if sufficiently large +enough fluxes are summed up, all terms arising from smaller n0,0 fluxes are canceled +out. This is how the blowup equation works for LSTs and is rather different from the +structure of the typical blowup equations for 4d/5d/6d SCFTs. +In particular, without the sum over the auxiliary flux n0,0, the above blowup +equation does not hold at all. This is related to the fact that the LSTs possess +mixed gauge anomalies in the presence of background fields such as ⃗B and ϵ1,2 for the +global and Lorentz symmetries which we need to activate to formulate a consistent +blowup equation and that we need to introduce the 2-form B0 and the counterterm +(2.31) associated to the auxiliary flux to cancel such mixed gauge anomalies. We +have checked this for a number of examples that we will discuss in detail in the next +section. Therefore, we propose that the auxiliary magnetic flux n0,0 must be taken +into account in the construction of the blowup equations for LSTs. The counterterm +(2.31) with the auxiliary 2-form field B0 is required in this sense. +Importantly, we can use the blowup equations, combining them with the modular +ansatz, to determine the elliptic genera of 2d worldsheet SCFTs on strings in LSTs. +To show this, let us now illustrate how to bootstrap the BPS spectra of LSTs using +the blowup equations and the modular properties of the elliptic genera. +– 13 – + +2.3 +Bootstrapping LSTs +We first review how to formulate a general ansatz for the elliptic genus of BPS strings +in 6d theories by exploiting its properties under the modular transformation. The +modular property of the elliptic genus defined in (2.1) is governed by the ’t Hooft +anomalies of the worldsheet SCFT. Under the modular transformation, the elliptic +genus transforms as [75], +Z⃗k +�aτ + b +cτ + d, +z +cτ + d +� += ϵ(a, b, c, d)cR−cL exp +� 2πic +cτ + df(z) +� +Z⃗k(τ, z) , +(2.37) +where ( a b +c d ) ∈ SL(2, Z), ϵ(a, b, c, d) is a phase factor, cL,R are chiral central charges of +the worldsheet SCFT, and z collectively denotes chemical potentials for the symmetries. +The modular anomaly f(z) is closely related with the anomaly polynomial I4 of the +2d SCFT [76, 77]. In fact, it agrees with the supersymmetric Casimir energy of the +2d SCFT defined in [78] which is given by an equivariant integral of the anomaly +polynomial I4, +f(z) = +� +eq +I4 . +(2.38) +The equivariant integration here can be implemented by the replacement rules for +the characteristic classes as +p1(T2) �→ 0 , +c2(l) �→ ϵ2 +− , +c2(r), c2(R) �→ ϵ2 ++ , +1 +2 Tr F 2 +a �→ Ka,ijφa,iφa,j . +(2.39) +Knowing the anomaly polynomial of the worldsheet SCFT and the modular transfor- +mation in (2.37), we can formulate an ansatz for the elliptic genus in terms of elliptic +functions. +The anomaly polynomial of the 2d SCFTs living on self-dual strings in 6d SCFTs +has been calculated in [79, 80] by using anomaly inflow mechanism. For a 6d SCFT +with an intersection form Ωαβ +cft, the anomaly polynomial of the worldsheet CFT on a +self-dual string with charge ⃗k = {kα} is +I4 = Ωαβ +cftkα +� +X4β + 1 +2kβχ4(T4) +� +, +(2.40) +where X4β is a 4-form defined in (2.6), χ4(T4) is the Euler class of the transverse +SO(4) = SU(2)l × SU(2)r Lorentz rotation which can be written as χ4(T4) = +c2(l) − c2(r) in terms of the second Chern classes for the SU(2)l × SU(2)r bundle. +The first Pontryagin class p1(T6) of the 6d tangent bundle in X4α is decomposed as +p1(T6) = p1(T2) − 2c2(l) − 2c2(r). +Similarly, the anomaly polynomials of 2d SCFTs on BPS strings in a number +of LSTs were calculated in [65]. We will generalize this computation and provide a +universal expression for the anomaly polynomials of the 2d SCFTs on strings in LSTs. +– 14 – + +The ’t Hooft anomalies on the 2d worldsheet of the self-dual strings in the 6d +SCFTs embedded in a LST should be the same as (2.40). However, there is another +contribution to the ’t Hooft anomalies coming from the full winding strings in the +LST. This extra contribution can be captured by integrating the mixed gauge anomaly +8-form Imixed +8 +on the full winding string background [74]. Let us define the number +of full winding strings κ ∈ Z as a maximal integer satisfying kα − κℓα ≥ 0 for all α. +We propose that the anomaly polynomial of the 2d SCFT on strings in a little string +theory is +I4 = Ωαβkα +� +X4β + 1 +2kβχ4(T4) +� ++ κX4,0 . +(2.41) +Here, Ωαβ is the Dirac pairing of the N-dimensional string charge lattice and X4,0 +is defined in (2.29). We can use this anomaly polynomial to compute the modular +anomaly f(z) in (2.37) for the worldsheet SCFTs for strings with a charge ⃗k. +A function which transforms as (2.37) under the modular transformation is known +as Jacobi form3. In the language of Jacobi forms, (2.37) implies that elliptic genus +has weight 0 and its indices are fixed by ’t Hooft anomaly coefficients of the global +symmetries on the worldsheet theory. To write down an ansatz for the elliptic genus +of ⃗k strings using the Jacobi forms, we need to use the modular property in (2.37) +and the pole structure of Z⃗k. +We propose a modular ansatz for the elliptic genus for the strings in LSTs, which +will be of the form +Z⃗k = +1 +η(τ)2|cL−cR| +Φ⃗k(τ, ϵ±, φ, m) +� +α Dcm +α (τ, ϵ±)Dgα +α (τ, ϵ±, φ) +� +1 +Dbulk +κ +(τ, ϵ±, m0) +� +, +(2.42) +where φ and m collectively denote chemical potentials for gauge and flavor symmetries, +respectively. This is an extension of the ansatzes introduced in [65, 76, 77, 81, 82]. +Here the denominator factors in the ansatz will be fixed by pole structure expected +for the moduli space of strings, which we will explain now. +Firstly, the factor η(τ)2|cL−cR|, where η(τ) is the Dedekind eta function defined +in (A.5), fixes the leading behavior of the elliptic genus in q-expansion which is +determined by the vacuum Casimir energy of the 2d SCFT. +The second factor of the form +Dcm +α (τ, ϵ±) = +kα +� +s=1 +ϕ−1,1/2(τ, sϵ1)ϕ−1,1/2(τ, sϵ2) , +(2.43) +is the contribution coming from the transverse motions of the strings [83, 84], where +ϕ−1,1/2 is the weight −1 and index 1/2 Jacobi form +ϕ−1,1/2(τ, z) = iθ1(τ, z) +η(τ)3 +, +(2.44) +3We summarize the definition, terminologies, and properties of Jacobi forms in Appendix A +– 15 – + +with the Jacobi theta function θ1(τ, z) given in (A.11). Notice that the leading +order of ϕ−1,1/2(τ, z) in q-expansion is O(1), so that this does not change the leading +behavior of the elliptic genus in q-expansion. +The third factor, Dgα +α , arises from the bosonic zero modes of instantons for the +6d gauge algebra gα. For instance, when the gauge algebra is gα = su(2), we have +DA1 +α = +kα +� +s=1 +s−1 +� +l=0 +ϕ−1,1/2((s + 1)ϵ+ + (s − 1 − 2l)ϵ− ± e · φ) , +(2.45) +where e is the positive root of su(2) and ϕ(x ± y) ≡ ϕ(τ, x + y)ϕ(τ, x − y). For a +general gauge algebra gα, we use an embedding su(2) ⊂ gα which maps three SU(2) +generators to generators T a=1,2,3 +e +associated with a positive root e of gα [85, 86]. This +embedding gives the denominator factor Dgα +α as [65] +Dgα +α = +� +e∈R+ +gα +DA1 +⌊kα/ξe⌋,e , +(2.46) +where R+ +gα is the set of positive roots of gα, DA1 +k,e is (2.45) by replacing an su(2) +positive root with a given root of gα, ⌊·⌋ is the floor function and +tr +� +T a +e T b +e +� += ξeδab , +ξe = +� +� +� +1 +if e is a long root or g = An, Dn, En +2 +if e is a short root and g = Bn, Cn, F4 +3 +if e is a short root and g = G2 +. +(2.47) +Lastly, some LSTs have a denominator factor Dbulk +κ +, which depends on the winding +number κ, as mentioned in [65]. This factor is associated to certain full winding string +states that are decoupled from the 6d LST, which means that these states do not +possess dynamical gauge charges, and presumably escape to the bulk spacetime in +which the LST is embedded. In the examples we present in section 3.2, for example, +the LSTs can be embedded in 10d heterotic string theories and the modular ansatz +for strings in these theories includes a denominator factor of the form +Dbulk +κ += +κ +� +s=1 +ϕ−1,1/2(±sλ(m0 − ϵ+)) , +(2.48) +where m0 is the chemical potential for the SU(2)m ⊂ SU(2)R × SU(2)m rotational +symmetry transverse to the 6d spacetime of the LSTs. These are chosen to match the +ADHM constructions for the strings. Specifically, the value of λ is set to 1 for SO(32) +heterotic LST in section 3.2.2 and to 2 for E8 × E8 heterotic LST in section 3.2.1. +However, we find that it is possible to select alternative denominator factors that do +not alter the dynamical string spectrum, while leading to different full winding string +states that decouple from the LST. For example, as we will show in section 3.2.1, +the same string spectrum for the E8 × E8 heterotic LST can be obtained by using a +– 16 – + +different denominator factor with λ = 1. We postulate that, for a generic LST, it is +always possible to choose the factor Dbulk +κ +to be either trivial or in the form specified +in (2.48) with a certain λ ∈ Z, provided that a modular ansatz of the form (2.42) +can be established. This modular ansatz will be consistent with the dynamical string +spectrum of the LST, though the decoupled states that are independent of dynamical +gauge symmetries may vary. +After factoring out the denominator factors, the numerator Φ⃗k in the modular +ansatz (2.42) starts at q0 order in q-expansion and becomes a Weyl-invariant Jacobi +form whose weight and indices are fixed by the ’t Hooft anomalies of a given theory +and the structure of the denominator in the modular ansatz. For every simple Lie +algebra, the Weyl-invariant Jacobi forms with given weight and index can be written +as a linear combination of finite generators [87, 88]. Thus, the numerator is given +by the finite linear combination of the generators ϕkjl,mjl(zl) of the Weyl invariant +Jacobi forms with weight kjl and index mjl for l’th symmetry algebra gl: +Φ⃗k = +� +i +CiE +a(i) +4 +4 +E +a(i) +6 +6 +� +j,l +ϕkjlmjl(zl)b(i) +jl , +(2.49) +where zl denotes a chemical potential for l-th symmetry, Ci ∈ C, and E4 and E6 +denote the Eisenstein series of weight 4 and 6, respectively. The exponents a(i) +4,6 and b(i) +jl +are constrained by the condition requiring that the elliptic genus Z⃗k be transformed +as (2.37) under the modular transformation. They are thus non-negative integers +satisfying two conditions: +4a(i) +4 +6a(i) +6 + +� +j,l +kjlb(i) +jl −|cL−cR|+ +� +α +2kα+ +� +α +� +e∈R+ +gα +⌊kα/ξe⌋ +� +s=1 +2s−(weight(Dbulk +κ +)) = 0 , +(2.50) +and +f(z) = +� +j,l +mjlb(i) +jl ⟨zl, zl⟩gl − 1 +2 +� +α +kα +� +s=1 +s2(ϵ2 +1 + ϵ2 +2) +(2.51) +− 1 +2 +� +α +� +e∈R+ +gα +⌊kα/ξe⌋ +� +s=1 +((s + 1)ϵ+ + (s − 1 − 2l)ϵ− ± e · φ)2 − (index(Dbulk +κ +)) +for each i, where ⟨·, ·⟩gl is a symmetric bilinear form for gl defined by its Killing form. +The index l runs for all symmetries of the 2d worldsheet SCFT, while gα denotes +gauge symmetry for α-th node. Hence the modularity fixes the elliptic genus up to +finitely many constants Ci in (2.49). +The unknown constants Ci’s can be fixed by imposing the GV-invariant ansatz +(2.3) of the elliptic genus and by solving the blowup equation. Note that the modular +ansatz (2.42) for � +α kα > 1 can have higher order poles at ϵ1 = 0 and ϵ2 = 0 arising +– 17 – + +from the center of mass contribution (2.43) for generic Ci’s. However, the single letter +index in the Plethystic exponential of the GV-invariant ansatz can have only simple +poles at ϵ1 = 0 and ϵ2 = 0. This imposes strong constraints on Ci’s. +Furthermore, we demand that the partition function satisfies the blowup equation. +In contrast to the 5d/6d SCFT cases, the blowup equations for LSTs involve a +divergent summation over an auxiliary magnetic flux n0,0, as explained in the previous +subsection. Due to this structure, it seems that the partition function of an LST +cannot be determined solely by solving the blowup equations and the GV-invariant +ansatz (2.3). However, the modular ansatz in (2.42) places further constraints on +the partition function and, by combining it with the blowup equations, it should be +feasible to completely determine the partition functions of LSTs in Kähler parameter +expansion. We will demonstrate this with several interesting examples in the next +section. +3 +Examples +In this section, the partition functions of several low rank LSTs are calculated. We +first compute elliptic genera of strings using the ADHM constructions. We then +construct the blowup equations for the partition functions of the LSTs and verify +that the results from the ADHM constructions satisfy the blowup equations. Lastly, +we formulate the modular ansatz for the elliptic genera of strings in the LSTs and fix +the unknown coefficients in the ansatz by solving the blowup equations. We show +that the partition functions obtained through this method are consistent with the +results obtained using ADHM constructions. +3.1 +ˆA1 LSTs +Our first example is the little string theories on N parallel NS5-branes in type II +string theories in gravity decoupling limit introduced in [56, 57]. In the IIA theory, +the little string theory is the N = (2, 0) LST with N tensor multiplets. This LST +is realized in F-theory by an elliptically fibered Calabi-Yau threefold whose base +surface contains a loop of N rational curves of self-intersection number −2 [22]. The +intersection matrix Ωαβ of the −2 curves is given by the minus of the Cartan matrix +of the affine Lie algebra ˆAN−1 = A(1) +N−1. On the other hand, the LST in the IIB +theory is the N = (1, 1) Yang-Mills theory with U(N) gauge group which is realized +in F-theory by an elliptic CY 3-fold with a base containing a genus one curve of +self-intersection number 0. These two LSTs in IIA and in IIB, which we call ˆAN−1 +LSTs, are related via T-duality under a circle compactification. In this subsection, we +consider the partition functions and the blowup equations of these LSTs for N = 2. +– 18 – + +3.1.1 +IIA picture +Let us first consider the N = (2, 0) ˆA1 little string theory for 2 NS5-branes in type +IIA string theory. This theory has two tensor multiplets (for one dynamical tensor +field and one free tensor field) with the intersection form +Ωαβ = +�−2 2 +2 −2 +� +. +(3.1) +The index part of the partition function is factorized as +ZIIA +GV = ZIIA +pert · ZIIA +str , +(3.2) +where the perturbative partition function is given by the contributions coming from +two N = (2, 0) tensor multiplets +ZIIA +pert = PE +�� +√p1p2 +(1 − p1)(1 − p2) +� +M + M −1� +− +p1 + p2 +(1 − p1)(1 − p2) +� 2q +1 − q +� +. +(3.3) +Here M = e2πim is the fugacity for the SU(2)m ⊂ SU(2)R × SU(2)m rotational +symmetry of the R4 plane transverse to the NS5-branes. +The partition function ZIIA +str is from the strings carrying tensor charges defined as +ZIIA +str = +� +k1,k2≥0 +Qk1 +�e2πiw +Q +�k2 +ZIIA +(k1,k2), +(3.4) +where Q ≡ e2πi(2φ1,0−2φ2,0) is the fugacity for the dynamical tensor charge and w is +the chemical potential for the winding number. We will now study two distinct +methods for calculating the elliptic genus ZIIA +(k1,k2): the ADHM construction based on +2d gauged linear sigma model (GLSM) on the strings, and the blowup approach with +the modular ansatz. +GLSM +We start with the brane construction studied in [63]. The brane construction +for the ˆA1 LST is depicted in Figure 1(a) and (b). Here, we compactify the 9-th +direction on a circle, and put two NS5-branes extended along 012345 directions at +x9 = φ1,0 and x9 = φ2,0, respectively. The strings in the LST arise from the k1 +D2-branes and k2 D2-branes stretched between two NS5-branes. We also put a single +D6-brane, which becomes trivial in the M-theory uplift, to explicitly provide U(1)m +symmetry in the 2d GLSM. See [63] for a detailed study of this brane configuration. +The 2d GLSM on D2-branes has U(k1)×U(k2) gauge symmetry, SU(2)l ×SU(2)r +symmetry which rotates 2345 directions, SO(3) symmetry for 678 directions, and +U(1)m symmetry. At low energy, we expect the SO(3) and U(1)m symmetries to be +enhanced to SU(2)R × SU(2)m. There are an N = (0, 4) vector multiplet (A(i) +µ , λ ˙αA ++(i)) +and adjoint hypermultiplets (a(i) +α ˙β, λαA +−(i)) for each gauge node, bifundamental twisted +– 19 – + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9(S1) +NS5 +• +• +• +• +• +• +D2 +• +• +• +D6 +• +• +• +• +• +• +• +(a) +NS5 +NS5 +k1 D2’s +k2 D2’s +x9 +∥ +∥ +(b) +Field +Type +U(k1) × U(k2) +U(1)m +(A(i) +µ , λ ˙αA ++(i)) +vector +(adj, 1), (1, adj) +(a(i) +α ˙β, λαA +−(i)) +hyper +(adj, 1), (1, adj) +(ϕ(i) +A , χ ˙α +−(i)) +twisted hyper +(k1, k2), (k1, k2) ++1 +(χα ++(i)) +Fermi +(k1, k2), (k1, k2) ++1 +(q(i) +˙α , ψA(i) +− +) +hyper +(k1, 1), (1, k2) +(Ψ(i) ++ ) +Fermi +(k1, 1), (1, k2) ++1 +(˜Ψ(i) ++ ) +Fermi +(k1, 1), (1, k2) +−1 +(c) +Figure 1: (a) and (b) are brane configurations for ˆA1 LST in the type IIA string +theory where the x9-direction is compactified on a circle. (c) is the N = (0, 4) matter +contents in the 2d GLSM on the worldsheet of strings. +hypermultiplets (ϕ(i) +A , χ ˙α +−(i)) and Fermi multiplets (χα ++(i)) from D2-D2 string modes, +and hypermultiplets (q(i) +˙α , ψA(i) +− +) and Fermi multiplets (Ψ(i) ++ ), (˜Ψ(i) ++ ) from the D2-D6 +string modes. Here, i = 1, 2 denotes each gauge node, ± represent 2d chirality of +fermions, {α, β, · · · }, { ˙α, ˙β, · · · } and {A, B, · · · } are doublet indices for the SU(2)l, +SU(2)r, and SU(2)R, respectively. We summarize the matter content of the 2d GLSM +in Figure 1(c). +The gauge theory description for the 2d worldsheet theory allows us to express the +elliptic genus by a contour integral of 1-loop determinants from the supermultiplets, +and the contour integral can be evaluated by using the JK-residue prescription as +discussed in [75, 89]. The result is [63] +ZIIA +(k1,k2) = +� +{Y1,Y2},|Yi|=ki +2 +� +i=1 +� +(a,b)∈Yi +θ1(τ, E(a,b) +i,i+1 − m + ϵ−)θ1(τ, E(a,b) +i,i−1 + m + ϵ−) +θ1(τ, E(a,b) +i,i ++ ϵ1)θ1(τ, E(a,b) +i,i +− ϵ2) +, (3.5) +with +E(a,b) +i,j += (Yi,a − b)ϵ1 − (Y T +j,b − a)ϵ2 , +(3.6) +where Y1 and Y2 are Young diagrams, and Yi,a and Y T +i,b are the length of a-th row and +b-th column of Yi, respectively. +– 20 – + +Modularity +The modular properties of the elliptic genus can be obtained from the +anomalies of the 2d worldsheet CFT. The chiral fermions in the GLSM contribute to +the anomaly polynomial as +λ ˙αA ++(i) + λαA +−(i) → +2 +� +i=1 +2k2 +i +�c2(r) + c2(R) +2 +− c2(l) + c2(R) +2 +� +, +(3.7) +χ ˙α +−(i) + χα ++(i) → 4k1k2 +c2(l) − c2(r) +2 +, +(3.8) +ψA(i) +− ++ Ψ(i) ++ + ˜Ψ(i) ++ +→ (k1 + k2) +� +c2(R) + 1 +4 Tr F 2 +m +� +, +(3.9) +where Fm is the field strength for U(1)m symmetry. The same anomaly polynomial +can also be obtained from the anomaly inflow given in (2.41): +I4 = −(k1 − k2)2(c2(l) − c2(r)) + (k1 + k2) +� +−c2(R) + 1 +4 Tr F 2 +m +� +. +(3.10) +Hence, the modular anomaly of the (k1, k2) elliptic genus is +� +eq +I4 = (k1 − k2)2(−ϵ2 +− + ϵ2 ++) + (k1 + k2)(−ϵ2 ++ + m2) , +(3.11) +where we use the replacement rule in (2.39). +We can then establish a modular ansatz for (k1, k2)-string elliptic genus as +ZIIA +(k1,k2) = +Φ(k1,k2)(τ, ϵ±, m) +�k1 +s1=1 ϕ−1,1/2(s1ϵ1,2) · �k2 +s2=1 ϕ−1,1/2(s2ϵ1,2) +. +(3.12) +The numerator Φ(k1,k2) can be written in terms of the Eisenstein series E4, E6 and +the SU(2) Weyl invariant Jacobi forms ϕ−2,1, ϕ0,1 for ϵ± and m as we explained in +(2.49): +Φ(k1,k2) = +� +i +C(k1,k2) +i +E +a(i) +4 +4 +E +a(i) +6 +6 +ϕ−2,1(ϵ+)b(i) +1 ϕ0,1(ϵ+)b(i) +2 ϕ−2,1(ϵ−)b(i) +3 +·ϕ0,1(ϵ−)b(i) +4 ϕ−2,1(m)b(i) +5 ϕ0,1(m)b(i) +6 . +(3.13) +We need to determine the unknown coefficients in the modular ansatz for the +numerator Φ(k1,k2). For this, we first impose the consistency conditions (2.50) and +(2.51) and then use the GV-invariant ansatz (2.3). For instance, let us consider +(k1, k2) = (1, 0) case. By using (2.50) and (2.51), the numerator has weight −2 and +the modular anomaly f(z) = ϵ2 ++ + m2. Then the ansatz reduces to +Φ(1,0) = C(1,0) +1 +ϕ0,1(ϵ+)ϕ−2,1(m) + C(1,0) +2 +ϕ−2,1(ϵ+)ϕ0,1(m) , +(3.14) +where C(1,0) +1 +and C(1,0) +2 +are unknown constants. Now by expanding Z(1,0) in terms of +q = e2πiτ up to q1 order and comparing it with the GV-invariant form (2.3), one can +– 21 – + +find that BPS state degeneracies N d +jl,jr appearing in the (1, 0)-string elliptic genus +can be non-negative integers only if +C(1,0) +1 += −C(1,0) +2 +∈ Z/12 , +C(1,0) +1 +≥ 0 . +(3.15) +Similarly, Φ(1,1) has weight −4 and the modular anomaly f(z) = 2ϵ2 +− + 2m2, and thus +it can be written with 4 unknown constants as +Φ(1,1) = C(1,1) +1 +ϕ0,1(ϵ−)2ϕ−2,1(m)2 + C(1,1) +2 +ϕ−2,1(ϵ−)ϕ0,1(ϵ−)ϕ−2,1(m)ϕ0,1(m) ++ C(1,1) +3 +ϕ−2,1(ϵ−)2ϕ0,1(m) + C(1,1) +4 +E4ϕ−2,1(ϵ−)2ϕ−2,1(m)2 . +(3.16) +In order to have only simple poles at ϵ1 = 0 and ϵ2 = 0 in (1, 1)-order after taking +Plethystic logarithm as (2.3), the coefficient C(1,1) +i +’s should satisfy +C(1,1) +1 += +� +C(1,0) +1 +�2 +, C(1,1) +2 += 2C(1,0) +1 +C(1,0) +2 +, C(1,1) +3 += +� +C(1,0) +2 +�2 +, C(1,1) +4 += 0 . +(3.17) +Therefore, all the coefficients are fixed by one coefficient C(1,0) +1 +. +We can perform a similar computation for (k0, k2) = (2, 0), which has 44 unknown +constants in the modular ansatz. Requiring the partition function has correct GV- +invariant form (2.3) at this order, we can express all C(2,0) +i +in terms of C(1,0) +1 +. Moreover, +we find only two solutions at this order: one is C(1,0) +1 += 0 which leads to the trivial +solution ZIIA +(1,0) = ZIIA +(1,1) = ZIIA +(2,0) = 0, and another one is C(1,0) +1 += 1/12. The latter +non-trivial solution reproduces the result (3.5) from the ADHM computation at +(k1, k2) = (1, 0), (1, 1), (2, 0). +Furthermore, we also check that 110 unknown constants in the (k1, k2) = (2, 1) +modular ansatz can be completely fixed by requiring the GV-invariant ansatz (2.3). +We report the results in Table 1. In the table, we write an ordered list of C(k1,k2) +i +, where +C(k1,k2) +i +appears earlier than C(k1,k2) +j +in the list if (a(i) +4 , a(i) +6 , b(i) +1 , · · · , b(i) +6 ) in the ansatz +(3.13) appears before (a(j) +4 , a(j) +6 , b(j) +1 , · · · , b(j) +6 ) in an ascending order4. For instance, in +the case of Φ(1,1), we have +(a(i) +4 , a(i) +6 , b(i) +1 , · · · , b(i) +6 ) = +� +� +� +� +� +� +� +� +� +(0, 0, 0, 0, 0, 2, 2, 0) +(i = 1) +(0, 0, 0, 0, 1, 1, 1, 1) +(i = 2) +(0, 0, 0, 0, 2, 0, 0, 1) +(i = 3) +(1, 0, 0, 0, 2, 0, 2, 0) +(i = 4) +(3.18) +4We define the ascending order as follows. Suppose the modular ansatz is given as (2.49), where +we label weights and indices of the Jacobi forms as j1 < j2 if kj1l < kj2,l or kj1l = kj2l, mj1l < mj2l. +We also define a set of the exponents in the ansatz as +L(i) := +� +a(i) +1 , a(i) +2 , b(i) +j1,l1, · · · , b(i) +jN1,l1, · · · , b(i) +j1,ln, · · · , b(i) +jNn,ln +� +. +Then, if we have (L(i))1 = (L(j))1, ..., (L(i))s−1 = (L(j))s−1, and (L(i))s < (L(j))s, we order L(i) and +L(j) as {L(i), L(j)}. In this way, we fix the ordering of L(i), and we define their set {L(1), ..., L(N)} +that we call ascending order. The ordering of Ci follows the ordering of L(i). +– 22 – + +(k1, k2) +� +C(k1,k2) +i +� +(1, 0) +1 +22·3{1, −1} +(1, 1) +1 +24·32{1, −2, 1, 0} +(2, 0) +1 +215·38 {1, −1, −1, 1, 0, 0, 40, −40, −40, 40, 0, 0, −32, 32, 32, −32, 0, −15, 15, 15, −15, 0, 0, 24, −24, +−24, 24, 0, 0, 0, −45, 45, 45, −45, 0, 0, 0, 0, 0, 27, −27, −27, 27, 0} +(2, 1) +1 +217·39 {1, −5, 4, 0, 2, 2, −4, −3, 3, 0, −16, 8, 8, 0, −32, 40, −8, 48, −48, 48, −48, 8, −40, 32, 0, −8, −8, +16, 0, 96, −96, −128, 64, 64, 0, 128, −160, 32, 0, 6, 6, −12, 0, −12, 24, −24, 12, −9, 9, −18, 18, 6, 6, −12, +3, −3, 0, −24, 24, 0, 0, 0, 48, −48, 24, −24, 0, −48, 48, 0, 0, 0, 0, 9, −9, 36, −18, −18, −54, 54, −27, 27, +−36, 72, −72, 36, 0, 36, −18, −18, 0, 0, 0, 0, 0, 0, 0, −81, 81, 108, −54, −54, 0, −108, 135, −27, 0, 0, 0, 0} +Table 1: Coefficients C(k1,k2) +i +in the modular ansatz of N = (2, 0) ˆA1 LST. +When we look at a(i) +4 , a(4) +4 +is the biggest value, so C(1,1) +4 +is the last element. Similarly, +we find b(i) +3 , b(1) +3 +< b(2) +3 +< b(3) +3 += b(4) +3 , so C(1,1) +1 +is the first element, and C(1,1) +2 +is the +second element. Therefore, the ascending order of {Ci} in this case is +{C(1,1) +1 +, C(1,1) +2 +, C(1,1) +3 +, C(1,1) +4 +}. +(3.19) +Blowup equation +Finally, we consider the blowup equation for the (2,0) ˆA1 LST. +As explained in the section 2.2, the tree level contribution to the effective prepotential +consists of two parts. The first one is from the Green-Schwarz term for the dynamical +tensor multiplet and the second one is the contribution from the auxiliary 2-form +field B0 to cancel the mixed gauge-global anomalies. We can write the effective +prepotential as +E = +1 +ϵ1ϵ2 +� +τ(φ1,0 − φ2,0)2 + (φ1,0 − φ2,0)(−m2 + ϵ2 ++) +� ++ E(0) +tree , +(3.20) +where φ1,0 − φ2,0 is the scalar vacuum expectation value (VEV) of the dynamical +tensor multiplet. The second contribution E(0) +tree from the auxiliary 2-form field B0 is +given by +E(0) +tree = +1 +ϵ1ϵ2 +� +−2m2 + 2ϵ2 ++ +� +φ0,0 , +(3.21) +where φ0,0 is the auxiliary scalar associated with the B0 field. +To formulate the blowup equation, we have to sum over magnetic fluxes for both +the dynamical tensor field and the auxiliary 2-form field which can be realized by +shifting the parameters as +φ1,0 − φ2,0 → φ1,0 − φ2,0 + n1,0ϵ1,2 , +φ0,0 → φ0,0 + n0,0ϵ1,2 , +n1,0, n0,0 ∈ Z . (3.22) +– 23 – + +We do not turn on the background magnetic fluxes for τ and w: Bτ = Bw = 0. We +propose the blowup equation for this LST as +Λ ˆZIIA +str = +� +n1,n2∈Z +(−1)n1+n2q(n1−n2)2(M√p1p2)n1+n2 ˆZIIA(N) +str +ˆZIIA(S) +str +, +(3.23) +where n1 ≡ n0,0 + n1,0 and n2 ≡ n0,0. We absorbed the perturbative part of the +partition function into Λ as it is independent of the parameters φ0,0, φ1,0 − φ2,0. +To begin with, we will demonstrate how the known elliptic genera, as given in +(3.5), can be a solution to the blowup equation, although this equation becomes +singular along the summation direction n1 = n2, as mentioned in section 2.2. At +(k1, k2) = (1, 0) order, the blowup equation (3.23) is given by +� +n1,n2∈Z +F(n1, n2) := +� +n1,n2∈Z +(−1)n1+n2q(n1−n2)2(M√p1p2)n1+n2 +(3.24) +· +� +p2(n1−n2) +1 +ˆZIIA(N) +(1,0) ++ p2(n1−n2) +2 +ˆZIIA(S) +(1,0) +− ˆZIIA +(1,0) +� += 0 , +where we choose Λ = � +k Λke2πikw with +Λ0 = +� +n1,n2∈Z +(−1)n1+n2q(n1−n2)2(M√p1p2)n1+n2 . +(3.25) +Suppose we consider only magnetic fluxes (n1, n2) = (0, 0). Then (3.24) becomes +� +(n1,n2)=(0,0) +F(n1, n2) = +� 1 +M 2 + (1 + p1)(1 + p2) +M√p1p2 ++ +� +2 + p1 + p−1 +1 ++ p2 + p−1 +2 +� ++ M(1 + p1)(1 + p2) +√p1p2 ++ M 2 +� +q + O(q2), +(3.26) +in the double expansion of q and M. Now we add the contributions coming from the +magnetic fluxes |n1,2| ≤ 1. We then find that M 0 and M ±1 terms at q1 order are all +canceled and the remaining terms are +� +|n1,2|≤1 +F(n1, n2) = +� +1 +M 4p1p2 ++ (1 + p1)(1 + p2) +M 3(p1p2)3/2 ++ 1 + p1 + p2 +M 2p1p2 ++ M 2(p1 + p2 + p1p2) ++ M 3(1 + p1)(1 + p2)√p1p2 + M 4p1p2 +� +q + O(q2) . +(3.27) +Again, if we consider the summation of the magnetic fluxes up to |n1|, |n2| ≤ 2, M ±2 +and M ±3 terms are canceled and only higher order terms with M ±4,±5,±6 remain at +q1 order. In this way, if we sum over sufficiently large magnetic fluxes, every order of +the Kähler parameter expansion is satisfied. Using the elliptic genera (3.5) and +Λ = eπiw/12 +η(w) Λ0 , +(3.28) +– 24 – + +we checked that such cancellation occurs up to k1, k2 ≤ 2 string numbers and q3 order. +Now, we will solve the blowup equation and determine the unknown coefficients +in the modular ansatz. For Z(k,0) and Z(0,k), we can use the elliptic genera for k +M-strings in [90] and they satisfy the blowup equations for the M-string theory as +discussed in [50, 53]. The (k1, k2) = (1, 1) order in the blowup equation is independent +of dynamical Kähler parameter and thus we cannot fix four unknown coefficients +in the modular ansatz at this order. What we can determine is Λ at this order +expressed as τ, m, ϵ1,2, and the unknown constants in the ansatz. Next, we solve the +(k1, k2) = (2, 1) order in the blowup equation which now contains dynamical Kähler +parameter φ1,0 − φ2,0. We need to fix 4 + 110 undetermined coefficients arising from +(k1, k2) = (1, 1), (2, 1) elliptic genera. For this we substitute the modular ansatz and +Λ1 into the blowup equation at (k1, k2) = (2, 1) order, and solve it order by order in +the q and M double expansion as previously described. This allows us to determine +all 4 + 110 unknown coefficients as well as the Λ1 factor. The result is in perfect +agreement with Table 1. We expect that higher order elliptic genera can be similarly +calculated. +3.1.2 +IIB picture +The LST theory on two NS5-branes in type IIB string theory is the N = (1, 1) U(2) +Yang-Mills theory. The partition function of this LST is factorized as +ZIIB +GV = ZIIB +pert · ZIIB +str , +(3.29) +where the perturbative contribution coming from the U(2) vector multiplet and an +adjoint hypermultiplet is given by +ZIIB +pert = PE +� +− +1 + p1p2 +(1 − p1)(1 − p2) +� +Q2 + 2q + qQ−2� +1 +1 − q +� +· PE +� +√p1p2 +(1 − p1)(1 − p2) +� +Q2 + 2q + qQ−2�� +M + M −1� +1 +1 − q +� +, +(3.30) +where Q = e2πiφ1 is the SU(2) gauge fugacity and M = e2πim is the fugacity for +the SU(2)m symmetry of the adjoint hypermultiplet. The partition function of the +instanton strings is given by +ZIIB +str = +∞ +� +k=0 +e2πikwZIIB +k +, +(3.31) +where the little string tension w is identified with the square of the inverse gauge +coupling 1/g2 +YM in the low energy Yang-Mills theory. +GLSM +Upon applying S-duality, the system of NS5-branes and F1-strings in the +type IIB string theory is transformed into a system of the D1- and D5-branes. +– 25 – + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +2 D5 +• +• +• +• +• +• +k D1 +• +• +(a) +U(k) +adj +adj +adj +U(2) +(b) +Field +Type +U(k) +U(2) +(Aµ, λ ˙αA ++ ) +vector +adj +(aα ˙β, λαA +− ) +hyper +adj +(ϕAa, λ ˙αa +− ) +twisted hyper +adj +(λαa ++ ) +Fermi +adj +(q ˙α, ψA +−) +hyper +k +2 +(Ψa +−) +Fermi +k +2 +(c) +Figure 2: (a) A brane configuration of the ˆA1 LST in the type IIB string theory +which consists of 2 D5-branes and k D1-branes. (b) The 2d N = (0, 4) gauge theory +description for the k D1-branes, where a circle represents gauge symmetry, a square +means flavor symmetry, and solid, dashed and zigzag lines denote hypermultiplets, +Fermi multiplets and twisted hypermultiplets, repectively. (c) The N = (0, 4) matter +content of the 2d gauge theory. +For k-instanton strings, we consider a configuration where k D1-branes are bound +to 2 D5-branes as illustrated in Figure 2(a). The 2d theory on the D1-branes is +described by a N = (4, 4) U(k) gauge theory with U(2) flavor symmetry. The theory +also has an SO(4) = SU(2)l × SU(2)r symmetry which rotates the 2345 directions, +and an SO(4) = SU(2)R × SU(2)m which rotates the 6789 directions. This brane +configuration is studied in [63, 91], and we summarize the 2d gauge theory description +and its matter content in N = (0, 4) language in Figure 2(b) and (c). In Figure 2(c), +we denote by {a, b · · · } doublet indices for SU(2)m, and other indices have been +already introduced in Section 3.1.1. +Based on the 2d gauge theory description, the elliptic genus of the LST can +be computed by evaluating the JK-residue of the contour integral of the 1-loop +determinants from all the supermultiplets. The result for k-strings is [63] +ZIIB +k += +� +|Y1|+|Y2|=k +2 +� +i,j=1 +� +s∈Yi +θ1(τ, Eij(s) + m − ϵ−)θ1(τ, Eij(s) − m − ϵ−) +θ1(τ, Eij(s) − ϵ1)θ1(τ, Eij(s) + ϵ2) +, +(3.32) +where Y1 and Y2 are Young diagrams, s is a box in the Young diagram, and +Eij(s) = ai − aj − ϵ1hi(s) + ϵ2vj(s), +(3.33) +– 26 – + +with a1 = −a2 = φ1. Here, hi(s) and vj(s) are the arm length and leg length of a box +s in Yi and Yj, respectively. +Since the IIA LST and the IIB LST are related by the T-duality, they should +have the same BPS spectra when placed on a spatial circle. This means that the +partition function ZIIA +GV of the type IIA LST is same as ZIIB +GV for the type IIB LST under +the exchange τ and w up to extra factors which are independent of the dynamical +parameter. More precisely, the following relation has been checked explicitly by +expanding both sides in terms of w, q, Q and M in [63]: +ZIIA +GV +��τ↔w +φ1,0−φ2,0→φ1 = ZIIB +GV · q1/24 +η(τ) . +(3.34) +Modularity +The modular properties of the elliptic genus can be read off from the +anomalies of the 2d chiral matters given in Figure 2(c). The chiral fermions contribute +to the 2d anomaly polynomial as +λ ˙αA ++ + λαA +− +→ 2k2 +�c2(r) + c2(R) +2 +− c2(l) + c2(R) +2 +� +, +(3.35) +λαa ++ + λ ˙αa +− +→ 2k2 +�c2(l) + c2(m) +2 +− c2(r) + c2(m) +2 +� +, +(3.36) +Ψa ++ + ψA +− → 2k(c2(m) − c2(R)) , +(3.37) +where c2(m) is the second Chern class of the SU(2)m symmetry. Summing up these +contributions gives the anomaly polynomial of the 2d gauge theory +I4 = 2k +� +−c2(R) + 1 +4 Tr F 2 +m +� +. +(3.38) +This indeed agrees with the anomaly inflow result in (2.41), which takes into account +the contribution from the counterterm in (2.31). This serves as indirect evidence to +support the use of the counterterm in (2.31) for cancelling the mixed gauge-global +anomaly of the 6d LST. +From the modular anomaly f(z) = +� +I4 = 2k(m2 − ϵ2 ++), we can set a modular +ansatz for the k instanton string as +ZIIB +k += +Φk(τ, ϵ±, 2φ1, m) +�k +s=1 ϕ−1,1/2(sϵ1,2) �s−1 +l=0 ϕ−1,1/2((s + 1)ϵ+ + (s − 1 − 2l)ϵ− ± 2φ1) +. +(3.39) +The numerator Φk can be written in terms of the Eisenstein series E4(τ), E6(τ) and +the SU(2) Weyl invariant Jacobi forms for ϵ±, 2φ1 and m. One can explicitly check +that the elliptic genus (3.32) has the same denominator structure as that in (3.39). +We summarize the coefficients in the modular ansatz in Table 2 obtained by comparing +two expressions (3.32) and (3.39). The ordering of the coefficients is ascending order +with respect to {ϵ+, ϵ−, 2φ1, m} for k = 1 as defined in footnote 4. Here, for k = 2, +we set ϵ+ = 0 for simplicity and the order of coefficients in the modular ansatz is +ascending order with respect to {ϵ−, 2φ1, m}. +– 27 – + +k +� +C(k) +i +� +1 +1 +29·35 {1, −1, −1, 1, 3, 0, −3, 0, 0, 12, 0, −12, 8, 4, −8, −4, 0, −3, 0, 3, −9, −3, 9, 3, +−3, 0, 3, 0, 0, −9, 0, 9} +2 +(ϵ+ = 0) +1 +216·39 {−2, 4, −2, −1, 8, −10, 7, 6, −6, −4, −5, −9, 5, 10, 4, −5, 0, 8, 4, −4, 14, +−24, −14, 16, 16, −28, 20, −24, −12, 28, −10, 0, −8, 20, 8, −10, 16, 80, −80, 0, +−64, 16, 0, 0, −16, 48, 0, 0, −3, −9, 0, −6, 24, −6, −39, 63, −30, 0, 30, −24, 15, +18, −6, −30, −12, 15, 0, −24, −108, 84, −48, 108, 48, 0, 24, 36, −108, 0, 24, −36, +0, 0, 0, 0, −9, 36, 18, 9, 0, −72, −9, 18, −72, 63, 18, −36, 36, 0, −9, 9, 0, −18, +−36, 72, 0, 36, −54, 27, 27, −81, 27, −54, 54, 0, −27, 27, 0, 0, 0, 0, 0} +Table 2: Coefficients C(k) +i +in the modular ansatz of N = (1, 1) ˆA1 LST. +Blowup equation +We can fix the unknown constants in the modular ansatz using +the blowup equation. To begin with, let us evaluate the effective prepotential. The +1-loop prepotential from the SU(2) vector multiplet, an adjoint hypermultiplet, and +their KK towers is given by +ϵ1ϵ2E1−loop = 1 +12 +� +n∈Z +� +|nτ ± 2φ1|3 − |nτ ± 2φ1 + m|3� ++ϵ2 ++φ1 = (−m2+ϵ2 ++)φ1. (3.40) +Here, we use the zeta function regularization for the infinite sum. Then the effective +prepotential is given by +E = +1 +ϵ1ϵ2 +(−m2 + ϵ2 ++)φ1 + E(0) +tree , +E(0) +tree = +1 +ϵ1ϵ2 +� +wφ2 +1 + (−2m2 + 2ϵ2 ++)φ0,0 +� +, +(3.41) +where the first term in E(0) +tree is from the SU(2) gauge kinetic term and φ0,0 is an +auxiliary scalar for the non-dynamical 2-form field. One notices that under the +reparametrization w → τ, φ1 → φ1,0 − φ2,0, the effective prepotential is the same as +the type IIA prepotential in (3.20). +To formulate the blowup equation, we choose magnetic fluxes for φ1, φ0,0, m, τ +and w as +n1 ∈ Z , +n0,0 ∈ Z , +Bm = 1/2 , +Bτ = Bw = 0 . +(3.42) +Since the effective prepotential in (3.41) and the elliptic genus in (3.32) are the same +as those for the type IIA picture up to reparametrization and overall factor, the same +blowup equation should hold for the partition function in the type IIB picture: +Λ ˆZIIB +str = +� +n0,0,n1∈Z +(−1)n1e−2πiV ˆZIIB(N) +pert +ˆZIIB(S) +pert +ˆZIIB +pert +ˆZIIB(N) +str +ˆZIIB(S) +str +. +(3.43) +We checked that this blowup equation holds for up to 2-strings and the third order in +q-expansion. We also checked that inserting the 1-string modular ansatz (3.39) into +the blowup equation and solving it allows us to determine all 32 unknown constants +given in Table 2 within the ansatz. +– 28 – + +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +M9 +• +• +• +• +• +• +• +• +• +• +M5 +• +• +• +• +• +• +M2 +• +• +• +(a) +O8− + 8D8 +O8− + 8D8 +NS5 +k1 D2’s +k2 D2’s +(b) +Figure 3: (a) Branes for the E8 × E8 LST in the M-theory setup. (b) The E8 × E8 +LST realized in type IIA string theory. +3.2 +Heterotic LSTs +The second example is the N = (1, 0) LSTs on N parallel NS5-branes in the E8 × E8 +and SO(32) heterotic string theories which we call rank N heterotic LSTs [56, 57]. +Again, these two LSTs are T-dual to each other under a circle compactification. In +this subsection, we study the elliptic genera and the blowup equations of the rank 1 +heterotic LSTs. +3.2.1 +E8 × E8 picture +The E8 × E8 heterotic LST is the worldvolume theory on a single M5-brane placed +between two M9-branes at each end of the interval S1/Z2. Under the circle reduction, +the M5-brane and the M9-branes reduce to an NS5-brane and two sets of O8− + 8D8- +branes located at two ends of the interval as illustrated in Figure 3 [92]. This theory +can also be realized in F-theory by two −1 curves Σ1 and Σ2 in the base surface of +an elliptic CY3 with the intersection matrix given by [22] +Ωαβ = +�−1 1 +1 −1 +� +. +(3.44) +The partition function of this LST can be factorized into the perturbative part +ZHE +pert for a single tensor multiplet and the contribution from strings ZHE +str as +ZHE +GV = ZHE +pert · ZHE +str = ZHE +pert · +� +k1,k2≥0 +Qk1 +�e2πiw +Q +�k2 +ZHE +(k1,k2) , +(3.45) +where Q ≡ e2πi(φ1,0−φ2,0) and φ1,0 − φ2,0 is the scalar VEV for the dynamical tensor +multiplet. +GLSM +The E8 × E8 LST contains non-perturbative strings arising from the D2- +branes stretched between the D8-branes, the O8−-plane and the NS5-brane in Fig- +ure 3(b). The worldvolume theory on the D2-branes at low energy can be described by +a 2d N = (0, 4) gauge theory. For (k1, k2)-strings, the gauge group is O(k1) × O(k2). +There are an N = (0, 4) vector multiplet and a symmetric hypermultiplet coming from +– 29 – + +O(k1) +SO(16) +O(k2) +SO(16) +sym +sym +(a) +Field +Type +O(k1) × O(k2) +SO(16) × SO(16) +(A(i) +µ , λ ˙αA ++(i)) +vector +(adj, 1), (1, adj) +(a(i) +α ˙β, λαA +−(i)) +hyper +(sym, 1), (1, sym) +(ϕA, χ ˙α +−) +twisted hyper +(k1, k2) +(χα ++) +Fermi +(k1, k2) +(Ψ(i) +l ) +Fermi +(k1, 1), (1, k2) +(16, 1), (1, 16) +(b) +Figure 4: Quiver diagram (a) and matter content (b) of 2d N = (0, 4) gauge theory +for E8 × E8 LST. Here, solid, dashed and zigzag lines denote hypermultiplets, Fermi +multiplets and twisted hypermultiplets, repectively and i = 1, 2 labels each gauge +node. +the D2-D2 string modes for each gauge node, the Fermi multiplets in the bifundamen- +tal representations of O(k1) × SO(16) and O(k2) × SO(16) coming from the D2-D8 +string modes, and the O(k1) × O(k2) bifundamental Fermi multiplets and twisted +hypermultiplets from the strings between two adjacent D2-branes. These multiplets +form representations of the SU(2)l × SU(2)r global symmetry which rotates 2345 +directions, and those of the SO(3) ∼ SU(2)R rotational symmetry for 678 directions. +In the strong coupling limit, the 678 directions and the M-theory circle become an +R4, so we expect the SO(3) symmetry enhances to SO(4) = SU(2)R × SU(2)m0. We +summarize the matter content of the 2d theory in Figure 4. When k1 = 0 or k2 = 0, +this 2d gauge theory reduces to that for self-dual strings in the 6d E-string theory +studied in [92, 93]. +We can compute Z(k1,k2) of the 2d gauge theory using the localization method +[75, 89]. Here, we give explicit expressions of the elliptic genera up to (k1, k2) = +(k2, k1) = (2, 1) order. The contour integral expressions for the elliptic genera and +the detailed computations are presented in Appendix B.1. When k2 = 0, the elliptic +– 30 – + +genera reduce to those for the E-strings obtained in [93]: +ZHE +(1,0) = −1 +2 +4 +� +I=1 +�8 +l=1 θI(ml) +η6θ1(ϵ1)θ1(ϵ2) , +(3.46) +ZHE +(2,0) = +1 +4η12θ1(ϵ1)θ1(ϵ2) +4 +� +I=1 +� �8 +l=1 θI(ml ± ϵ1 +2 ) +θ1(2ϵ1)θ1(ϵ2 − ϵ1) + +�8 +l=1 θI(ml ± ϵ2 +2 ) +θ1(2ϵ2)θ1(ϵ1 − ϵ2) +� ++ +4 +� +I=1 +4 +� +J=I+1 +θσ(I,J)(0)θσ(I,J)(2ϵ+) �8 +l=1 θI(ml)θJ(ml) +4η12θ1(ϵ1,2)2θσ(I,J)(ϵ1)θσ(I,J)(ϵ2) +, +(3.47) +where θI are the Jacobi theta functions defined in (A.11), m1,··· ,8 are chemical poten- +tials for the SO(16) global symmetry, and σ(I, J) is defined as +σ(I, J) = σ(J, I) , +σ(I, I) = 0 , +σ(1, I) = I , +σ(2, 3) = 4 , +σ(2, 4) = 3 , +σ(3, 4) = 2 . +(3.48) +Here we use a shorthand notation θI(x±y) = θI(x+y)θI(x−y). For (k1, k2) = (1, 1), +ZHE +(1,1) = 1 +4 +4 +� +I,J=1 +�8 +l=1 θI(ml) · �16 +l=9 θJ(ml) +η12θ1(ϵ1)2θ1(ϵ2)2 +θσ(I,J)(±m0 + ϵ−) +θσ(I,J)(±m0 − ϵ+) , +(3.49) +where ml=9,··· ,16 are chemical potentials for the other SO(16) symmetry and m0 is a +chemical potential for SU(2)m0. The (k1, k2) = (2, 1)-string elliptic genus is +ZHE +(2,1) = 1 +4Z(0) +(2,1) + 1 +8 +4 +� +K=1 +4 +� +I 0. In the last expression, we keep only the terms dependent +on the dynamical Kähler parameter φi’s. We also evaluate the perturbative partition +function (3.80) in this chamber. With the tree level contributions and the contributions +from the mixed Chern-Simons terms, the full effective prepotential is given by +E = +1 +ϵ1ϵ2 +� +F − ϵ2 +1 + ϵ2 +2 +48 +(4φ1 − 4φ2) + ϵ2 ++(φ1 + φ2) +� ++ E(0) +tree , +(3.93) +where +E(0) +tree = +1 +ϵ1ϵ2 +� +w(φ2 +1 − φ1φ2 + φ2 +2) + φ0 +� +3ϵ2 ++ − 5 +2m2 +1 − 1 +2m2 +2 +�� +, +(3.94) +with w ∼ 1/g2 +YM and an auxiliary scalar VEV φ0 ≡ φ0,0. Also, because of the 6d +mixed anomaly free condition given in (3.89), we impose the condition +m2 = 7m1 +(3.95) +in the effective prepotential (3.93). This is compatible with the 2d mixed anomaly +free condition (3.84). +The blowup equation can be constructed with a set of magnetic fluxes +n0 ∈ Z , +n1 ∈ Z + 2/3 , +n2 ∈ Z + 1/3 , +Bm1 = 1/6 , +Bτ = Bw = 0 . +(3.96) +We propose that the blowup equation of the form given in (2.18) with these inputs is +satisfied for the partition function of the SU(3) LST. We have checked that the elliptic +genera computed using the 2d gauge theory satisfy this blowup equation order by +order in the expansion with respect to the fugacities e2πiw, q = e2πiτ, t = e2πi(2φ1−φ2), +and u = e2πi(−φ1+2φ2), up to 2-strings, q1, t1 and u1 orders. +We also attempted to solve the blowup equation with the help of the modular +ansatz in (3.90). By utilizing the blowup equation and modular ansatz, we were able +to find a BPS spectrum up to q1, t−1 and u1 order which mathches with the ADHM +computation given in (3.83). This fixes 419 unknown coefficients in the modular +ansatz, which are compatible with those listed in Table 8. We expect that higher order +computations of blowup equation will fix the remaining unknowns in the modular +ansatz. +– 48 – + +4 +Conclusion +In this paper, we have proposed the blowup equations for six-dimensional little string +theories (LSTs), and demonstrated how our proposal works in some cases. In order to +formulate the blowup equations, we have found that we need to introduce an auxiliary +2-form field to cancel the mixed gauge-global anomalies and also take into account +the summation over its magnetic fluxes on the blown-up P1 as well as the fluxes for +the dynamical tensor and gauge symmetries. Although the flux sum for the auxiliary +2-form field in the blowup equation is divergent, which is essentially because the +auxiliary 2-form field has no quadratic kinetic term and it is thus non-dynamical, we +have found that the blowup equation still makes sense as a Laurant series expansion +in terms of Kähler parameters, and we can even use it to determine the BPS spectra +of strings in the LSTs with the help of modular ansatz. As concrete examples, we have +computed the elliptic genera of strings in ˆA1 type LSTs in IIA/IIB string theories, +LSTs in E8 × E8 and SO(32) heterotic string theories, and an rank-2 LST with +SU(3) guage symmetry and 1sym + 1Λ2 hypermultiplets. We then checked that +these elliptic genera satisfy the blowup equations, and conversely, that the unknown +coefficients of their modular ansatz can be fixed by solving the blowup equations. +There are some interesting extensions of the results in this paper. First, it +would be quite interesting to generalize the blowup formalism into supergravity +theories. The blowup equations for little string theories may suggest the possibility +of this generalization because elliptic genera of the string worldsheet theories in some +supergravity theories such as 9d/10d heterotic string theories are related with the +elliptic genera of LSTs through RG-flows by higgsings. A key difference between +supergravity theories and LSTs or SCFTs is that all symmetries in supergravity +theories are gauged. As a result, we need to turn on dynamical magnetic fluxes for +all the symmetries in the theory, and the Λ factor in the blowup equation can only +depend on the Ω-deformation parameters. We leave this generalization as a future +work. +Another extension of the current work is the consideration of twisted compactifi- +cations of little string theories. The blowup formalism for 5d Kaluza-Klein theories +resulting from 6d SCFTs compactified on a circle with automorphism twists has been +previously explored in [50]. It is straightforward to extend this approach to derive +the blowup equations for twisted compactifications of LSTs by simply replacing the +intersection form Ωαβ of the tensor nodes and the Killing forms Kij for the gauge +algebras in the blowup equation for untwisted LSTs with their twisted counterparts. +One potential use of this formulation is to confirm T-dualities between LSTs including +twists along the T-dual circle. For example, the SU(3) gauge theory with 1sym+1Λ2 +we discussed in section 3.3 is expected to be T-dual to another little string theory +with a twist [103], which is due to the presence of the symmetric hypermultiplet. +The blowup equations for twisted LSTs may provide a more rigorous method for +– 49 – + +identifying and verifying such dualities. +As another generalization, one can also study little string theories with super- +symmetric defects. Various BPS defects in superconformal field theories have been +widely studied. For instance, the partition functions of 5d/6d field theories in the +presence of the codimension 4 defects were investigated in [55] in the context of the +blowup formalism. It should be straightforward to extend this approach to the study +of LSTs coupled to codimension 4 defects, offering a concrete method for analyzing +the dynamics of these defects within the LSTs. +Recently, a systematic method for calculating the partition functions of LSTs +engineered by NS5-branes on D- and E-type singularities using the topological vertex +formalism was proposed in [104]. The resulting partition functions for D-type LSTs +were found to be consistent with those obtained using the elliptic genus computation +in [64], while the partition functions for E-type LSTs represent new results. It would +be valuable to verify these proposed partition functions using the blowup equations. +Acknowledgments +We are grateful to Sung-Soo Kim and Kimyeong Lee for valuable discussions. HK, +MK and YS thank APCTP for its hospitality during completion of this work. HK also +thanks the Simons Center for Geometry and Physics, Stony Brook University for the +hospitality and partial support during the final stage of this work at the workshops +“2022 Simons Summer Workshop” and “Geometry of (S)QFT”. The research of HK, +MK and YS is supported by Samsung Science and Technology Foundation under +Project Number SSTF-BA2002-05 and by the National Research Foundation of Korea +(NRF) grant funded by the Korea government (MSIT) (No. 2018R1D1A1B07042934). +Some of the computations that were conducted using mathematica were carried out on +the computer sushiki at Yukawa Institute for Theoretical Physics in Kyoto University. +A +Elliptic functions +In this appendix, we summarize definintions and properties of the modular forms and +Jacobi forms used in this paper. +A.1 +Modular forms +Let H = {z ∈ C | ℑz > 0} be the upper half plane of the complex plane, τ ∈ H be +the complex structure of the torus and q = e2πiτ. A modular form of weight k is a +function f : H → C satisfying +f +�aτ + b +cτ + d +� += (cτ + d)kf(τ) , +�a b +c d +� +∈ SL(2, Z) . +(A.1) +– 50 – + +An example of the modular form is the Eisenstein series defined by +E2k(τ) = +1 +2ζ(2k) +� +(m,n)̸=(0,0) +1 +(m + nτ)2k = 1 + +(2πi)2k +ζ(2k)(2k − 1)! +∞ +� +n=1 +σ2k−1(n)qn , (A.2) +where ζ(s) is the Riemann zeta function and σk(n) = � +d|n dk is the divisor function. +E2k(τ) with k > 1 are the holomorphic modular forms of weight 2k, while E2(τ) is +only quasi-modular: +E2 +�aτ + b +cτ + d +� += (cτ + d)2E2(τ) − 6i +π c(cτ + d) . +(A.3) +Two Eisenstein series E4(τ) and E6(τ) generate the ring of holomorphic modular +forms M∗(SL(2, Z)) = � +k≥0 M2k(SL(2, Z)), where M2k(SL(2, Z)) is the space of +weight 2k modular forms. In other words, M2k(SL(2, Z)) can be written as +M2k(SL(2, Z)) = +� +4a+6b=2k +CE4(τ)aE6(τ)b . +(A.4) +As a related function with the Eisenstein series, we define the Dedekind eta +function as +η(τ) = q1/24 +∞ +� +n=1 +(1 − qn) . +(A.5) +Its 24th power ∆(τ) = η(τ)24 = (E4(τ)3 − E6(τ)2)/1728 is a weight 12 modular form +called modular discriminant and η(τ) itself has following modular transformation +properties: +η(τ + 1) = eπi/12η(τ) , +η(−1/τ) = +√ +−iτη(τ) . +(A.6) +A.2 +Jacobi forms +There is a generalization of the modular forms including additional fugacities. A +function ϕk,m : H × C → C is called a Jacobi form [105] if it has two transformation +properties +ϕk,m +�aτ + b +cτ + d, +z +cτ + d +� += (cτ + d)ke +2πimcz2 +cτ+d ϕk,m(τ, z) for +�a b +c d +� +∈ SL(2, Z) , (A.7) +ϕk,m(τ, z + λτ + µ) = e−2πim(λ2τ+2λz)ϕk,m(τ, z) +for λ, µ ∈ Z , +(A.8) +and a Fourier expansion of the form +ϕk,m(τ, z) = +� +n,r +c(n, r)qne2πirz , +(A.9) +– 51 – + +where k ∈ Z is called the weight and m ∈ Z≥0 is called the index or level of the Jacobi +form. When m = 0, ϕk,m is independent of z and reduces to a modular form of weight +k. ϕk,m is called a holomorphic Jacobi form if c(n, r) = 0 unless 4mn ≥ r2, a cusp +Jacobi form if c(n, r) = 0 unless 4mn > r2, and a weak Jacobi form if c(n, r) = 0 +unless n ≥ 0. +Let Jk,m be a space of weak Jacobi forms of weight k and level m. The ring of +weak Jacobi form J∗,∗ = � +k,m Jk,m is freely generated over the ring of modular forms +M∗(SL(2, Z)), whose generators are +ϕ−2,1(τ, z) = −θ1(τ, z)2 +η(τ)6 +, +ϕ0,1(τ, z) = 4 +4 +� +i=2 +θi(τ, z)2 +θi(τ, 0)2 , +(A.10) +where θi(τ, x) are Jacobi theta functions defined by +θ1(τ, x) = −i +� +n∈Z +(−1)nq +1 +2 (n+1/2)2yn+1/2 , +θ2(τ, x) = +� +n∈Z +q +1 +2 (n+1/2)2yn+1/2 , +θ3(τ, x) = +� +n∈Z +q +n2 +2 yn , +θ4(τ, x) = +� +n∈Z +(−1)nq +n2 +2 yn , +(A.11) +for y = e2πix. In other words, any weak Jacobi form ϕk,m can be written as +Jk,m ∋ ϕk,m(τ, z) = +� +4a1+6a2−2a3=k +a3+a4=m,ai∈Z≥0 +CaiE4(τ)a1E6(τ)a2ϕ−2,1(τ, z)a3ϕ0,1(τ, z)a4 +(A.12) +for some Cai ∈ C. We also frequently use +ϕ−1,1/2(τ, z) = iθ1(τ, z) +η(τ)3 , +(A.13) +which satisfies ϕ−1,1/2(τ, z)2 = ϕ−2,1(τ, z). +The notion of weak Jacobi forms is further generalized to Weyl invariant Jacobi +forms [87]. Let g be a Lie algebra of rank l, hC ∼= Cl be the complexification of the +Cartan subalgebra, W be its Weyl group, Q∨ be the coroot lattice, and P be its weight +lattice. Denote ⟨·, ·⟩ a Killing form on hC normalized to 2 for the shortest coroot. A +Weyl invariant Jacobi form of weight k and index m is a function ϕk,m : H × hC → C +satisfying following conditions. +(i) Weyl-invariance: for w ∈ W, +ϕk,m(τ, wz) = ϕk,m(τ, z) . +(A.14) +(ii) Modularity: for ( a b +c d ) ∈ SL(2, Z), +ϕk,m +�aτ + b +cτ + d, +z +cτ + d +� += (cτ + d)k exp +� πimc +cτ + d⟨z, z⟩ +� +ϕk,m(τ, z) . +(A.15) +– 52 – + +g +(−k, m) +Al +(0, 1), (j, 1) for 2 ≤ j ≤ l + 1 +Bl +(2j, 1) for 0 ≤ j ≤ l +Cl +(0, 1), (2, 1), (4, 1), (2j, 2) for 3 ≤ j ≤ l +Dl +(0, 1), (2, 1), (4, 1), (l, 1), (2j, 2) for 3 ≤ j ≤ l − 1 +E6 +(0, 1), (2, 1), (5, 1), (6, 2), (8, 2), (9, 2), (12, 3) +E7 +(0, 1), (2, 1), (6, 2), (8, 2), (10, 2), (12, 3), (14, 3), (18, 4) +F4 +(0, 1), (2, 1), (6, 2), (8, 2), (12, 3) +G2 +(0, 1), (2, 1), (6, 2) +Table 9: Weights and indices for the fundamental Weyl invariant Jacobi forms +(iii) Quasi-periodicity: for λ, µ ∈ Q∨, +ϕk,m(τ, z + λτ + µ) = exp(−πim[⟨λ, λ⟩τ + 2⟨λ, z⟩])ϕk,m(τ, z) . +(A.16) +(iv) Fourier expansion: +ϕk,m(τ, z) = +∞ +� +n=0 +� +ℓ∈P +c(n, ℓ)qne2πi⟨ℓ,z⟩ . +(A.17) +The weak Jacobi forms defined above are g = A1 case. +Let Jk,m(g) be the space of the g Weyl invariant Jacobi forms with weight k and +index m. Then, for a simple Lie algebra except for E8, the bigraded ring, +J∗,∗(g) = +� +k,m∈Z +Jk,m(g) +(A.18) +is freely generated by l + 1 fundamental Weyl invariant Jacobi forms over the ring of +modular forms M∗(SL(2, Z)). The Wirthmüller’s theorem [87] provides weights and +indices for fundamental Weyl invariant Jacobi forms of simple Lie algebras except for +E8 as we list in Table 9. Although the theorem does not give explicit form of the +Jacobi forms, generators of Weyl invariant Jacobi forms for each Lie algebra have been +studied in many literatures [106–111]. The E8 is exceptional case for Wirthmüller’s +theorem, but its Weyl invariant Jacobi forms are also studied recently [88, 112–114]. +See also [65, 115] for a review in physics liturature. Here, we give a construction of +Weyl invariant Jacobi forms used in this paper. +Let us consider g = Al. The weight −k Jacobi form ϕAl +k ∈ J−k,1(Al) is given by +ϕAl +k = Zl+1−k +l+1 +� +j=1 +iθ1(xj) +η3 +������ xj=0 +, +(k = 0, 2, 3, · · · , l + 1) +(A.19) +– 53 – + +where +Z = +1 +2πi +� l+1 +� +j=1 +∂ +∂xj ++ π2 +3 E2(τ) +l+1 +� +j=1 +xj +� +. +(A.20) +The orthogonal basis xj’s are related with the Dynkin basis φi’s by x1 = φ1, xj = +−φj−1 + φj for 2 ≤ j ≤ l and xl+1 = −φl. In particular, we use +ϕA2 +3 += (χ3 − χ3) + (χ6 − χ6 + 7χ3 − 7χ3)q + O(q2), +(A.21) +ϕA2 +2 += 1 +2 +� +(6 − χ3 − χ3) + (42 + 6χ8 − χ6 − χ6 − 13χ3 − 13χ3)q + O(q2) +� +, +ϕA2 +0 += 1 +4 +� +(18 + χ3 + χ3) + (342 + 18χ8 + χ6 + χ6 − 83χ3 − 83χ3)q + O(q2) +� +, +for g = A2 in section 3.3 to write the modular ansatz for the SU(3) + 1sym + 1Λ2 +LST, where χR denotes character of SU(3) for representation R.5 +Next, to study the Dl Jacobi forms, we first consider the Bl Jacobi forms. The +generators of Bl Jacobi forms ϕBl +2j ∈ J−2j,1 can be computed from the generating +function +l� +j=1 +iθ1(v − xi) +η3 +iθ1(v + xi) +η3 += +� +iθ1(v) +η3 +�2l +l +� +j=0 +℘(2j−2)(v) +(2j − 1)! ϕBl +2j (x1, · · · , xl) , +(A.22) +where j = 0 term in the summation is understood as ϕBl +0 (x1, · · · , xl), and ℘ is the +Weierstraß ℘ function defined as +℘(z) = θ3(0)2θ2(0)2 +4 +θ4(z)2 +θ1(z)2 − 1 +12 +� +θ3(0)4 + θ2(0)4� +. +(A.23) +Then the l − 3 generators of Dl Jacobi forms with index 2 is identified with Bl Jacobi +forms: +ϕDl +−k,2 = ϕBl +k +(k = 6, 8, · · · , 2l − 2) , +(A.24) +where ϕDl +−k,2 ∈ J−k,2(Dl) and ϕBl +k ∈ J−k,1(Bl). The index 1 generators are +ϕDl +−n,1 = +l� +j=1 +θ1(xj) +η3 +, +ϕDl +−4,1 = 1 +η12 +��l +j=1 θ3(xj) +θ3(0)l−4 +− +�l +j=1 θ4(xj) +θ4(0)l−4 +− +�l +j=1 θ2(xj) +θ2(0)l−4 +� +, +ϕDl +−2,1 = θ3(0)4 + θ4(0)4 +η12 +��l +j=1 θ3(xj) +θ3(0)l−4 +− +�l +j=1 θ4(xj) +θ4(0)l−4 ++ +2 �l +j=1 θ2(xj) +θ2(0)l−4 +� +− 3θ2(0)4 +η12 +��l +j=1 θ3(xj) +θ3(0)l−4 ++ +�l +j=1 θ4(xj) +θ4(0)l−4 +� +, +ϕDl +0,1 = 1 +η12 +��l +j=1 θ3(xj) +θ3(0)l−12 +− +�l +j=1 θ4(xj) +θ4(0)l−12 +− +�l +j=1 θ2(xj) +θ2(0)l−12 +� +, +(A.25) +5Note that ϕA2 +0 +in our paper is −6ϕ0 defined in Appendix B of [77]. +– 54 – + +where ϕDl +−k,1 ∈ J−k,1(Dl). These level 1 Jacobi forms are used to construct the 1-string +elliptic genus of the SO(32) heterotic LST in subsection 3.2.2. +Lastly, we review the E8 Jacobi forms. The bigraded ring J∗,∗(E8) for the E8 +Weyl invariant Jacobi forms are contained in a polynomial algebra over M∗(SL(2, Z)) +generated by nine functions [88]: +J∗,∗(E8) ⊊ M∗(SL(2, Z))[A1, A2, A3, A4, A5, B2, B3, B4, B6] , +(A.26) +where [112] +A1 = ΘE8(τ, x) = 1 +2 +4 +� +k=1 +8 +� +j=1 +θk(τ, xj) , +A4 = ΘE8(τ, 2x) , +(A.27) +An = +n3 +n3 + 1 +� +ΘE8(nτ, nx) + 1 +n4 +n−1 +� +k=0 +ΘE8( τ+k +n , x) +� +(n = 2, 3, 5) , +B2 = 32 +5 +� +e1(τ)ΘE8(2τ, 2x) + 1 +24e3(τ)ΘE8( τ +2, x) + 1 +24ΘE8( τ+1 +2 , x) +� +, +B3 = 81 +80 +� +h(τ)2ΘE8(3τ, 3x) − 1 +35 +2 +� +k=0 +h( τ+k +3 )2ΘE8( τ+k +3 , x) +� +, +B4 = 16 +15 +� +θ4(2τ, 0)4ΘE8(4τ, 4x) − 1 +24θ4(2τ, 0)4ΘE8(τ + 1 +2, 2x) +− 1 +210 +3 +� +k=0 +θ2( τ+k +2 , 0)4ΘE8( τ+k +4 , x) +� +, +B6 = 9 +10 +� +h(τ)2ΘE8(6τ, 6x) + 1 +24 +1 +� +k=0 +h(τ + k)2ΘE8( 3τ+3k +2 +, 3x) +− 1 +35 +2 +� +k=0 +h( τ+k +3 )2ΘE8( 2τ+2k +3 +, 2x) − +1 +24 · 35 +5 +� +k=0 +h( τ+k +3 )2ΘE8( τ+k +6 , x) +� +. +Here, +e1(τ) = 1 +12 +� +θ3(τ, 0)4 + θ4(τ, 0)4� +, +e2(τ) = 1 +12 +� +θ2(τ, 0)4 − θ4(τ, 0)4� +, +(A.28) +e2(τ) = 1 +12 +� +−θ2(τ, 0)4 − θ3(τ, 0)4� +, +h(τ) = θ3(2τ, 0)θ3(6τ, 0) + θ2(2τ, 0)θ2(6τ, 0) , +An and Bn have index n and weight 4 and 6, repectively, and normalized such that +An(τ, 0) = E4(τ) and Bn(τ, 0) = E6(τ). They are used to construct modular ansatz +of E8 × E8 LST in subsection 3.2.1. +B +Derivation of elliptic genera +In this appendix, we present the details for elliptic genus computations of E8 × E8 +heterotic LST, SO(32) heterotic LST and SU(3) + 1sym + 1Λ2 LST using the 2d +ADHM constructions for the moduli spaces of (instanton) strings. +– 55 – + +B.1 +Elliptic genus of E8 × E8 heterotic LST +We can evaluate the elliptic genera of the rank 1 E8 × E8 heterotic LST from the 2d +N = (0, 4) gauge theory description given in Figure 4. The elliptic genus is given by +the integration of 1-loop determinants of supermultiplets in 2d gauge theory over +flat connections of the O(k1) × O(k2) gauge group. Note that we also have to sum +over disconnected sectors of the flat connections corresponding to the disconnected +components of the orthogonal gauge group. For a O(k) group with k ≥ 3, there +are at most ⌊k/2⌋ complex moduli uI and in total eight disconnected sectors for +flat connections, while O(2) has seven sectors consist of one continuous complex +modulus and six discrete holonomies, and O(1) has four discrete sectors [93]. In total, +(k1, k2)-string elliptic genus is given by +Z(k1,k2) = +� +I1,I2 +1 +|W (I1)| · |W (I2)| +1 +(2πi)r1+r2 +� +Z1−loop , +(B.1) +where I1 and I2 represents disconnected sectors of O(k1) and O(k2) flat conections, +W (I1,2) are corresponding Weyl group factors and r1,2 are number of continuous +complex moduli. The integration contour is chosen by Jeffery-Kirwan residue (JK- +residue for short) prescription as discussed in [75, 89]. The 1-loop determinant Z1−loop +is the collection of following 1-loop determinants +Z(j) +vec = +� rj +� +I=1 +2πη2duI +i +iθ1(2ϵ+) +η +�� +� � +e∈Rj +iθ1(e · u) +η +iθ1(2ϵ+ + e · u) +η +� +� , +Z(j) +sym,hyp = +� +w∈symj +(iη)2 +θ1(ϵ1,2 + w · u) , +Z(j) +fund,Fermi = +� +w∈fundj +lj+7 +� +l=lj +iθ1(ml + w · u) +η +, +Zbifund = +� +w∈bifund +θ1(±m0 + ϵ− + w · u) +θ1(±m0 − ϵ+ + w · u) , +(B.2) +for j = 1, 2, where Rj, symj and fundj denotes root system, symmetric and funda- +mental representation of SO(kj), repectively, bifund is the bifundamental represen- +tation of SO(k1) × SO(k2) and (l1, l2) = (1, 9). The details of the contour integral +for O(k) gauge group are explained in [93], and we will use some of their results. +(1,0)-string +The O(1) gauge group consists of four discrete flat connections labelled +by uI = 0, 1 +2, τ+1 +2 , τ +2. For each sector, the 1-loop determinant is +ZI +(1,0) = +(iη)2 +θ1(ϵ1 + 2u)θ1(ϵ2 + 2uI) +8 +� +l=1 +iθ1(ml + uI) +η +. +(B.3) +Thus, the (1, 0)-string elliptic genus is +Z(1,0) = −1 +2 +4 +� +I=1 +�8 +l=1 θI(ml) +η6θ1(ϵ1)θ1(ϵ2) , +(B.4) +– 56 – + +where 1/2 is the Weyl group factor. +(2,0)-string +The O(2) gauge group has one continuous flat connection and six +discrete flat connections. The contribution from the continuous sector is +Z(0) +(2,0) = +1 +2πi +� 2πη2du +i +iθ1(2ϵ+) +η +(iη)6 +θ1(ϵ1,2)θ1(ϵ1,2 ± 2u) +8 +� +l=1 +iθ1(ml ± u) +η +. +(B.5) +The JK-residue comes from u = − ϵ1,2 +2 + uI, where uI = 0, 1 +2, τ+1 +2 , τ +2. In total, we +compute6 +Z(0) +(2,0) = +1 +2η12θ1(ϵ1)θ1(ϵ2) +4 +� +I=1 +� �8 +l=1 θI(ml ± ϵ1 +2 ) +θ1(2ϵ1)θ1(ϵ2 − ϵ1) + +�8 +l=1 θI(ml ± ϵ2 +2 ) +θ1(2ϵ2)θ1(ϵ1 − ϵ2) +� +. +(B.6) +The six discrete sectors are +Z(I,J) +(2,0) = iθ1(uI + uJ) +η +iθ1(2ϵ+ + uI + uJ) +η +· +(iη)6 +θ1(ϵ1,2 + 2uI)θ1(ϵ1,2 + 2uJ)θ1(ϵ1,2 + uI + uJ) +8 +� +l=1 +iθ1(ml + uI,J) +η +, +(B.7) +where (I, J) = (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) labels the six sectors of flat con- +nections for (u1, u2, u3, u4) = (0, 1 +2, τ+1 +2 , τ +2). These sectors can be rewritten as +Z(I,J) +(2,0) = θσ(I,J)(0)θσ(I,J)(2ϵ+) �8 +l=1 θI(ml)θJ(ml) +η12θ1(ϵ1,2)2θσ(I,J)(ϵ1)θσ(I,J)(ϵ2) +, +(B.8) +where +σ(I, J) = σ(J, I) , +σ(I, I) = 0 , +σ(1, I) = I , +σ(2, 3) = 4 , +σ(2, 4) = 3 , +σ(3, 4) = 2 . +(B.9) +After dividing it by the Weyl group factor, the (2, 0)-string elliptic genus is given by +Z(2,0) = 1 +2Z(0) +(2,0) + 1 +4 +4 +� +I=1 +4 +� +J=I+1 +Z(I,J) +(2,0) . +(B.10) +(1,1)-string +Let u1 and u2 label the flat connections for two O(1) gauge groups. +As we explained in the (1,0)-string case above, there are four distinct flat connections +labelled by uI = 0, 1 +2, τ+1 +2 , τ +2 for each O(1) gauge group. The 1-loop determinant is +Z(I,J) +(1,1) = +(iη)4 +θ1(ϵ1 + 2uI,J)θ1(ϵ2 + 2uI,J) +θ1(±m + ϵ− + uI + uI) +θ1(±m − ϵ+ + uI + uJ) +· +� 8 +� +l=1 +iθ1(ml + uI) +η +�� 16 +� +l=9 +iθ1(ml + uJ) +η +� +, +(B.11) +6The overall sign is chosen by requiring the GV-invariant structure (2.3). +– 57 – + +where I, J = 1, 2, 3, 4 when uI,J = 0, 1 +2, τ+1 +2 , τ +2, repectively. By dividing it by the Weyl +group factor, the (1, 1)-string elliptic genus is +Z(1,1) = 1 +4 +4 +� +I,J=1 +�8 +l=1 θI(ml) · �16 +l=9 θJ(ml) +η12θ1(ϵ1)2θ1(ϵ2)2 +θσ(I,J)(±m0 + ϵ−) +θσ(I,J)(±m0 − ϵ+) . +(B.12) +(2,1)-string +To compute the (2, 1)-string elliptic genus, we need to consider all +combinations of O(2) and O(1) flat connections. First, from the continuous sector of +O(2) and four discrete sectors of O(1), we have +Z(0) +(2,1) = +1 +2πi +� 2πη2du1 +i +iθ1(2ϵ+) +η +(iη)6 +θ1(ϵ1,2)θ1(ϵ1,2 ± 2u1) +(iη)2 +θ1(ϵ1 + 2uJ)θ1(ϵ2 + 2uJ) +· θ1(±m + ϵ− + u1 + uJ)θ1(±m + ϵ− − u1 + uJ) +θ1(±m − ϵ+ + u1 + uJ)θ1(±m − ϵ+ − u1 + uJ) +(B.13) +· +� 8 +� +l=1 +iθ1(ml ± u1) +η +�� 16 +� +l=9 +iθ1(ml + uJ) +η +� +, +where uJ = 0, 1 +2, τ+1 +2 , τ +2 labels O(1) flat connections. Then Z(0) +(2,1) is given by sum of +following two JK-residues: +• u1 = − ϵ1,2 +2 + uI for uI = 0, 1 +2, τ+1 +2 , τ +2 +4 +� +I,J=1 +− �8 +l=1 θI(ml ± ϵ1 +2 ) · �16 +l=9 θJ(ml) +2η18θ1(ϵ1,2)2θ1(2ϵ1)θ1(ϵ2 − ϵ1) +θσ(I,J)(±m0 + ϵ1 − ϵ2 +2 ) +θσ(I,J)(±m0 − ϵ1 − ϵ2 +2 ) + (ϵ1 ↔ ϵ2) +(B.14) +• u1 = ±m + ϵ+ − uJ +4 +� +I=1 +− �8 +l=1 θI(ml ± (m0 + ϵ+)) · �16 +l=9 θI(ml) +η18θ1(ϵ1,2)θ1(2m0)θ1(2m0 + 2ϵ+)θ1(2m0 + 2ϵ+ + ϵ1,2) + (m0 → −m0) +(B.15) +Next, there are combinations of six discrete sectors for O(2) and four discrete sectors +for O(1). If we denote (uI, uJ) = (0, 1 +2), (0, τ+1 +2 ), (0, τ +2), ( 1 +2, τ+1 +2 ), ( 1 +2, τ +2), ( τ+1 +2 , τ +2) as the +O(2) discrete flat connections and uK = 0, 1 +2, τ+1 +2 , τ +2 as the O(1) flat connections, the +1-loop determinant is +Z(I,J,K) +(2,1) += iθ1(uI + uJ) +η +iθ1(2ϵ+ + uI + uJ) +η +(iη)6 +θ1(ϵ1,2 + 2uI,J)θ1(ϵ1,2 + uI + uJ) +· +(iη)2 +θ1(ϵ1,2 + 2uK) +θ1(±m + ϵ− + uI + uK)θ1(±m + ϵ− + uJ + uK) +θ1(±m − ϵ+ + uI + uK)θ1(±m − ϵ+ + uJ + uK) +· +� 8 +� +l=1 +iθ1(ml + uI,J) +η +�� 16 +� +l=9 +iθ1(ml + uK) +η +� +. +(B.16) +– 58 – + +Then we get +Z(I,J,K) +(2,1) += − θσ(I,J)(0)θσ(I,J)(2ϵ+) +η18θ1(ϵ1,2)3θσ(I,J)(ϵ1,2) +θσ(I,K)(±m0 + ϵ−)θσ(J,K)(±m0 + ϵ−) +θσ(I,K)(±m0 − ϵ+)θσ(J,K)(±m0 − ϵ+) +· +8 +� +l=1 +θI(ml)θJ(ml) · +16 +� +l=9 +θK(ml) . +(B.17) +By dividing it by the Weyl group factor, the (2, 1)-string elliptic genus can be written +as +Z(2,1) = 1 +4Z(0) +(2,1) + 1 +8 +4 +� +K=1 +4 +� +I400 J cm-2) could be obtained in a steel with +a ferrite/martensite laminated microstructure. The softer +and more ductile ferrite lamella were also able to transmit +the stress deeper into the material which explains why the +tips of the ferrite lamellae away from the fracture surface +also experienced significant plastic strain (Figure 1c). A +schematic of the described process in the Low C, 22 ◦C +sample is shown in Figure 5a. +Therefore, the energy expended during crack propaga- +tion in the Low C steel was used to transform the austen- +ite to martensite within the TRIP zone, tear through a +ferrite/martensite laminate structure and deform the sur- +rounding ferrite lamella far away from the TRIP zone. The +austenite to martensite transformation in itself does not +absorb significant amounts of energy [38] but Song et al. +[39] suggested that the transformation helped relax the +6 + +Phase map +AusteniteKAM +Ferrite/martensite KAM +Y +α +(a)(b) +(c) +5 μm +20 μm +20 μm +5 μm +20 μm +(d) +(e) +(f) +um +20μum +20μm +5 μm +20μmFigure 4: APT results obtained from a needle containing a PAGB in +the Low C plate steel. (a) Concentration profile across an γ/α inter- +face. Full circles at 0 nm and 40 nm indicate the far-field composition +of ferrite and austenite respectively. Inset: Mn atom map and loca- +tion of cylinder used to measure the concentration profile within the +needle. (b) Concentration profile of tramp elements within the same +volume as (a). +stress at the crack tip suppressing void formation. Nev- +ertheless, a significant amount of energy expended during +crack propagation in the Low C steel was used to tear +through the laminate structure. +In the High C, 22 ◦C sample, the stress ahead of the +crack tip would similarly cause the austenite to trans- +form to martensite. However, due to the mixed morphol- +ogy of the ferrite phase and also a higher austenite frac- +tion, there may not always be bridging ferrite lamella to +blunt the crack tip. Therefore, large uninterrupted regions +of austenite could transform into martensite which might +subsequently cleave open. Austenite grains are also not +always kept seperate from each other, implying a large +amount of γ/γ grain boundaries. If a γ/γ boundary was +caught in the stress field, it will turn into a α′/α′ boundary +after transformation which might also cleave open. Both +of these factors may result in the crack being able to propa- +gate rapidly through the microstructure. Where the crack +Figure 5: Schematic of martensite transformation with crack propa- +gation in (a) Low C within a PAG and (b) High C steel across several +PAGs with different ferrite lamella orientations. Stress field ahead +of the crack tip indicated by the dotted circle. +N.B. the PAG in +the Low C steel is ferritic with austenite lamella and the other way +around in the High C steel. +was able to propagate rapidly, there would likely be very +little subsurface plastic deformation i.e. stress shielding +[40], as observed in Figures 3e-f. In certain areas, even the +austenite grains just below the fracture surface were pro- +tected from transformation. A schematic of the described +mechanism in the High C steel is shown in Figure 5b. The +facets observed in Figures 2i-j could therefore correspond +to the cleavage surfaces of the martensite grains in the +High C steel. +Therefore, the energy absorbed during crack propaga- +tion in the High C steel was consequently lower than the +Low C steel as the crack was able to propagate via brittle +fracture of large areas of connected martensite (previously +austenite) grains. This effect was coined the “brittle net- +work” effect by Jacques et al. [41] who similiarly found a +decrease in resistance to cracking in a steel with a larger +volume of high carbon retained austenite. Future medium +Mn alloy development should focus on isolating austenite +grains in order to improve resistance to cracking. +The morphology of the austenite and ferrite grains there- +fore appear to be a significant factor in the Charpy im- +pact performance of medium Mn steels. Song et al. [39] +showed in low alloy TRIP steels that the TRIP effect was +most beneficial when the austenite phase was in the form +of films between bainitic laths as compared to blocky is- +lands. +However, Han et al. +[26] found that the room +temperature Charpy impact performance was very similar +between fine grained equiaxed and lamellar microstruc- +ture variants, both having the same bulk composition and +austenite fraction. This suggests that microstructure may +not be the only factor influencing the Charpy impact per- +formance of medium Mn steels. +4.2. Effects of composition and segregation +Aside from differences in microstructure, the two in- +vestigated medium Mn steels had very different composi- +tions with the High C steel having a greater alloy content +in all major elements: Mn, Al, Si and C. While the ef- +fects of individual elements on the Charpy impact perfor- +7 + +(a) +16 +50 nm +14 +12 +Mn +Composition (at%) +Mr +10 +Y +8 +a +6 +Al +Si +2 +MWAAL +c +0 +10 +20 +30 +40 +Distance(nm) +(b) +1.2 +0 +1.0 +s +(%) +0.8 +p +Composition ( +0.6 +0.4 +0.2 +0.0 +0 +10 +20 +30 +40 +Distance(nm)(a) +(b) +a +000 +TRIP zone +Facetedregion +Shallow dimple regionmance have not been investigated in medium Mn steels, +C was expected to be the most significant element. The +ASM handbook [42] showed that increasing the C con- +tent generally leads to a higher Ductile-Brittle Transition +Temperature (DBTT) but a reduced upper shelf energy in +various ferritic/martensitic steels. On the other hand, in +fully austenitic TWIP steels, C has the effect of strength- +ening the austenite phase and improving the absorbed im- +pact energy [43]. In TWIP+TRIP-type Fe-Cr-Mn stain- +less steels, Hwang et al. [44] showed that there was no +significant difference in room temperature Charpy impact +energy between C contents of 0.2–0.4 wt%. +C also significantly influences the kinetics and extent +of the TRIP effect by stabilising the austenite phase, i.e. +increasing the resistance to deformation induced marten- +sitic transformation. The austenite stability of the High C +steel was consequently significantly higher than the Low C +steel (Table III). While both High C and Low C exhibited +the TRIP effect, it was difficult to quantify the extent of +the TRIP effect just below the fracture surface. Further- +more, due to stress shielding effects in the High C steel, +the extent of transformation could not be attributed to +composition alone. +Nevertheless, depending on the C content, the trans- +formed martensite will vary in hardness and therefore brit- +tleness [45, 46]. The strength of the transformed marten- +site, σα′ can be estimated using the equation [4, 47]: +σα′ (MPa) = 413 + 1720 XC +(2) +where XC is the C content in wt%. Based on the C con- +tent of the High C and Low C steels in Table III, σα′ in +the High C and Low C steel would be 2.3 GPa and 1.6 +GPa respectively. The martensite in the High C steel was +therefore expected to be very brittle [48]. Therefore, while +a stronger martensite might be preferable for higher tensile +strengths and resistance to necking from the perspective +of a tensile test (Figure 1e), it may not be as beneficial in +terms of crack resistance. +These results therefore show that the Charpy V-notch +impact properties of medium Mn steels appear to be TRIP- +limited. The morphology, composition, strength and duc- +tility of the martensite phase heavily influence the crack +propagation energy during the impact test. While not in- +vestigated in this study, the TWIP effect would therefore +only be expected to play a limited role. +On the other hand, there is a growing body of litera- +ture demonstrating segregation of elements to certain in- +terfaces such as PAGBs [26] or δ-ferrite boundaries [49] +leading to poor cohesion and reduced impact properties. +APT was conducted on the Low C sample and Figure 4 +shows a ferrite/austenite boundary in a needle lifted from +a PAGB. The results do not show any concentration spike +of Mn, C or any other tramp elements to the identified +boundary. This gives confidence that segregation does not +always occur in medium Mn steels. +Segregation of ele- +ments such as Mn and C also appears to be a time-related +issue. For medium Mn steels where segregation was identi- +fied [26, 31, 49], the IA duration was ≤ 1 h. In this study, +the Low C steel was intercritically annealed for 24 h to +replicate the batch annealing process. This suggests that +batch annealed medium Mn steels might be less suscepti- +ble to segregation related embrittlement. +5. Conclusions +The Charpy impact properties of two different medium +Mn steels with different microstructures, tensile properties +and compositions were compared. Several key findings are +shown below. +1. Both the Low C and High C steels exhibited the +TRIP effect along the fracture edge. However, the +Low C steel had a significantly higher absorbed Charpy +impact energy compared to the High C steel. The +reasons for which could be attributed to microstruc- +ture and C content. +2. A lamellar microstructure absorbs more energy dur- +ing crack propagation compared to a mixed equiaxed ++ lamellar microstructure by acting as a laminate +composite. The austenite within the stress field trans- +forms into martensite and reinforces the ferrite ma- +trix. +3. 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Acta Materialia, 2019; vol. 164, pp. 122–134. +10 + diff --git a/utAzT4oBgHgl3EQfPvvS/content/tmp_files/load_file.txt b/utAzT4oBgHgl3EQfPvvS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8937f09f4b813f39f8e57e84519601da5654dc1 --- /dev/null +++ b/utAzT4oBgHgl3EQfPvvS/content/tmp_files/load_file.txt @@ -0,0 +1,690 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf,len=689 +page_content='Carbon in solution and the Charpy impact performance of medium Mn steels TWJ Kwoka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' FF Worsnopa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' JO Douglasa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' D Dyea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='∗ aDepartment of Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Royal School of Mines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Prince Consort Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' London SW7 2BP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' UK bDepartment of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 77 Massachusetts Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' MA 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' USA Abstract Carbon is a well known austenite stabiliser and can be used to alter the stacking fault energy and stability against martensitic transformation in medium Mn steels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' producing a range of deformation mechanisms such as the Transforma- tion Induced Plasticity (TRIP) or combined Twinning and Transformation Induced Plasticity (TWIP + TRIP) effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, the effect of C beyond quasi-static tensile behaviour is less well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, two medium Mn steels with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2 wt% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 wt% C were designed to produce similar austenite fractions and stability and therefore tensile behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' These were processed to form lamellar and mixed equiaxed + lamellar microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The low C steel had a corrected Charpy impact energy (KV10) of 320 J cm-2 compared to 66 J cm-2 in the high C steel despite both having a ductility of over 35%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Interface segregation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' of tramp elements, was investigated as a potential cause and none was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Only a small amount of Mn rejection from partitioning was observed at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The fracture surfaces were investigated and the TRIP effect was found to occur more readily in the Low C Charpy specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore it is concluded that the use of C to promote TWIP+TRIP behaviour should be avoided in alloy design but the Charpy impact performance can be understood purely in terms of C in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Introduction Medium Mn steels (4–12 wt% Mn) are a relatively re- cent class of steels despite their conception in 1972 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Having been “rediscovered” as a leaner alternative to high Mn Twinning Induced Plasticity (TWIP) steels (16–30 wt% Mn), medium Mn steels have been shown to exhibit several different plasticity enhancing mechanisms such as the Transformation Induced Plasticity (TRIP) effect [2, 3] or a combined TWIP+TRIP effect [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Both mecha- nisms can be tailored through heat treatments and alloying to vary the strain hardening rate, leading to large elonga- tions to failure of over 50% [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' These tensile proper- ties make medium Mn steels very suitable materials for energy absorbing applications such as automotive crash pillars [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Current safety related automotive steels are designed to be either anti-intrusion or to crumple and absorb as much energy as possible in the event of a crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Hot stamp- ing or press hardening martensitic steels such as 22MnB5 are examples of anti-intrusion steels which were designed to be very strong and resist deformation [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Energy absorbing steels such as Dual Phase (DP) steels [12] are softer but significantly more ductile to allow the steel to crumple and fold, absorbing energy in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The opportunity for medium Mn steels, therefore, is to replace DP steels in the automotive Body in White (BIW) [8, 9] ∗Corresponding author Email address: ddye@ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='uk (D Dye) as they have equivalent or better tensile properties and are also potentially cheaper due to the omission of expensive alloying elements such as Cr, Nb and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The ability to exhibit the TWIP+TRIP effect upon de- formation, therefore, was of considerable academic inter- est due to the prospect of activating two powerful plastic- ity enhancing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Typically, TWIP+TRIP-type medium Mn steels do indeed exhibit larger elongations to failure compared to TRIP-type medium Mn steels (≥ 50% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' ≥ 25%) [4, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The activation of the TWIP+TRIP effect depends on the control of Stacking Fault Energy (SFE) and stability against transformation of the austen- ite phase in medium Mn steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In order to raise the SFE into the twinning regime, a large amount of C, typically more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 wt%, is needed while keeping the Mn con- tent within the “medium” range of between 3–12 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, our previous work [7] and the results by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [4] showed that the strengthening effect from twinning was very small compared to the TRIP effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' It was there- fore postulated that the large elongation in TWIP+TRIP- type medium Mn steels came from a very controlled TRIP effect due to the very stable and C-enriched austenite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Nevertheless, regardless of the strengthening contribu- tion from TWIP or TRIP, TWIP+TRIP-type medium Mn steels still have higher strengths (due to the higher C con- tent) and elongations than most TRIP-type medium Mn steels [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Since the energy absorbed during plastic de- formation is equal to the area under a tensile curve, it should also follow that TWIP+TRIP-type medium Mn steels would be more suitable for energy absorbing appli- Preprint submitted to arXiv January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='01190v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='mtrl-sci] 3 Jan 2023 cations than TRIP-type medium Mn steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Furthermore, the TWIP effect was also shown to be active at high strain rates up to approximately 2000 s−1 [15], while the TRIP effect is diminished at high strain rates due to adiabatic heating [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, it is possible that the TWIP ef- fect might begin to play a significant role at higher strain rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' High strain rate tests such as the Hopkinson pressure bar test would be able to provide very useful information but are relatively difficult to perform [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Alternatively, Charpy V-notch tests can also provide some insights into the failure mechanisms, tear resistance, notch toughness and energy absorption at high strain rates of up to 103 s−1 depending on the type of material [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In this study, the Charpy energies of two different medium Mn steels will be compared: a high C TWIP+TRIP-type medium Mn steel with a mixed equiaxed + lamellar microstructure, developed in previous work [7], and a novel low C TRIP- type medium Mn steel with a fully lamellar microstruc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This study aims to identify and compare the failure mechanisms in both steels in order to guide future alloy design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Experimental Two steel ingots, High C and Low C, were vacuum arc melted using pure elements and cast into ingots measuring approximately 75 mm × 23 mm × 23 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The composi- tions of both steels as measured using Inductively Coupled Plasma (ICP) and Inert Gas Fusion (IGF) are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Both steels were then homogenised at 1250 ◦C for 2 h in a vacuum furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The High C steel was hot rolled from 23 mm → 4 mm in thickness between 1000 ◦C and 850 ◦C in 8 passes at approximately 20% reduction per pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The rolled plate was quenched immediately af- ter the last pass and subsequently intercritically annealed at 750 ◦C for 20 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Low C steel was hot rolled from 23 mm → 6 mm in thickness between 1000 ◦C and 950 C in 6 passes at approximately 20% reduction per pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' After the last pass, the Low C steel was returned to the furnace at 600 ◦C for 30 min then allowed to furnace cool to room temperature to simulate a coiling cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Low C steel was then cold rolled from 6 mm → 4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' After cold rolling, the Low C steel was reheated to 950 ◦C for 5 min, water quenched and then intercritically annealed at 680 ◦C for 24 h (two-step heat treatment as described by Steineder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' For comparison with strip properties, another ingot of Low C steel was rolled between 1000 ◦C and 950 ◦C from 10 mm → 3 mm in 5 passes at approximately 20% reduc- tion per pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The strip was then cold rolled from 3 mm → 2 mm and heat treated in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The final thickness of the Low C strip after descaling was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The method to produce 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm strip of High C steel is described in previous work [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Subsized quarter thickness Charpy V notch samples (55 mm × 10 mm × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm) were obtained from both High C and Low C plates in the L-T orientation (notch fac- ing the plate transverse direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Three Charpy samples were tested at −196 ◦C, −40 ◦C and 22 ◦C each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Tensile samples with gauge dimensions of 19 mm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm were obtained from the High C and Low C plates and sheets using Electrical Discharge Machining (EDM) such that the tensile axis was parallel to the gauge length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Tensile testing was conducted using a nominal strain rate of 10−3 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Applied strain was measured with a clip-on extensometer up to 10% and the crosshead displacement thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Electron Backscattered Diffraction (EBSD) and Sec- ondary Electron Microscopy (SEM) was conducted on a Zeiss Sigma FEG-SEM with a Bruker EBSD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' For EBSD, a 750 nm step size, dwell time of 10–15 ms and an accelerating voltage of 20 kV were used to reduce the amount of unindexed patterns below 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Bruker ES- PRIT software was used to analyse the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Secondary Electron (SE) imaging was conducted using an accelerat- ing voltage of 5 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Atom Probe Tomography (APT) specimens were fabri- cated using site-specific Focused Ion Beam (FIB) liftout in a Thermo Fisher Scientific Helios 5 CX DualBeam micro- scope from regions that contained austenite/ferrite phase interfaces at a Prior Austenite Grain Boundary (PAGB) identified using EBSD [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Micron scale regions of ma- terial were then mounted onto pre-fabricated silicon posts (Cameca) and were cross sectioned to identify the specific nanoscale boundaries of interest [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' SEM high resolution imaging at 2 kV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='1 nA using immersion mode was used to guide the subsequent milling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 30 kV Ga+ annular milling was then used to create needle shaped specimens that contained such an interface close to the apex and the sample was then polished using 5 kV Ga+ ions prior to APT analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' APT analyses were carried out using a Cameca LEAP 5000 XR atom probe between a base temperature of 50 and 55 K in voltage pulsing mode with a pulse frequency of 200 kHz, a pulse fraction of 20% and detection rates be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Data acquired was analysed using the Integrated Visualization and Analysis Software (IVAS) in AP Suite 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='1 (Cameca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Peak overlaps such as Al+/ Fe2+ at 27 Da and Si+/ Fe2+ at 28 Da were resolved based on isotopic ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In this study, it is acknowledged that while the APT samples were lifted from a PAGB, any in- terfaces identified within the analysed volume cannot be guaranteed with absolute certainty to be a PAGB with- out conducting further analysis, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Transmission Kikuchi Diffraction (TKD) to confirm orientation relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Alloy design concept The High C steel developed in previous work [7] had a relatively low Mn content and therefore relied on a high C content as an alternate austenite stabiliser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The high C 2 Figure 1: EBSD phase map + image quality maps of (a) High C sheet, (b) Low C sheet, (c) High C plate, (d) Low C plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Red – austenite, green – ferrite, phase fractions given to the nearest %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Black lines indicate high angle grain boundaries and white lines in- dicate austenite Σ3 boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' (e) Tensile behaviour of High C and Low C sheet (dotted lines) and plate (red lines) material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Table I: Composition of the ingots used to produce High C and Low C plate steels in mass percent obtained using ICP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' and IGF for elements marked with †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Mn Al Si C† N† S† P Fe High C 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='002 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='005 Bal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Low C 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='001 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='005 Bal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' content was able to lower the Mn content needed to create a fully austenitic hot working temperature window, raise the SFE of the austenite into the TWIP+TRIP regime [21, 22] and slow TRIP kinetics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, when the C content in the Low C steel was reduced, the Mn content had to be increased in order to stabilise a sufficiently large austenitic hot working tem- perature window and raise the SFE into the lower end of the TWIP+TRIP regime [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' With the increase in Mn content, Al and Si content can also be reduced, both of which have an effect of balancing the SFE of the austenite phase [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' A fully lamellar microstructure, rather than a mixed equiaxed+lamellar microstructure in the High C steel, was chosen for the Low C steel as it was reported that soft polygonal ferrite was detrimental for hole expansion in multi-phase steels [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' A fully lamellar microstructure would help to reduce any differences in strength between polygonal and lamellar ferrite grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Charpy impact testing and characterisation Figure 1 shows the microstructures of both plate and sheet material from High C and Low C steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The High C steel was processed in a manner to produced a mixed equiaxed + lamellar microstructure comprising of both equiaxed and lamellar ferrite with lamellar austentie grains [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' On the other hand, the Low C steel was processed to produce a fully lamellar microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' When comparing between plate and sheet microstructures, it can be seen that the overall phase fractions and microstructure mor- phologies were preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, the grain size of the equiaxed ferrite and lamellar thicknesses were generally observed to be larger in the plate material of both steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The larger grain size could be attributed to a combination of lower hot rolling reduction ratio per pass and slower cooling rate in the plate material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' When comparing the tensile properties in Figure 1e, the plate material from both steels had a lower elonga- tion to failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The reduced elongation may be attributed to a larger prior austenite grain size in the plate material compared to the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Yield strength of the plate ma- terial was preserved within ±10% of the sheet material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Deformation behaviour of Low C plate was nearly iden- tical to Low C sheet but some deviation was observed in High C plate compared to High C sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The deviation could be attributed to grain size effects which may affect the austenite stability and therefore the TRIP response in medium Mn steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Nevertheless, while not perfectly identical, the plate versions of High C and Low C steels capture the essence of the tensile behaviour of their sheet counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Charpy impact energies from both High C and Low C steels are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Charpy impact energy from the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm thick subsized samples (KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5) in J were normalised (J cm-2) by dividing the impact en- ergy by the cross section area of the fracture surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='8 cm × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Charpy impact energy from the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 mm thick subsized sample can also be corrected to obtain the theoretical impact energy from a full sized 10 mm Charpy impact sample (KV10) using the correction method by Wallin [24, 25]: 3 a) High C sheet Y: 46% b Low C sheet Y: 30% a: 54% a: 70% ND RD 20 μm 20 μm High C plate y: 44% C LowCplate Y: 33% a: 56% a: 67% 20 um 20 μm (e) 1200 Extensometer 1000- removed (MPa) High C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5mm) HighC(4mm) Istress 800 600 LowC(4mm) LowC (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5mm) 400 200- 0- 0 10 20 30 40 50 60 70 Engineering strain (%)Figure 2: Stereomicrographs of postmortem Charpy samples of Low C steel tested at (a) 22 ◦C and (b) −196 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' SE micrographs of fracture surfaces of (c) Low C, 22 ◦C, showing dimples and ductile failure in the “valley” between the adjacent shear lips, (d) magnified view of (c), (e) Low C, −196 ◦C, showing cleavage facets, likely PAGBs circled in dotted red lines and (f) magnified view of white square in (e) showing fracture of individual lamellae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Stereomicrographs of postmortem Charpy samples of High C steel tested at (g) 22 ◦C and (h) −196 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' SE micrographs of fracture surfaces of (i) High C, 22 ◦C, showing microtears and (j) magnified view of the white square in (i), (k) High C, −196 C, showing smooth facets and (l) magnified view of (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 × 10 KV10 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 = 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5ef 1 + ef where f = 2(KV10/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 − 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='7) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3 (1) The theoretical full sized Charpy impact energy can then be normalised by dividing the impact energy by the fracture surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='8 cm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' It should be noted that there is a strong deviation from linearity between KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 and KV10, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 4×KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 ≈ KV10, when normalised KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 exceeds approximately 100 J cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This is likely to account for the increasing size of the shear lip which begins to form at higher Charpy impact energies [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' It is acknowledged that such normalisation and correction methods are not foolproof and should be interpreted qual- itatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Quantitative comparisons should only be made with other KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Figure 2 shows the fracture surfaces of the post mortem Charpy impact samples under stereo-optical microscopy and SE imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' For brevity, the Charpy samples will henceforth be referred to either Low or High C, followed by the test temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In Figure 2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' the Low C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 22 ◦C sample showed very prominent shear lips on the top and 4 (a) Low C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 22 °℃ (b) Shear lip Low C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' -196 °℃ Notch 广 Notch (c) (d) e 100 μm 20 μm 20 μm 5 μm (g) High C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 22 °C (h) High C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' -196C Shearlip Notch Notch Tearing (k) Shallow dimples Microtears Facet 20 μm 5 μm 20 μm 5 μmTable II: Normalised and corrected Charpy impact energies (KVB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' where B is the thickness of the Charpy impact sample) from High C and Low C steels tested at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Standard errors in parantheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Normalised – Charpy energy divided by cross sectional area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Corrected – KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 converted to KV10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' *Corr+norm– corrected and normalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 22 ◦C −40 ◦C −196 ◦C High C KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (J) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='7) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3) Normalised KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (J cm−2) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (4) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5) Corrrected KV10 (J) 53 17 6 Corr+norm KV10 (J cm−2) 66 21 8 Low C KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (J) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='7) NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4) Normalised KV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (J cm−2) 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5) NA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0) Corrected KV10 (J) 256 NA 4 Corr+norm KV10 (J cm−2) 320 NA 5 bottom edges indicating ductile failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' SE micrographs in Figures 2c-d also show a cup-and-cone type fracture surface, indicative of ductile failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This is in contrast to the High C, 22 ◦C sample (Figure 2g) where there was only a very small shear lip just behind the notch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' SE microscopy in Figures 2i-j show micro tears in the fracture surface as well as a mixed ductile/brittle failure mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Several facets were observed which indicate brittle fracture but also shallow dimples which show a limited degree of ductile failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Charpy impact samples tested at cryogenic tem- peratures showed brittle failure with a faceted fracture sur- face regardless of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In the Low C, −196 ◦C sample (Figures 2b, e-f), the facets had a corrugated ap- pearance which may correlate with the lamellar microstruc- ture within a prior austenite grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This might suggest that the crack may have propagated through a prior austen- ite grain, rather than along the PAGBs as shown by Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [26] in a Fe-7Mn-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='5Si-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='1C steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In the High C, −196 C sample, the facets were very smooth and equiaxed in morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' EBSD maps were obtained from the post mortem room temperature Charpy samples and shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In the Low C, 22 ◦C sample, the phase map (Figure 3a) showed a thin region of approximate 5–10 µm thick be- low the fracture surface where austenite was not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This strongly suggests that the austenite within this region (TRIP zone) had completely transformed into martensite, indicating that the TRIP effect was active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' It cannot be said definitively if the crack propagated along the PAGBs or across the prior austenite grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' When comparing the austenite and ferrite/martensite Kernal Average Misori- entation (KAM) maps in Figures 3b and c respectively, it was observed that the austenite lamellae were hardly de- formed, while the ferrite lamellae were plastically deformed at a significantly greater depth than the TRIP zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In the High C, 22 ◦C sample, the phase map in Fig- ure 3d did not show the same uniform subsurface TRIP zone and austenite was still observed very close to the fracture surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, the TRIP effect was still active in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In Figure 3d, martensite can be qualita- tively identified based on its larger blocky morphology and lower indexing quality (appears darker) compared to the surrounding ferrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, martensite could be iden- tified immediately below the fracture surface in the High C, 22 ◦C sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Figure 4 shows the results from an APT needle ob- tained from a PAGB in the Low C plate steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Unfortu- nately, the interface from the APT needle obtained from the High C plate steel was lost due to a microfracture event during analysis but compositions from both phases could still be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The compositions of austenite and ferrite phases from both needles, obtained via APT, are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' SFE was determined from the austen- ite compositions using the method by Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The martensite start (Ms) temperature was determined using the equation by Kaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [28] and the Md30 tempera- ture was determined using the equation by Angel [29] and Nohara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' From Table III, the austenite phase in both Low C and High C steels have SFE ranges within the TWIP+TRIP regime in medium Mn steels [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, the austenite phase in the High C steel was significantly more stable against transformation as seen by the lower Md30 temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' From Figure 4a, the austenite phase could be identified as the Mn and C enriched phase, while the ferrite phase could be identified as the Mn and C depleted but Al en- riched phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' It is noteworthy that Si was not observed to partition strongly to either phase although a slight enrich- ment of Si was observed at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This Si enrich- ment at the interface was also observed in previous work [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The Mn profile in the ferrite phase was relatively con- stant but in the austenite phase, the Mn content appeared to decrease with distance away from the interface moving into the grain interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This was likely due to the sluggish diffusion of Mn in FCC austenite under Partitioning Lo- cal Equilibrium (PLE) mode [32], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Mn at the interface struggles to diffuse into the austenite interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Figure 4b shows the concentration of tramp elements O, S and P within the same sampled region as Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' It could be observed that the austenite phase had a greater solubility for O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, there was no segregation of tramp elements to the grain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Elements such as B and N were not detected above noise background lev- els and therefore omitted from Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, from Figure 4, there was no significant segregation of solute, interstitial nor tramp elements to the PAGB within the APT needle obtained from the Low C plate steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Discussion Perhaps the most interesting result from this compara- tive study was that the Charpy impact energy of the Low C steel was significantly larger than the High C steel, despite having a lower yield strength, tensile strength and elonga- tion (Figure 1 and Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' While higher yield strengths 5 Figure 3: EBSD maps of the post mortem room temperature Charpy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Notch is towards the left of the micrograph and crack propagation direction is parallel to the TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' EBSD (a) Image Quality and Phase Map (IQ+PM), dotted line indicates the region where the austenite has almost fully transformed, (b) austenite KAM and (c) ferrite/martensite KAM maps of the fracture edge in Low C steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' EBSD (d) IQ+PM, (e) austenite KAM and (f) ferrite/martensite KAM maps of the fracture edge in High C steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Insets are high magnification maps of the respective areas bounded by the white box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Table III: APT composition analysis of the High C and Low C plate steels in wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' † C content determined by lever rule using C con- tent measured by IGF and phase fractions obtained using EBSD, assuming negligible C content in ferrite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' A lower Md30 temper- ature indicates a more stable austentite against deformation induced transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Mn Al Si C† SFE Ms Md30 (mJ m−2) (◦C) (◦C) High C γ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='12 41 53 19 Low C γ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='68 25 80 155 High C α 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='8 – – – – Low C α 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 – – – – tend to correlate with improved energy absorption in drop tower crush tests [33, 34], it seems that the same correla- tion does not exist between tensile properties and Charpy impact performance [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Of the total energy absorbed in a Charpy impact test, Sugimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [36] found that the energy expended to initiate a crack was relatively constant in medium Mn steels, regardless of strength, Mn content or volume fraction of austenite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Instead, it was the en- ergy expended to propagate the crack which dominated the total energy absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, any crack retard- ing or blunting mechanisms in the steel will be expected to greatly improve the Charpy impact performance of the steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Effects of microstructure In the Low C, 22 ◦C sample, a 5 – 10 µm TRIP zone was observed immediately beneath the fracture surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The austenite below the TRIP zone did not appear to be significantly deformed (Figures 3a–c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This suggests that the austenite transformed to martensite rather than un- dergo plastic deformation under the stress at the crack tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The microstructure ahead of the crack tip would then resemble a laminate composite comprising of alternating layers of soft ferrite reinforced by layers of hard marten- site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [37] showed that ultrahigh Charpy impact energies (>400 J cm-2) could be obtained in a steel with a ferrite/martensite laminated microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The softer and more ductile ferrite lamella were also able to transmit the stress deeper into the material which explains why the tips of the ferrite lamellae away from the fracture surface also experienced significant plastic strain (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' A schematic of the described process in the Low C, 22 ◦C sample is shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, the energy expended during crack propaga- tion in the Low C steel was used to transform the austen- ite to martensite within the TRIP zone, tear through a ferrite/martensite laminate structure and deform the sur- rounding ferrite lamella far away from the TRIP zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The austenite to martensite transformation in itself does not absorb significant amounts of energy [38] but Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [39] suggested that the transformation helped relax the 6 Phase map AusteniteKAM Ferrite/martensite KAM Y α (a)(b) (c) 5 μm 20 μm 20 μm 5 μm 20 μm (d) (e) (f) um 20μum 20μm 5 μm 20μmFigure 4: APT results obtained from a needle containing a PAGB in the Low C plate steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' (a) Concentration profile across an γ/α inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Full circles at 0 nm and 40 nm indicate the far-field composition of ferrite and austenite respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Inset: Mn atom map and loca- tion of cylinder used to measure the concentration profile within the needle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' (b) Concentration profile of tramp elements within the same volume as (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' stress at the crack tip suppressing void formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Nev- ertheless, a significant amount of energy expended during crack propagation in the Low C steel was used to tear through the laminate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In the High C, 22 ◦C sample, the stress ahead of the crack tip would similarly cause the austenite to trans- form to martensite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, due to the mixed morphol- ogy of the ferrite phase and also a higher austenite frac- tion, there may not always be bridging ferrite lamella to blunt the crack tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, large uninterrupted regions of austenite could transform into martensite which might subsequently cleave open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Austenite grains are also not always kept seperate from each other, implying a large amount of γ/γ grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' If a γ/γ boundary was caught in the stress field, it will turn into a α′/α′ boundary after transformation which might also cleave open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Both of these factors may result in the crack being able to propa- gate rapidly through the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Where the crack Figure 5: Schematic of martensite transformation with crack propa- gation in (a) Low C within a PAG and (b) High C steel across several PAGs with different ferrite lamella orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Stress field ahead of the crack tip indicated by the dotted circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' the PAG in the Low C steel is ferritic with austenite lamella and the other way around in the High C steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' was able to propagate rapidly, there would likely be very little subsurface plastic deformation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' stress shielding [40], as observed in Figures 3e-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In certain areas, even the austenite grains just below the fracture surface were pro- tected from transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' A schematic of the described mechanism in the High C steel is shown in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The facets observed in Figures 2i-j could therefore correspond to the cleavage surfaces of the martensite grains in the High C steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, the energy absorbed during crack propaga- tion in the High C steel was consequently lower than the Low C steel as the crack was able to propagate via brittle fracture of large areas of connected martensite (previously austenite) grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This effect was coined the “brittle net- work” effect by Jacques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [41] who similiarly found a decrease in resistance to cracking in a steel with a larger volume of high carbon retained austenite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Future medium Mn alloy development should focus on isolating austenite grains in order to improve resistance to cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The morphology of the austenite and ferrite grains there- fore appear to be a significant factor in the Charpy im- pact performance of medium Mn steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [39] showed in low alloy TRIP steels that the TRIP effect was most beneficial when the austenite phase was in the form of films between bainitic laths as compared to blocky is- lands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [26] found that the room temperature Charpy impact performance was very similar between fine grained equiaxed and lamellar microstruc- ture variants, both having the same bulk composition and austenite fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This suggests that microstructure may not be the only factor influencing the Charpy impact per- formance of medium Mn steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Effects of composition and segregation Aside from differences in microstructure, the two in- vestigated medium Mn steels had very different composi- tions with the High C steel having a greater alloy content in all major elements: Mn, Al, Si and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' While the ef- fects of individual elements on the Charpy impact perfor- 7 (a) 16 50 nm 14 12 Mn Composition (at%) Mr 10 Y 8 a 6 Al Si 2 MWAAL c 0 10 20 30 40 Distance(nm) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0 s (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='8 p Composition ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='0 0 10 20 30 40 Distance(nm)(a) (b) a 000 TRIP zone Facetedregion Shallow dimple regionmance have not been investigated in medium Mn steels, C was expected to be the most significant element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The ASM handbook [42] showed that increasing the C con- tent generally leads to a higher Ductile-Brittle Transition Temperature (DBTT) but a reduced upper shelf energy in various ferritic/martensitic steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' On the other hand, in fully austenitic TWIP steels, C has the effect of strength- ening the austenite phase and improving the absorbed im- pact energy [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In TWIP+TRIP-type Fe-Cr-Mn stain- less steels, Hwang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [44] showed that there was no significant difference in room temperature Charpy impact energy between C contents of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='4 wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' C also significantly influences the kinetics and extent of the TRIP effect by stabilising the austenite phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' increasing the resistance to deformation induced marten- sitic transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The austenite stability of the High C steel was consequently significantly higher than the Low C steel (Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' While both High C and Low C exhibited the TRIP effect, it was difficult to quantify the extent of the TRIP effect just below the fracture surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Further- more, due to stress shielding effects in the High C steel, the extent of transformation could not be attributed to composition alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Nevertheless, depending on the C content, the trans- formed martensite will vary in hardness and therefore brit- tleness [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The strength of the transformed marten- site, σα′ can be estimated using the equation [4, 47]: σα′ (MPa) = 413 + 1720 XC (2) where XC is the C content in wt%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Based on the C con- tent of the High C and Low C steels in Table III, σα′ in the High C and Low C steel would be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='3 GPa and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content='6 GPa respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The martensite in the High C steel was therefore expected to be very brittle [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Therefore, while a stronger martensite might be preferable for higher tensile strengths and resistance to necking from the perspective of a tensile test (Figure 1e), it may not be as beneficial in terms of crack resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' These results therefore show that the Charpy V-notch impact properties of medium Mn steels appear to be TRIP- limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The morphology, composition, strength and duc- tility of the martensite phase heavily influence the crack propagation energy during the impact test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' While not in- vestigated in this study, the TWIP effect would therefore only be expected to play a limited role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' On the other hand, there is a growing body of litera- ture demonstrating segregation of elements to certain in- terfaces such as PAGBs [26] or δ-ferrite boundaries [49] leading to poor cohesion and reduced impact properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' APT was conducted on the Low C sample and Figure 4 shows a ferrite/austenite boundary in a needle lifted from a PAGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The results do not show any concentration spike of Mn, C or any other tramp elements to the identified boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This gives confidence that segregation does not always occur in medium Mn steels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Segregation of ele- ments such as Mn and C also appears to be a time-related issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' For medium Mn steels where segregation was identi- fied [26, 31, 49], the IA duration was ≤ 1 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In this study, the Low C steel was intercritically annealed for 24 h to replicate the batch annealing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' This suggests that batch annealed medium Mn steels might be less suscepti- ble to segregation related embrittlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Conclusions The Charpy impact properties of two different medium Mn steels with different microstructures, tensile properties and compositions were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Several key findings are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Both the Low C and High C steels exhibited the TRIP effect along the fracture edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' However, the Low C steel had a significantly higher absorbed Charpy impact energy compared to the High C steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The reasons for which could be attributed to microstruc- ture and C content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' A lamellar microstructure absorbs more energy dur- ing crack propagation compared to a mixed equiaxed + lamellar microstructure by acting as a laminate composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' The austenite within the stress field trans- forms into martensite and reinforces the ferrite ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Austenite containing a high C content consequently transforms to high C martensite, which is strong but very brittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Formation of high C martensite might be beneficial in a tensile test but might be deletrious in a Charpy impact test especially if the martensite grains are able to form a continuous network in the microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Apart from Mn partitioning effects, segregation of solute, interstitial and tramp elements to the PAGB were not detected in the Low C steel which may be attributed to long intercritical annealing durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Acknowledgements TWJK gratefully acknowledges the provision of a stu- dentship from A*STAR, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' We gratefully acknowl- edge the Engineering and Physical Science Research Coun- cil for funding the Imperial Centre for Cryo Microscopy of Materials at Imperial College London (EP/V007661/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' References References [1] Miller RL.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Physical Metallurgy of Batch Annealed Medium-Mn Steels for Physical Metallurgy of Batch Annealed Medium-Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' In 20th Anniversary Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' 2019 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utAzT4oBgHgl3EQfPvvS/content/2301.01190v1.pdf'} +page_content=' [10] Taylor T, Clough A.' metadata={'source': 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a/xNE0T4oBgHgl3EQfcQCC/content/tmp_files/2301.02360v1.pdf.txt b/xNE0T4oBgHgl3EQfcQCC/content/tmp_files/2301.02360v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c249c82d83988f67d407c9f2d175ab5362bd168 --- /dev/null +++ b/xNE0T4oBgHgl3EQfcQCC/content/tmp_files/2301.02360v1.pdf.txt @@ -0,0 +1,2458 @@ +1 +Algorithm Unrolling-Based +Distributed Optimization for RIS-Assisted +Cell-Free Networks +Wangyang Xu, Jiancheng An, Member, IEEE, Hongbin Li, Fellow, IEEE, +Lu Gan, and Chau Yuen, Fellow, IEEE +Abstract +The user-centric cell-free network has emerged as an appealing technology to improve the next- +generation wireless network’s capacity thanks to its ability to eliminate inter-cell interference effectively. +However, the cell-free network inevitably brings in higher hardware cost and backhaul overhead as a +larger number of base stations (BSs) are deployed. Additionally, severe channel fading in high-frequency +bands constitutes another crucial issue that limits the practical application of the cell-free network. +In order to address the above challenges, we amalgamate the cell-free system with another emerging +technology, namely reconfigurable intelligent surface (RIS), which can provide high spectrum and energy +efficiency with low hardware cost by reshaping the wireless propagation environment intelligently. To this +end, we formulate a weighted sum-rate (WSR) maximization problem for RIS-assisted cell-free systems +by jointly optimizing the BS precoding matrix and the RIS reflection coefficient vector. Subsequently, we +transform the complicated WSR problem to a tractable optimization problem and propose a distributed +cooperative alternating direction method of multipliers (ADMM) to fully utilize parallel computing +resources. Inspired by the model-based algorithm unrolling concept, we unroll our solver to a learning- +based deep distributed ADMM (D2-ADMM) network framework. To improve the efficiency of the +D2-ADMM in distributed BSs, we develop a monodirectional information exchange strategy with a +small signaling overhead. In addition to benefiting from domain knowledge, D2-ADMM adaptively +learns hyper-parameters and non-convex solvers of the intractable RIS design problem through data- +driven end-to-end training. Finally, numerical results demonstrate that the proposed D2-ADMM achieve +W. Xu and L. Gan are with the School of Information and Communication Engineering, University of Electronic Science and +Technology of China, Chengdu, Sichuan, 611731, China (e-mail: wangyangxu@std.uestc.edu.cn; ganlu@uestc.edu.cn). +H. Li is with the Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, +USA (e-mail: hli@stevens.edu). +J. An and C. Yuen are with Engineering Product Development (EPD) Pillar, Singapore University of Technology and Design, +Singapore 487372, Singapore (e-mail: jiancheng an@sutd.edu.sg; yuenchau@sutd.edu.sg). +arXiv:2301.02360v1 [eess.SP] 6 Jan 2023 + +2 +around 210% improvement in capacity compared with the distributed noncooperative algorithm and +almost 96% compared with the centralized algorithm. +Index Terms +Cell-free system, reconfigurable intelligent surface, distributed cooperative design, algorithm un- +rolling. +I. INTRODUCTION +Next-generation wireless communication systems are expected to meet an even greater demand +for higher capacity, denser connectivity, and broader coverage with the advent of the internet of +everything [1]–[4]. The conventional communication network relies on cellular topology where +effective communication paradigms, such as small-cell network and cellular massive multiple- +input multiple-output (MIMO) are developed based on cell-centric principles [5], [6]. Specifically, +a single base station (BS) serves all users in the same cell while appropriate resource reuse +policies are adopted among different cells. As a result, users at the cell edge are more likely to +be disturbed by the uplink/downlink signals from other adjacent cells, resulting in the common +issue of inter-cell interference [7]. +It has been demonstrated that small-cell network can achieve better energy efficiency than +cellular massive MIMO in some typical scenarios by properly reducing the cell size [8], [9]. +However, as the cell density increases, the inter-cell interference will increase accordingly and +become the main bottleneck limiting the capacity of the cellular network [10]. Although cellular +massive MIMO is not affected by the inter-cell interference, the shadow fading due to blocking +will become a performance-limiting factor if a large number of antennas are centralizedly +configured on a single BS. Therefore, cellular massive MIMO’s coverage and network capacity +may be significantly deteriorated in some harsh environments [11]. +In sharp contrast to the aforementioned cell-centric networks, a user-centric network paradigm +known as the cell-free massive MIMO network has recently received significant attention as a +potential and cutting-edge substitute [7], [12], [13]. In a cell-free massive MIMO network, a large +number of antennas is spread on numerous BSs in a distributed form [14]. These BSs provide +service to a relatively small number of users within the same time-frequency domain. Since cell- +free massive MIMO removes the underlying cell edge, it does not cause inter-cell interference as +existing cellular networks. Although cell-free massive MIMO has many appealing advantages, + +3 +its application in higher frequency bands of future communication systems still has to overcome +issues related to severe transmission attenuation and coverage blind spots [15], [16]. In addition, +deploying a large number of BSs also brings prohibitive hardware cost and energy consumption. +Fortunately, a promising technology, reconfigurable intelligent surface (RIS), has recently been +introduced in various communication scenarios to significantly improve the system throughput +and spectrum/energy efficiency [17]–[20]. Specifically, an RIS is a metal panel equipped with +many low-cost passive elements. The phase shifts of these passive elements can be adjusted +to achieve intelligent manipulation of the wireless environment and enhance communication +quality-of-service (QoS) [21]. In view of the superiority of the RIS and cell-free systems, it is +of interest to develop an RIS-assisted cell-free approach for future wireless communications. +A. Prior Works +Downlink precoding is crucial to unleash the full potential of cell-free networks. Currently, +most existing precoding schemes for cell-free systems can be generally classified into non- +cooperative [7], [22], [23] and cooperative [13], [24]–[26]. Non-cooperative precoding assumes +that each BS can only utilize local channel state information (CSI) acquired through uplink +channel estimation, without performing any CSI exchange among BSs. Along this direction, +some rather simple strategies such as maximum ratio transmission (MRT) [7], local zero-forcing +(ZF) [22], and local minimum mean square error (MMSE) [23] designs have been employed for +precoding. Cooperative precoding including both centralized and distributed cooperative schemes +that perform joint precoding across all BSs achieves better system performance compared with +its non-cooperative counterpart. Specifically, in the centralized scheme, BSs upload their local +CSI to the centralized processing unit (CPU) through a specific backhaul link, based on which +the precoding matrices of all BSs are jointly designed and then distributed. Most existing works +on centralized precoding concentrate on developing precoding algorithms for the CPU, such as +the centralized ZF precoding [24], [25] and the centralized MMSE precoding [13]. Distributed +cooperative precoding distributes the computational load to multiple BSs, thus reducing the +computational burden of the CPU [27], [28]. The precoding of each BS is carried out locally +and updated based on the cross-term information exchange among different BSs to approach the +optimal performance of the centralized design. +Meanwhile, RIS reflection coefficient design has received much attention recently, e.g., [29]– +[32], by taking into account of various practical constraints and application backgrounds. How- + +4 +ever, the research on RIS-assisted cell-free network is still in its infancy stage [33]–[37]. Specif- +ically, the authors of [34] used the conjugate beamforming method with the local CSI to design +precoding vectors along with randomly adjusted RIS reflection coefficient vector to illustrate +the performance gain of the cell-free system. By assuming that BSs send their local CSI to the +CPU, the authors of [35], [36] adopted alternating optimization algorithms to jointly design the +BS precoding and the RIS reflection coefficient vector. Note that the works mentioned above are +based on either a non-cooperative scheme, which requires no CSI exchange but yields inferior +performance, or a centralized scheme, which trades system complexity for better performance. +Although recently [37] proposed a distributed cooperative optimization method for RIS-assisted +cell-free system, it has to perform a set of iterations at different BSs without fully taking +advantage of the distributed parallel computing capabilities of cell-free systems. +Also, most existing research efforts on RIS-assisted cell-free systems are focused on developing +iterative optimization algorithms, which are based on some sophisticated models derived from +the underlying physical processes or through handcrafting [38], [39]. On the contrary, deep +learning (DL) methods attempt to automatically infer model information and network parameters +directly from training data [19], [40]. Therefore, DL is very promising for scenarios where the +environment is complex and the system model is challenging to be constructed explicitly [41]. +In addition, the number of layers of most neural networks is much fewer than the number +of iterations incurred by typical iterative algorithms, which allows DL methods to attain a +faster inference speed. Nevertheless, neural networks are often trained as a “black-box” with +poor interpretability and lack essential domain knowledge that is beneficial for generalization. +Therefore, combining conventional iterative algorithms and raw data-driven DL has become +a new surge of research. Recently, an appealing concept called algorithm unrolling has been +proposed, which unrolls iteration-based algorithms into learning-based neural network structures +[42]–[51]. Such a unfolding process can not only integrate domain knowledge but also learn +complex mapping functions and hyper-parameters from input data. Specifically, each step in the +traditional iterative algorithm is unrolled into a layer or a block of the neural network. Different +network layers or blocks are cascaded to form a holistic neural network framework for solving +the original problem more efficiently. The algorithm unrolling methods have shown advantages +in many application domains, such as computational imaging [48], [49], speech processing [50], +and remote sensing [51]. + +5 +TABLE I +CONTRAST OF THE PROPOSED D2-ADMM TO EXISTING WORKS FOR RIS-ASSISTED CELL-FREE SYSTEMS +Features +Proposed +[34] +[35] +[36] +[37] +Optimization objective +WSR +ASR +EE +WSR +WSR +BS precoding design +DL +Local MRT +IA +PDS +ADMM +RIS passive beamforming design +DL +Random +IA +PDS +MM +Centralized design +× +× +✓ +✓ +× +Distributed design +Noncooperative +× +✓ +× +× +× +Cooperative +✓ +× +× +× +✓ +Convergence speed +Fast +N/A +Moderate +Moderate +Slow +WSR: weighted sum-rate; ASR: average sum-rate; EE: energy efficiency; IA: inner approximation; +PDS: primal-dual subgradient; MM: majorization-minimization; +B. Contributions +Targeting RIS-assisted cell-free systems, we design a fully distributed joint BS precoding and +RIS reflection coefficient optimization scheme based on the alternating direction method of mul- +tipliers (ADMM). Furthermore, we unroll the proposed solver to a learning-based neural network +to attain better convergence and system performance. More specifically, the main contributions +of this paper in contrast with existing works are shown in Table I and further summarized as +follows: +• We propose a distributed RIS-assisted cell-free system, where multiple energy-efficient RISs +are deployed to assist in the downlink communications from a set of distributed BSs to +multiple users (UEs). A distributed cooperative BS precoding and RIS reflection coefficient +design scheme is developed to make full use of distributed computing resources. +• Furthermore, we propose a distributed design based on ADMM that iteratively updates +the corresponding auxiliary variables, BS precoding, RIS reflection coefficient vectors, and +multipliers involved. The proposed design considers the consensus problem when separately +designing the RIS reflection coefficients at each BS in parallel. +• We unroll the proposed distributed ADMM design into a learning-based deep distributed +ADMM (D2-ADMM) neural network structure, which consists of a cascade of multiple +neural blocks. Each neural block is designed by unfolding a single iteration of the proposed +distributed ADMM design. Moreover, an effective monodirectional information exchange + +6 +strategy with a small information exchange overhead is proposed for implementing our +algorithm. In addition to obtaining deterministic variable updating strategies from domain +knowledge, D2-ADMM adaptively learns hyper-parameters and non-convex solvers of the +RIS design problem through data-driven end-to-end training. Furthermore, D2-ADMM re- +quires only a few neural blocks to reach convergence thanks to the strong inferential +capability of DL. +• Finally, we elaborate on the training and implementation of the proposed algorithm. Numer- +ical results demonstrate that the proposed algorithm has faster convergence, less computa- +tional complexity, and better performance compared with various traditional algorithms. +C. Organization and Notations +The rest of this paper is organized as follows. Section II introduces the system model and for- +mulates the joint precoding and RIS reflection design problem in RIS-assisted cell-free systems. +In Sections III, we propose a distributed ADMM-based design by maximizing the weighted +sum-rate. Section IV presents a D2-ADMM neural network structure and a monodirectional +information exchange strategy to design the BS precoding and the RIS reflection coefficients. +Numerical results are provided in Section V. Finally, we conclude the paper in Section VI. +Notations: In this paper, scalars are denoted by italic letters. Vectors and matrices are denoted +by bold-face lower-case and upper-case letters, respectively. The superscripts (·)T and (·)H +represent the operations of transpose and Hermitian transpose. |·| denotes the absolute value +of a real number. ∥·∥ denotes the 2-norm of a vector or a matrix. Re {x} and Im {x} denote the +real and imaginary parts of the complex number x, respectively. diag (·) denotes the diagonal +operation. The distribution of a circularly symmetric complex Gaussian (CSCG) with mean v +and variance σ is denoted as ∼ CN(v, σ). log2(·) represents the logarithmic function. C denotes +the set of complex values. S denotes the set of symmetric positive definite matrices. +II. SYSTEM MODEL AND PROBLEM FORMULATION +This section starts by introducing the system model of the RIS-assisted cell-free system. In +order to design the BS precoding and RIS reflection coefficient vector, a practical weighted +sum-rate (WSR) maximization problem is formulated. + +7 +A. System Model +In this paper, we consider a downlink RIS-assisted cell-free system, as illustrated in Fig. +1, where multiple BSs and RISs are deployed in a distributed arrangement to serve all UEs +cooperatively. The sets of BSs, RISs, and UEs are defined as B = {1, 2, · · · B}, R = {1, 2, · · · R}, +and K = {1, 2, · · · K}, respectively. The number of antennas of each BS and UE is Nt and 1, +respectively. Each RIS is equipped with a rectangular metasurface having N passive reflecting +elements. +As shown in Fig. 1, each RIS builds one virtual channel consisting of the BS-RIS and RIS-UE +channels between a BS and a UE to assist in the downlink communication. In this paper, the +BS-RIS channel between the b-th BS and the r-th RIS is denoted by Gb,r ∈ CN×Nt. The RIS-UE +channel between the r-th RIS and the k-th UE is denoted by vH +r,k ∈ C1×N. Moreover, the direct +channel between the b-th BS and the k-th UE is denoted by hH +b,k ∈ C1×Nt. We consider that the +proposed system operates in the mmWave band, where Gb,r, vr,k, and hb,k are described by the +Saleh-Valenzuela model [52], which are expressed as +Gb,r = +� +NtN +LG +LG +� +l=1 +βb,r,laP (ψb,r,l, ςb,r,l) aH +L (χb,r,l) , +vr,k = +� +N +Lv +Lv +� +l=1 +βr,k,laP (ψr,k,l, ςr,k,l) , +hb,k = +� +Nt +Lh +Lh +� +l=1 +βb,k,laL (ψb,k,l) +(1) +respectively, where LG, Lv, and Lh denote the multi-path number of Gb,r, vr,k, and hb,k, +respectively. ψ∗,∗,l(ς∗,∗,l), and χ∗,∗,l denote the azimuth (elevation) angles of arrival (AoAs), and +azimuth angles of departure (AoDs), where ∗ represents the index of the BS, the RIS element, and +the UE, respectively. βb,r,l ∼ CN(0, PLb,r,l), βr,k,l ∼ CN(0, PLr,k,l), and βb,k,l ∼ CN(0, PLb,k,l) +denote the corresponding complex-valued path gain, where PL represents the path loss. Besides, +aL (ς) and a (ψ, ς) denote the array response vectors of uniform linear array (ULA) and uniform +planar array (UPA), which are defined as +aL (ψ) = +1 +√NL +[1, · · ·, ejπnl sin ψ, · · ·, ejπ(NL−1) sin ψ]T, +(2) +aP (ψ, ς) = +1 +√ +NxNy +� +1, · · ·, ejπ(nx sin ψ sin ς+ny cos ς), · · ·, ejπ((Nx−1) sin ψ sin ς+(Ny−1) cos ς)�T, +(3) + +8 +BS 1 +BS b +BS B +RIS 1 +RIS r +RIS R +UE 1 +UE k +UE K +, +b r +G +, +H +b k +h +, +H +r k +v +Fig. 1. The downlink RIS-assisted cell-free system. +respectively, where NL and nl denote the total antenna number and the antenna index of ULA; +Nx, Ny, nx, and ny represent the horizontal antenna number, the vertical antenna number, the +horizontal antenna index, and the vertical antenna index of UPA, respectively. +In the downlink transmission, the transmitted symbol xb ∈ CNt×1 at the b-th BS is defined as +xb = +K +� +k=1 +wb,ksk, b ∈ B, +(4) +where sk is the transmitted symbol for the k-th UE. Thus, we have s = [s1, s2, · · · , sK]T ∈ CK×1 +representing the transmitted symbol vector that satisfies E[ssH] = IK; wb,k ∈ CNt×1 denotes the +precoding vector at the b-th BS for the k-th UE. +At each UE, the received signal component corresponding to one BS includes two parts, +one is that directly propagated from the BS to the UE, while the other is that superimposing +mutiple signal copies reflected by R RISs. Hence, the received signal component at the k-th UE +corresponding to the b-th BS can be expressed as +yb,k = +� +hH +b,k + +R +� +r=1 +vH +r,kΘH +r Gb,r +� +K +� +k=1 +wb,ksk += +� +hH +b,k + θHVH +k Gb +� +xb += ˆhH +b,kxb, +(5) +where Θr = diag([ejϕr,1, ejϕr,2, · · · , ejϕr,N]T) ∈ CN×N is the reflection coefficient matrix of +the r-th RIS; ϕr,n is the phase shift imposed by the n-th element of the r-th RIS; θ +∆= +e +� +j[ϕ1,1,··· ,ϕ1,N,ϕ2,1,··· ,ϕR,N] +T � +∈ CNR×1 denotes the phase shift vector of R RISs; Vk +∆= diag([vT +1,k, vT +2,k, + +9 +· · · , vT +R,k]) ∈ CNR×NR represents the equivalent channel from R RISs to the k-th UE; Gb = +[GT +b,1, GT +b,2, · · · , GT +b,R]T ∈ CNR×Nt denotes the equivalent channel from the b-th BS to R RISs; +ˆhb,k is the composite channel from the b-th BS to the k-th UE, incorporating one direct and NR +reflected channels. +We assume that all BSs are synchronized to ensure joint service for all UEs in the same +time-frequency resource block. Therefore, the received signal at the k-th UE is the superposition +of the signals transmitted from all BSs, which can be expressed as +yk = +B +� +b=1 +yb,k + zk += +B +� +b=1 +K +� +j=1 +ˆhH +b,kwb,jsj + zk += +B +� +b=1 +ˆhH +b,kwb,ksk +� +�� +� +Desired signal ++ +B +� +b=1 +K +� +j=1,j̸=k +ˆhH +b,kwb,jsj +� +�� +� +Interference of other UEs ++zk, +(6) +where zk ∼ CN(0, δ2 +k) denotes the additive white Gaussian noise (AWGN). Without loss of +generality, we assume that all UEs have the same noise power, i.e., δ2 +k = δ2, ∀k ∈ K. +B. Problem Formulation +Based on the signal model expressed in (6), the signal-to-interference-plus-noise ratio (SINR) +ζk of the k-th UE can be written as +ζk = +���� +B� +b=1 +ˆhH +b,kwb,k +���� +2 +K +� +j=1,j̸=k +���� +B� +b=1 +ˆhH +b,kwb,j +���� +2 ++ δ2 +. +(7) +To evaluate the performance of the RIS-assisted cell-free system, the WSR is given as +WSR = +K +� +k=1 +ωklog2 (1 + ζk), +(8) +where ωk > 0 is the weight of the k-th UE, which indicates the priority of different UEs. + +10 +In this paper, we endeavor to maximum the WSR of the RIS-assisted cell-free system by +designing the BS precoding W = {wb,k|∀b ∈ B, ∀k ∈ K} and the RIS reflection coefficient +vector θ. Mathematically, the optimization problem can be formulated as +P0 : +max +θ,W +WSR = +K +� +k=1 +ωklog2 (1 + ζk) +(9a) +s.t. +K +� +k=1 +∥wb,k∥2 ≤ Pb,max, ∀b ∈ B, +(9b) +|θr,b| = 1, ∀r ∈ R, ∀n ∈ N, +(9c) +where (9a) is the WSR objective function; (9b) is the power constraints of BSs, where Pb,max +denotes the maximum transmit power budget at the b-th BS. Constraint (9c) represents that the +amplitude of reflection coefficient of each RIS remains constant in this paper. +Remark 1: Although the centralized algorithm can achieve the optimal solution to P0 [36], +it requires collecting the local CSI of all BSs for joint optimization at the CPU. This inevitably +increases both the CSI feedback and control signaling overhead as well as the computational +complexity of the CPU. Therefore, we aim to develop a distributed algorithm to solve P0 by +spreading the computational load to the distributed BSs. +Therefore, the distributed optimation problem of P0 is rewritten as +P1 : +max +θ,W +WSR = +K +� +k=1 +ωklog2 (1 + ζk) +(10a) +s.t. (9b), (9c), +θb = θ¯b, ∀b ∈ B, ∀¯b ∈ Fb, +(10b) +where (10b) is the consensus constraint, which means that θb optimized at adjacent BSs should +be consistent. Fb represents the index set of the adjacent BSs that can exchange information +with the b-th BS. Specifically, the b-th BS requires utilizing the information from the adjacent +BSs when designing the RIS reflection coefficients. Then, it sends its local information to the +adjacent BSs until the RIS reflection coefficients on all BSs reach a consensus. +Remark 2: Here we highlight that the optimization of Wb and θb in a distributed system +are distinctly different. Specifically, the downlink precoding matrix Wb is unique for different +BSs. By contrast, θb optimized by different BSs correspond to the same RIS and need to be +appropriately fused into a single reflection coefficient vector, which is known as the consensus + +11 +problem in distributed systems. Although the centralized algorithm does not involve the consensus +problem, the distributed optimization strategy is more effective and practical considering the +distributed deployment of BSs as well as the limited backhaul capacity. +III. ADMM-BASED DISTRIBUTED OPTIMIZATION +In this section, we propose a distributed ADMM-based design for effectively optimizing the +precoder and reflection phase shifts in the practical RIS-assisted cell-free system. Specifically, +we first convert the non-convex P1 into a tractable form P2. Then, we propose a distributed +ADMM design to solve P2. +A. A Tractable Form of P1 +Observe form (10a) that P1 is a non-convex optimization problem due to the coupling of the +optimization variables W and θ and the consensus constraint (10b). Therefore, we transform P1 +into a tractable problem by applying the Lagrangian dual transform and the quadratic transform, +which are summarized in Lemmas 1 and 2, respectively. +Lemma 1 (Lagrangian dual transform): Given a sum-of-logarithmic-ratios problem, expressed +as [53] +max +x +D +� +d=1 +ωdlog2 +� +1 + Qd (x) +Fd (x) +� +(11a) +s.t. x ∈ χ, +(11b) +where ωd is a nonnegative weight; Qd (x) is a nonnegative function that satisfies Qd (x) ≥ 0; +Fd (x) is a positive function with Fd (x) > 0; x is the optimization variable, and χ denotes a +nonempty constraint set. Moving the ratio from inside of the logarithm to the outside, (11a) can +be rewritten as +min +x,γ +D +� +d=1 +ωd +� +γd − log2 (1 + γd) − (1 + γd) Qd (x) +Qd (x) + Fd (x) +� +(12a) +s.t. x ∈ χ, γd ≤ Qd(x) +Fd(x) , +(12b) +where γ = [γ1, γ2, · · · γD]T is the auxiliary variable vector. + +12 +Lemma 2 (Quadratic transform): Given a sum-of-functions-of-ratio problem for the multidi- +mensional and complex cases, expressed as [54] +min +x +D +� +d=1 +¯fd +� +Qd (x) F −1 +d +(x) QH +d (x) +� +(13a) +s.t. x ∈ χ, +(13b) +where function Qd (x) : Cd1 → Cd2, Fd (x) Cd1 → Sd2×d2, x, and constraint χ ⊆ Cd1. Let ¯fd (·) +denotes a monotonically nondecreasing function, problem (13) can be transformed to +min +x,η +D +� +d=1 +¯fd +� +2Re {ηdQd (x)} − η2 +dFd (x) +� +(14a) +s.t. x ∈ χ, ηd ∈ Cd1, +(14b) +where η = [η1, η2, · · · ηD]T denotes the auxiliary variable vector. +Therefore, by using the Lagrangian dual transform, P1 can be reformulated as +LP1 : +min +θ,W,γ f1 (θ, W, γ) +(15a) +s.t. (9b), (9c), (10b), +where f1 (θ, W, γ) is the new objective function via Lemma 1, which is described in (16). +Besides, γ = [γ1, γ2, · · · γK]T represents the auxiliary variable vector. +f1 (θ, W, γ) = +K +� +k=1 +ωk +� +� +� +� +�γk − log2 (1 + γk) − +(1 + γk) +���� +B� +b=1 +ˆhH +b,kwb,k +���� +2 +K +� +k=1 +���� +B� +b=1 +ˆhH +b,kwb,k +���� +2 ++ δ2 +� +� +� +� +�. +(16) +Then we use the quadratic transform shown in Lemma 2 to decouple the numerator and the +denominator of the fraction in LP1 to further simplify the optimization. Consequently, LP1 can +be transformed as +LP2 : +min +θ,W,γ,η f2 (θ, W, γ, η) +(17a) +s.t. (9b), (9c), (10b), +where f2 (θ, W, γ, η) is given in (18); η = [η1, η2, · · · ηK]T denotes the auxiliary variable vector. +f2 (θ, W, γ, η) = +K +� +k=1 +(|ηk|2 +K +� +j=1 +����� +B +� +b=1 +ˆhH +b,jwb,k +����� +2 ++ |ηk|2δ2 + ωkγk +− 2 +� +(1 + γk) ωk +B +� +b=1 +Re +� +ηkˆhH +b,kwb,k +� +− ωklog2 (1 + γk) . +(18) + +13 +Next, we rewrite problem LP2 in its augmented Lagrangian form, expressed as +L (θ, W, γ, η, λ) = f2 (θ, W, γ, η) + +B +� +b=1 +µb +� K +� +k=1 +∥wb,k∥2 − Pb,max +� +� +�� +� +Power constraint ++ +B +� +b=1 +I (θb) +� +�� +� +Feasible constraint ++ +B +� +b=1 +ρb +2 +����θb − θ¯b + λb +ρb +���� +2 +� +�� +� +Consensus constraint +. +(19) +where λ = +� +λb ∈ CN×1|∀b ∈ B +� +is the Lagrange multiplier and ρb > 0. I (·) represents a +feasible function such that I (θb) = 0 for |θb| = 1 and I (θb) = ∞ for |θb| ̸= 1. As a result, the +final tractable form of P1 is given as +P2 : +min +θ,W,γ,η,λ L (θ, W, γ, η, λ) +(20) +B. Proposed Distributed Design Based on ADMM +To solve the problem P2, we propose a distributed design based on ADMM [55], [56]. The +proposed design iteratively designs the local BS precoding and RIS reflection coefficient vectors +at each BS. Specifically, the i-th iteration at the b-th BS can be expressed as +γi = arg min +γ f1 +� +θi−1, Wi−1, γ +� +, +(21a) +ηi = arg min +η L +� +θi−1, Wi−1, γi, η, λi−1� +, +(21b) +Wi +b = arg min +Wb L +� +θi−1, ¯ +Wi−1 +b +, Wb, γi, ηi, λi−1� +, +(21c) +θi +b = arg min +θb L +� +¯θ +i−1 +b +, θb, ¯ +Wi−1 +b +, Wi +b, γi, ηi, λi−1� +, +(21d) +λi +b = λi−1 +b ++ ρb +� +θi +b − θi +¯b +� +, +(21e) +where Wb = {wb,k|∀k ∈ K} denotes the local precoding at the b-th BS; ¯ +Wb = W\Wb and +¯θb = θ\θb are the downlink precoding and the RIS reflection coefficient vector of other BSs +except the b-th BS; Observe from (21) that γ, η, Wb, θb, and λb are updated locally in sequence. +Next, we give the solutions to problems (21a)-(21d) one by one. Note that P2 is an equivalence +problem to LP1, which means that the optimal γ of LP1 is equal to ones of P2. Besides, it is +easier to solve LP1 than P2 for optimal γ. Therefore, we solve LP1 for optimal γ. + +14 +1) The Solver of (21a): For problem (21a), f1 (θ, W, γ) is a convex function for γ with fixed +θ and W. Therefore, the optimal γk can be obtained by taking ∂(f1(γ)) +∂γk += 0. Thus, we have +γ† +k = +���� +B� +b=1 +ˆhH +b,kwb,k +���� +2 +K +� +j=1,j̸=k +���� +B� +b=1 +ˆhH +b,kwb,j +���� +2 ++ δ2 += +|ϖk,k + ϑk,k|2 +K +� +j=1,j̸=k +|ϖk,j + ϑk,j|2 + δ2 +, +(22) +where ϖk,j = +B� +b=1 +hH +b,kwb,j and ϑk,j = +B� +b=1 +θHVH +k Gbwb,j are two defined cross-term information, +which contain the information of all BSs. Note that {ϖk,j|∀k, j ∈ K} and {ϑk,j|∀k, j ∈ K} are +then exchanged among different BSs to achieve the goal of cooperative design. +2) The Solver of (21b): For problem (21b), we note that only f2 (θ, W, γ, η) in (19) is +dependent on η. Therefore, problem (21b) can be reformulated as +η = arg min +η +K +� +k=1 +� +�|ηk|2 +� +� +K +� +j=1 +����� +B +� +b=1 +ˆhH +b,jwb,k +����� +2 ++ δ2 +� +� +− +� +(1 + γk) ωk +� B +� +b=1 +� +ηkˆhH +b,kwb,k + η∗ +kwH +b,kˆhb,k +��� +. +(23) +The optimal η† +k can also be obtained by taking ∂(f2(η)) +∂η∗ +k += 0, which is expressed as +η† +k = (ϖk,k + ϑk,k)∗� +(1 + γk) ωk +K +� +j=1 +|ϖk,j + ϑk,j|2 + δ2 +. +(24) +3) The Solver of (21c): Given a set of tentative values of other variables, we have +L (wb,k) = +K +� +k=1 +(|ηk|2 +K +� +j=1 +� +� +������ +� +� +B +� +b′̸=b +ˆhH +b′,jwb′,k +� +� wH +b,kˆhb,j +������ +2 ++ +��hH +b,jwb,k +��2 +� +� +− +� +(1 + γk) ωkRe +� +ηkˆhH +b,kwb,k +� +) + µb∥wb,k∥2 + C1. +(25) +where C1 is defined by +C1 = +K +� +k=1 +(|ηk|2 +K +� +j=1 +B +� +b′̸=b +���ˆhH +b′,jwb′,k +��� +2 +K +� +j=1 +−2 +� +(1 + γk) ωk +B +� +b′̸=b +Re +� +ηkˆhH +b′,kwb′,k +� ++|ηk|2δ2 + ωkγk +− ωklog2 (1 + γk)) + +B +� +b′̸=b +µb′ +� K +� +k=1 +��wb′,k +��2 − Pb,max +� ++ µb +� +� +K +� +k′̸=k +��wb,k′��2 − Pb,max +� +� ++ +B +� +b=1 +I (θb) + +B +� +b=1 +ρb +2 +����θb − θ¯b + λb +ρb +���� +2 +. +(26) + +15 +Similarly, the optimal w† +b,k can be obtained by taking +∂(L(wb,k)) +∂wH +b,k += 0, which is given as +w† +b,k = +� +(1 + γk) ωkη∗ +kˆhb,k − Ωb,k +� +µb + |ηk|2 K +� +j=1 +ˆhb,jˆhH +b,j +� , +(27) +where Ωb,k = |ηk|2 K +� +j=1 +ˆhb,k(ϖj,k + ϑj,k − ˆhH +b,jwb,k); µb is a normalized factor used to scale wb,k +for satisfying the total power constraint, which needs to be dynamically updated in each iteration. +We apply the power normalization approach below to bypass the update of µb in D2-ADMM. +Specifically, wl +b,k is scaled by +w† +b,k = +� +Pb,maxw† +b,k +� +K +� +k=1 +���w† +b,k +��� +2 +. +(28) +4) The Solver of (21d): For problem (21d), we first rewrite (19) in a more intuitive form as +L (θb) = θH +b Sθb − 2Re +� +θH +b Z +� ++ +B +� +b=1 +I (θb) + C2, +(29) +where S, Z, and C2 are independent of θb and given in (30), (31), and (32) respectively. Note +that (29) is a non-convex problem due to the feasible constraint. To solve this problem, we build +a neural block based on DL, and the details will be discussed in Section IV. +S = +K +� +k=1 +|ηj|2 +K +� +j=1 +VH +j Gbwb,kwH +b,kGH +b Vj + ρb +2 . +(30) +Z = +K +� +k=1 +|ηk|2 +K +� +j=1 +VH +j Gbwb,k +� +wH +b,kˆhb,j − ϖ∗ +j,k − ϑ∗ +j,k − wH +b,khb,j +� ++ +� +(1 + γk) ωkηkVH +k Gbwb,k. +(31) +C2 = +K +� +k=1 +� +�|ηk|2 +K +� +j=1 +� +� +B +� +b′̸=b +���ˆhH +b′,jwb′,k +��� +2 ++ 2Re +� +� +�hH +b,jwb,k +� +� +B +� +b′̸=b +wH +b′,kˆhb′,j +� +� +� +� +� + +��hH +b,jwb,k +��2 +� +� +− +� +(1 + γk) ωk +B +� +b′̸=b +Re +� +ηkˆhH +b′,kwb′,k +� ++ ωkγk − ωklog2 (1 + γk) + |ηk|2δ2 +� +� ++ +B +� +b′̸=b +ρb′ +2 +����θb′ − θ¯b′ + λb′ +ρb′ +���� +2 ++ρb +2 +���� +λb +ρb +− θ¯b +���� +2 +. +(32) + +16 +Ini-Block +Mid- +Block1 +Mid- +Block2 +Mid- +Block3 +Out- +Block +BS b +BS b+1 +, +, +, +b +k +b k +G +V h +1 +1 +, +, +b +b k + +w +2 +2 +, +b +b + + +0 +b + +0 +, +b k +w +1 +1 +1 +1 +, +, +l +l +b +b k + +− +− +w +1 +1 +, +l +l +b +b + + +1 +1 +, +, +l +l +b +b k + +w +2 +2 +, +l +l +b +b + + +1 +1 +1 +1 +, +l +l +b +b + + ++ ++ +3 +3 +1 +1 +, +, +l +l +b +b k + +− +− +w +2 +2 +1 +1 +, +, +l +l +b +b k + +− +− +w +2 +2 +, +, +l +l +b +b k + +w +2 +2 +1 +1 +, +l +l +b +b + + ++ ++ +3 +3 +, +l +l +b +b + + +3 +3 +, +, +l +l +b +b k + +w +1 +1 +, +, +L +L +b +b k + +− +− +w +3 +3 +1 +1 +, +l +l +b +b + + ++ ++ +, +L +L +b +b + + +, +, +L +L +b +b k + +w +1 +1ˆ +ˆ , +b +b +  +1 +1ˆ +ˆ , +l +l +b +b + + +2 +2ˆ +ˆ , +l +l +b +b + + +3 +3ˆ +ˆ , +l +l +b +b + + +3 +3 +1 +1 +ˆ +ˆ +, +l +l +b +b + + ++ ++ +2 +2 +1 +1 +ˆ +ˆ +, +l +l +b +b + + ++ ++ +1 +1 +1 +1 +ˆ +ˆ +, +l +l +b +b + + ++ ++ +1 +1 +1 +1 +ˆ +ˆ +, +b +b + + ++ ++ +BS b-1 +{ +B +1 +1 +b + + +1 +1 +l +b + + +1l +b + +2l +b + +3l +b + +3 +1 +l +b + + +2 +1 +l +b + + +1 +b + +Fig. 2. The structure of D2-ADMM. +IV. D2-ADMM: A LEARNING-BASED ALGORITHM UNROLLING METHOD +This section proposes a D2-ADMM neural network structure to design the BS precoding and +the RIS reflection coefficient by unfolding the proposed distributed ADMM design. Furthermore, +an efficient monodirectional information exchange strategy is proposed to link different BSs to +improve the performance of our distributed designs. Finally, we elaborate on the training and +the implementation of D2-ADMM. +A. Structure of Deep Distributed ADMM +The proposed distributed ADMM design iteratively updates auxiliary variables, BS precoding, +RIS reflection coefficient vectors, and multipliers. However, it has high computational complexity +since the conventional distributed ADMM may take hundreds or thousands of iterations to achieve +convergence. System performance and convergence are additionally hampered by the requirement +to manually choose crucial hyper-parameters, such as the power normalized factor {µb|∀b ∈ B} +and the penalty factor {ρb|∀b ∈ B}. To overcome these shortcomings, we unfold the proposed +distributed ADMM design into the D2-ADMM to learn the hype-parameters {ρb|∀b ∈ B} auto- +matically and bypass {µb|∀b ∈ B}. Besides, we create a neural block called θ-Block to solve +the complicated problem (21d). The specific D2-ADMM structure is illustrated in Fig. 2. +As shown in Fig. 2, a total of B D2-ADMM are respectively implemented at B BSs. A D2- +ADMM is composed of L ≥ B cascaded neural blocks. Each neural block is designed according +to one iteration of the distributed ADMM design, which means that a neural block is equivalent to +a single iteration in traditional iterative algorithms. The (l−1)-st neural block’s output constitutes + +17 +I-Layer +A-Layer +W-Layer +I1-Layer + + -Layer + +0 +b + +0 +, +b k +w +1 +1 +, +b +b +  +2 +2 +, +b +b + + +1 +1 +, +, +, +b k +b k + + +In +1 +1 +b + + +1 +b + +1 +b + +1 +1 +1 +1 +ˆ +ˆ +, +b +b + + ++ ++ +1 +1ˆ +ˆ , +b +b +  +1 +, +b k +w +1 +, +b k +w +1 +b + +-Block +, +, +, +b +k +b k +G +V h +Fig. 3. The structure of Ini-Block relying on algorithm unrolling. +the input of the l-th neural block. The input of the first neural block is initialized, and the last +neural block outputs the optimized BS precoding matrix and RIS reflection coefficient vectors. +More specifically, we have five different kinds of neural blocks, namely the initialization neural +block (Ini-Block), the middle neural block 1 (Mid-Block1), the middle neural block 2 (Mid- +Block2), the middle neural block 3 (Mid-Block3), and the output neural block (Out-Block). For +the sake of illustration, we give the schematic diagram of the Ini-Block in Fig. 3. The structures +of other neural blocks are based on the Ini-Block by replacing or pruning certain parts. Ini-Block +initializes the network, which includes a cross-term information initialization layer (I-Layer), an +auxiliary variable update layer (A-Layer), a BS precoding update layer (W-Layer), an RIS update +block (θ-Block), a multiplier update layer (λ-Layer), and a cross-term information exchange layer +1 (I1-Layer). The first neural block of D2-ADMM is an Ini-Block. The 2nd to (B−1)-st network +blocks of D2-ADMM are created as the Mid-Block1, which contains an A-Layer, a W-Layer, +a θ-Block, a λ-Layer, and a I1-Layer. The B-th neural block of D2-ADMM is Mid-Block2, +which has a similar structure as Mid-Block1, except that I1-Layer is replaced with a cross-term +information exchange layer 2 (I2-Layer). Moreover, the (B + 1 ∼ L − 1)-st neural blocks are +Mid-Block3, which is constructed similarly as Mid-Block2 with the exception of using a cross- +term information exchange layer 3 (I3-Layer). The last neural block of D2-ADMM is referred +to Out-Block, which consists of an A-Layer, a W-Layer, and a θ-Block. +Next, we will discuss the structure and function of each layer and the θ-Block. + +ConvCony +2Cony +318 +1) Auxiliary Variable Update Layer (A-Layer): A-Layer updates two auxiliary variables, γ +and η, according to (22) and (24). To reflect the iteration order, we rewrite (22) and (24) as +γl +b,k = +��ϖl +b,k,k + ϑl +b,k,k +��2 +K +� +j=1,j̸=k +��ϖl +b,k,j + ϑl +b,k,j +��2 + δ2 +, +(33) +ηl +b,k = +� +ϖl +b,k,k + ϑl +b,k,k +�∗�� +1 + γl +b,k +� +ωk +K +� +j=1 +��ϖl +b,k,j + ϑl +b,k,j +��2 + δ2 +, +(34) +respectively, where ϖl +b,k,j and ϑl +b,k,j denote the cross-term information of the l-th neural block +for the b-th BS; γl +b,k, and ηl +b,k are the two auxiliary variables of the l-th neural block for the b-th +BS. +2) BS Precoding Update Layer (W-Layer): According to (27), W-Layer updates the BS +precoding matrix W as +wl +b,k = +�� +1 + γl +b,k +� +ωb,k +� +ηl +k +�∗ˆhl−1 +b,k − Ωl−1 +b,k +K +� +j=1 +��ηl +b,j +��2ˆhl−1 +b,j +� +ˆhl−1 +b,j +�H +, +(35) +where Ωl−1 +b,k = +K +� +j=1 +��ηl +b,j +��2ˆhl−1 +b,k (ϖl +b,j,k+ϑl +b,j,k−(ˆhl−1 +b,j )Hwl−1 +b,k ) and ˆhl−1 +b,j = (hH +b,j + (θl−1 +b +) +HVH +k Gb)H. +In order to satisfy the power constraint, wl +b,k can be rewrited as +wl +b,k = +� +Pb,maxwl +b,k +� +K +� +k=1 +��wl +b,k +��2 +. +(36) +3) RIS Update Block (θ-Block): As previously mentioned, (29) is a non-convex function that +is challenging to solve by conventional methods. Therefore, we introduce the θ-Block, which +aims to exploit the inference ability of DL to solve this problem. θ-Block is composed of multiple +convolutional layers. Specifically, we first rewrite (29) as +∠ (θb) = fθ (S, Z) , +(37) +where fθ denotes a non-linear function that applied as the solver of problem (29). +We then use multiple convolutional layers to approximate this complicated non-linear function +fθ. Since the neural network is more amenable with real-valued data, we first convert S and Z +into real-valued sequences Inθ as the input of the θ-Block, expressed as follows. +Inθ = [Re {S} , Im {S} , Re {Z} , Im {Z}] . +(38) + +19 +Therefore, the working principle of θ-Block can be expressed as +∠ (θb) = fC,U (· · · fC,u (· · · fC,1 (Inθ|υ1) |υu) |υU) , +(39) +where fC,u is the u-th convolutional layer; U denotes the number of convolutional layers; υu is +the parameter set of the u-th convolutional layer. In this paper, we empirically choose U = 3 +which is sufficient for our problem. +Note that in the proposed architecture, the parameters of each convolutional layer can be +automatically learned through end-to-end training. +4) Multiplier Update Layer (λ-Layer): The multipliers are updated through this layer using +the following strategy +λl +b = λl−1 +b ++ ρb +� +θl +b − θl +¯b +� +, +(40) +where ρb is a learnable parameter; θl +¯b is the RIS reflection coefficient vector exchanged from +the ¯b-th BS. +5) Cross-Term Information Initialization Layer (I-Layer): Again, in the cooperative design +of distributed RIS-assisted cell-free systems, CSI sharing is necessary among BSs. However, +considering the security and the excessive overhead associated with direct CSI exchange, we +define {ϖb,k,j, ϑb,k,j|∀b ∈ B; ∀k, j ∈ K} as two types of necessary cross-information in A-layer, +W-Layer, and θ-Block. +I-Layer initializes the local cross-term information, which is expressed as +� +� +� +ˆϖ0 +b,k,j = 0, +ˆϑ0 +b,k,j = 0, +(41a) +� +� +� +ϖ1 +b,k,j = hH +b,kw0 +b,k, +ϑ1 +b,k,j = +� +θ0 +b +�HVH +k Gbw0 +b,k, +(41b) +where ˆϖ0 +b,k,j and ˆϑ0 +b,k,j are two initialized cross-term information, which will be sent to the +adjacent BSs; ϖ1 +b,k,j and ϑ1 +b,k,j denote two cross-term information, which will be used for updating +the next neural block; w0 +b,k and θ0 +b are initialized randomly. +6) Cross-Term Information Layer 1 (I1-Layer): I1-Layer includes two processes. The process +1 is to send the updated cross-term information to the adjacent BSs, expressed as (42a), where +ˆϖl +b,k,j and ˆϑl +b,k,j are two cross-term information that needs to be shared with the adjacent BSs. +Moreover, ˆϖl−1 +¯b,k,j and ˆϑl−1 +¯b,k,j represent two cross-term information symbols that are received from +the adjacent BSs. θl +b and wl +b,k denote the l-th update of the RIS reflection coefficient vector and + +20 +the BS precoding vector, respectively. In the process 2, the cross-term information required for +updating the next neural block will be determined based on the received cross-term information +from the adjacent BSs, as demonstrated in (42b), where ϖl+1 +b,k,j and ϑl+1 +b,k,j denote the two cross- +term information symbols that are required for updating the (l + 1)-st neural block. I1-Layer is +configured for the (1 ∼ B−1)-st neural blocks. +� +� +� +ˆϖl +b,k,j = ˆϖl−1 +¯b,k,j + hH +b,kwl +b,k, +ˆϑl +b,k,j = ˆϑl−1 +¯b,k,j + +� +θl +b +�HVH +k Gbwl +b,k, +(42a) +� +� +� +ϖl+1 +b,k,j = ˆϖl +¯b,k,j + hH +b,kwl +b,k, +ϑl+1 +b,k,j = ˆϑl +¯b,k,j + +� +θl +b +�HVH +k Gbwl +b,k. +(42b) +7) Cross-Term Information Layer 2 (I2-Layer): I2-Layer has the similar process 1 but distinct +process 2 as I1-Layer. Specifically, the updates of the b-th BS in the first neural block is included +in the cross-term information needed for updating the (B + 1)-st neural block. Thus, we have +to eliminate the obsolete updates from the cross-term information and add the B-th update to +guarantee that only the new update is included. The specific process 2 is expressed as follows +� +� +� +ϖB+1 +b,k,j = ˆϖB +¯b,k,j − hH +b,kw1 +b,j + hH +b,kwB +b,j, +ϑB+1 +b,k,j = ˆϑB +¯b,k,j − +� +θ1 +b +�HVH +k Gbw1 +b,j + +� +θB +b +�HVH +k GbwB +b,j. +(43) +Therefore, I2-Layer is only exploited in the B-th neural block. +8) Cross-Term Information Layer 3 (I3-Layer): When l ≥ B + 1, both the cross-term infor- +mation to be sent to the adjacent BSs and the cross-term information used for updating the next +neural block need to eliminate obsolete updates. Therefore, the two processes of I3-Layer can +be described as� +� +� +ˆϖl +b,k,j = ˆϖl−1 +¯b,k,j − hH +b,kwl−B+1 +b,j ++ hH +b,kwl +b,j, +ˆϑl +b,k,j = ˆϑl−1 +¯b,k,j − +� +θl−B+1 +b +�HVH +k Gbwl−B+1 +b,j ++ +� +θl +b +�HVH +k Gbwl +b,j, +(44a) +� +� +� +ϖl+1 +b,k,j = ˆϖl +¯b,k,j − hH +b,kwl−B+2 +b,j ++ hH +b,kwl +b,j, +ϑl+1 +b,k,j = ˆϑl +¯b,k,j − +� +θl−B+2 +b +�HVH +k Gbwl−B+2 +b,j ++ +� +θl +b +�HVH +k Gbwl +b,j. +(44b) +We deploy I3-Layer in the (B + 1 ∼ L − 1)-st neural blocks. +B. Information Exchange Strategy +Next, we elaborate on the proposed information exchange strategy. To safeguard the informa- +tion privacy of different BSs and reduce the proposed system’s information exchange overhead, + +21 +BS 1 +BS 2 +BS b +BS B +2 +2ˆ +ˆ , +l +l +  +2 +l + +1 +l + +1 +1ˆ +ˆ , +l +l +  +l +B + +1 +l +b + + +3 +l + +3 +3ˆ +ˆ , +l +l +  +ˆ +ˆ , +l +l +b +b +  +l +b + +1 +1 +ˆ +ˆ +, +l +l +b +b + + ++ ++ +ˆ +ˆ , +l +l +B +B + + +Fig. 4. The proposed information exchange strategy. +we define two types of cross-term information used for the update at each BS. The updating of +each neural block needs to guarantee the integrality and timeliness of the cross-term information, +as demonstrated by the updating process of the I1-layer, the I2-layer, and the I3-layer. In most +existing distributed information exchange strategies, each BS often receives information shared +by multiple BSs [35], [36]. This exchange strategy will reduce the integrality and timeliness of +the cross-term information defined in our paper, affecting the convergence and performance of +the system. Therefore, we propose an effective monodirectional information exchange strategy, +assuming all BSs have a monodirectional topology, as illustrated in Fig. 4. +Each BS performs a monodirectional information exchange with two adjacent BSs through +a dedicated link. For instance, the b-th BS receives cross-term information from the (b + 1)-st +BS and sends its cross-term information to the (b − 1)-st BS. Such a strategy requires at least +B exchanges to ensure the integrality of the cross-term information. As the iteration proceeds, +the timeliness of the cross-term information is guaranteed by replacing the obsolete information +with the latest information. The specific cross-term information processing are completed at the +I1-Layer, I2-Layer, and I3-Layer. +In addition to exchanging cross-term information, we also need to exchange the RIS reflection +coefficient vectors updated by each neural block among various BSs to update the multiplier +λ. Therefore, the b-th BS needs to send { ˆϖl +b,k,j|∀k, j ∈ K} (K2 dimension), {ˆϑl +b,k,j|∀k, j ∈ K} +(K2 dimension), and θl +b (RN dimension) in the l-th neural block. As a consequence, the total +dimension of exchanged data in the practical RIS-assisted cell-free system is B(L − 1)(2K2 + +RN), which is significantly reduced compared with that exchanging CSI directly. + +22 +RIS +(50,-50,3) +BS +UE +(100,-50,3) +(150,-50,3) +(200,-50,3) +(75,0,1.5) +(75,10,6) +(125,10,6) +x(m) +y(m) +(0,0,0) +Fig. 5. The 3D scenario of the RIS-assisted cell-free system. +C. Training of D2-ADMM +In this section, we give the specific training and practical application methods of the proposed +D2-ADMM. The input to D2-ADMM at the b-th BS is its local CSI, the initialized w0 +b,k and +θ0 +b, while the output is the optimized wL +b,k and θL +b . Then the parameters of θ-Layer and ρb in +D2-ADMM are updated through an end-to-end training. The loss function for training is set as +fLoss = 1 +Q +Q +� +q=1 +B +� +b=1 +��θq,b − θq,¯b +��2 +� +�� +� +Consensus error +− WSRq, +(45) +where Q is the sample number of one training batch. +By minimizing the loss function fLoss, the consensus error is minimized while maximizing +WSR. It is worth noting that the training process is completed on a single CPU. After complet- +ing the training, we deploy B D2-ADMMs to the corresponding BSs for practical distributed +implementation. +V. NUMERICAL RESULTS +This section provides simulation results to demonstrate the effectiveness of our proposed D2- +ADMM framework for the RIS-assisted cell-free system. +A. Simulation Setup +We consider a typical RIS-assisted cell-free system 3D scenario shown in Fig. 5. In this +scenario, the b-th BS is deployed at (200× b +B, −50, 3) m. Without loss of generality, we consider + +23 +R = 2 RISs, which are deployed at (75, 10, 6)m and (125, 10, 6) m. K UEs served by B BSs +are randomly distributed in a circular area with a center at (75, 0, 1.5) m, a radius of 5m, and +a height of 1.5 m. The number of antennas at each BS is set to Nt = 2. Given the location +information of each device, the corresponding channel can be determined by (1). In this setup, +we assume that the multi-path number of each channel is 3 (1 LoS, 2 NLoS), and their AOAs +and AODs are chosen randomly in the range [− π +2, π +2]. Likewise, all BSs have the same maximum +transmit power, i.e., Pb,max = P. The received noise power is set to δ2 = −80 dBm. +To better demonstrate the performance of the proposed D2-ADMM, we consider several +representative benchmarks, as listed below. +• Centralized: Assuming that all BSs send their local CSI to the central CPU for the centralized +design of the BS precoding matrix and the RIS reflection coefficient vectors [36]. +• MRT Random θ: A distributed design method, where the RIS reflection coefficient vector +is randomly configured, and the precoding of BS is designed as the conjugate of local CSI +[7]. +• MRT Comb MaxAO: A distributed algorithm that maximizes the channel gain of cascaded +channels for configuring the RIS, and the design of BS precoding is the same as MRT +Random θ. +• Local ZF Comb MaxAO: This distributed algorithm has the same design of RIS as MRT +Comb MaxAO and exploits the local ZF algorithm proposed in [57] for optimizing the BS +precoding matrix. +B. Training Performance of D2-ADMM +In order to show the convergence of D2-ADMM, we first conduct experiments to evaluate +various indicators in the training process of D2-ADMM, as shown in Figs. 6(a)-6(c), where we +set B = 4, N = 50, K = 4, P = 30 dBm. +Specifically, Fig. 6(a) illustrates the training loss of the D2-ADMM under the different number +of neural blocks. It can be seen that the D2-ADMM training loss can converge as the training +proceeds. In addition, the final convergent training loss gap for different L is negligible when +L ≥ 6. Furthermore, Fig. 6(b) shows the fluctuation of the consensus error of D2-ADMM +against different L. From Fig. 6(b), we can observe that the consensus error of D2-ADMM can +converge to a minimal value as the training proceeds, and varied L does not severely impact the +convergence result of the consensus error. The WSR in the training phase against the number + +24 +0 +50 +100 +150 +200 +-24 +-22 +-20 +-18 +-16 +-14 +-12 +-10 +-8 +-6 +(a) +0 +50 +100 +150 +200 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +(b) +0 +50 +100 +150 +200 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +(c) +Fig. 6. (a) The training loss of D2-ADMM; (b) The consensus error of D2-ADMM in the training process; (c) The WSR of +D2-ADMM in the training process; +0 +10 +20 +30 +40 +0 +5 +10 +15 +20 +25 +30 +35 +Fig. 7. The WSR comparison against transmit power P, where B = 4, Nt = 2, N = 50, K = 4. +of neural blocks is plotted in Fig. 6(c), demonstrating that D2-ADMM can gradually converge +to performance comparable to the centralized algorithm as the training progresses. Moreover, +D2-ADMM converges more quickly as the number of neural blocks increases. Again, the final +convergence performance reaches saturation when L ≥ 6 in the simulation setups considered. +By comparing Fig. 6(a)-6(c), it can be concluded that the performance of D2-ADMM can +converge nearly to that of the centralized algorithm and gradually saturate as the number of +neural blocks L grows. Considering the tradeoff between the number of neural blocks and +system performance, we provide a empirical selection criterion for the number of neural blocks +as L = B + 2. +C. Performance of D2-ADMM under Various Setups +This section presents the performance comparison of D2-ADMM and benchmark algorithms +under various setups. + +25 +10 +20 +30 +40 +50 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +Fig. 8. The WSR comparison against the number of RIS elements N, where B = 4, Nt = 2, K = 4, P = 30 dBm. +In Fig. 7, we compare the WSR against the transmit power P of different algorithms when +B = 4, N = 50, K = 4. According to the conclusions given in Section V-A, we choose +L = 6 to balance the computational complexity and system performance. As shown in Fig. 7, +the WSR of all algorithms increases as P increases. The centralized algorithm performs the best +because it perfectly utilizes the CSI of all BSs. D2-ADMM is demonstrated to have comparable +performance, e.g., 95.6% when P = 30 dBm, to the centralized algorithm. The MRT Rand +θ algorithm performs the worst because the unoptimized RIS reflection coefficient does not +attain any benefits. Since Local ZF Comb MaxAO and MRT Comb MaxAO algorithms are non- +distributed algorithms without incorporating all BSs for system design, they suffer from severe +performance penalty compared with the proposed D2-ADMM, e.g., the WSR by applying the +D2-ADMM attains about 213% WSR improvement compared with the Local ZF Comb MaxAO +when P = 30 dBm. +Fig. 8 shows the performance comparison between D2-ADMM and benchmarks for different +number of RIS elements N, where B = 4, K = 4, P = 30 dBm. Observe from Fig. 8 +that the centralized algorithm, the D2-ADMM, and the local ZF Comb MaxAO algorithms +improve as N increases. However, MRT Comb MaxAO and MRT rand θ algorithms hardly +benefit from increasing the number of RIS elements. Besides, D2-ADMM outperforms the other +three distributed design algorithms, e.g., 223% compared with the Local ZF Comb MaxAO when +N = 30, and can attain comparable performance, e.g., 96.6% when N = 30, to the centralized +method. +Next, we show the WSR of various algorithms versus the number of UEs in Fig. 9, where + +26 +1 +2 +3 +4 +5 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +Fig. 9. The WSR comparison against the number of UE K, where B = 4, Nt = 2, N = 50, P = 30 dBm. +B = 4, N = 50, P = 30 dBm. The centralized algorithm, the D2-ADMM, and the Local +ZF Comb MaxAO algorithm increase with K thanks to the spatial multiplexing gain brought +by the increased number of UEs. Again, the D2-ADMM can perform as well as the centralized +algorithm, e.g., about 96.5% when K = 5, and better than the Local ZF Comb MaxAO algorithm, +e.g., about 216% when K = 5. When only a single UE is served, the performance of the other +four algorithms is the same except for the MRT rand θ algorithm since the inter-user interference +disappears in this situation. However, as a larger number of UEs access into the network, the MRT +Comb MaxAO algorithm’s performance declines, due to the fact that the distributed algorithm +fails to suppress the inter-user interference. +Finally, we evaluate the D2-ADMM algorithm’s performance against other benchmarks by +considering various numbers of BSs B in Fig. 10, where N = 50, K = 4, P = 30 dBm. +Again, the D2-ADMM can achieve comparable performance, e.g., about 96.2% when B = 5, to +the centralized algorithm with different B. The performance of D2-ADMM also increases as B +increases since that more BSs can provide more power for UEs. +VI. CONCLUSION +In this paper, we considered a RIS-assisted cell-free system that can boost communication +capacity and overcome the drawbacks of conventional cellular networks. To jointly design the +downlink precoding of BSs and the reflection phase shifts of RISs, we proposed a distributed +cooperative design based on ADMM, which can fully utilize the parallel computing resources. +Subsequently, we developed a neural network framework, D2-ADMM, by unrolling each iteration + +27 +4 +6 +8 +10 +12 +5 +10 +15 +20 +25 +30 +35 +40 +Fig. 10. The WSR comparison against the number of BS B, where Nt = 2, N = 50, K = 4, P = 30 dBm. +of the proposed distributed cooperative design, to automatically learn hyper-parameters and non- +convex RIS solvers through end-to-end training. Compared with conventional iterative algorithms, +D2-ADMM has a faster convergence speed. Moreover, we proposed an effective monodirectional +information exchange strategy to attain the cooperative design of all BSs with a small exchange +overhead. 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Wireless Commun., vol. 19, no. 7, pp. 4758–4774, Jul. 2020. + diff --git a/xNE0T4oBgHgl3EQfcQCC/content/tmp_files/load_file.txt b/xNE0T4oBgHgl3EQfcQCC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6768c577c311ee60525f972f6c35f74efae24943 --- /dev/null +++ b/xNE0T4oBgHgl3EQfcQCC/content/tmp_files/load_file.txt @@ -0,0 +1,1175 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf,len=1174 +page_content='1 Algorithm Unrolling-Based Distributed Optimization for RIS-Assisted Cell-Free Networks Wangyang Xu, Jiancheng An, Member, IEEE, Hongbin Li, Fellow, IEEE, Lu Gan, and Chau Yuen, Fellow, IEEE Abstract The user-centric cell-free network has emerged as an appealing technology to improve the next- generation wireless network’s capacity thanks to its ability to eliminate inter-cell interference effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' However, the cell-free network inevitably brings in higher hardware cost and backhaul overhead as a larger number of base stations (BSs) are deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Additionally, severe channel fading in high-frequency bands constitutes another crucial issue that limits the practical application of the cell-free network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In order to address the above challenges, we amalgamate the cell-free system with another emerging technology, namely reconfigurable intelligent surface (RIS), which can provide high spectrum and energy efficiency with low hardware cost by reshaping the wireless propagation environment intelligently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To this end, we formulate a weighted sum-rate (WSR) maximization problem for RIS-assisted cell-free systems by jointly optimizing the BS precoding matrix and the RIS reflection coefficient vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Subsequently, we transform the complicated WSR problem to a tractable optimization problem and propose a distributed cooperative alternating direction method of multipliers (ADMM) to fully utilize parallel computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Inspired by the model-based algorithm unrolling concept, we unroll our solver to a learning- based deep distributed ADMM (D2-ADMM) network framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To improve the efficiency of the D2-ADMM in distributed BSs, we develop a monodirectional information exchange strategy with a small signaling overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In addition to benefiting from domain knowledge, D2-ADMM adaptively learns hyper-parameters and non-convex solvers of the intractable RIS design problem through data- driven end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Finally, numerical results demonstrate that the proposed D2-ADMM achieve W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Xu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Gan are with the School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China (e-mail: wangyangxu@std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ganlu@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Li is with the Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA (e-mail: hli@stevens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' An and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Yuen are with Engineering Product Development (EPD) Pillar, Singapore University of Technology and Design, Singapore 487372, Singapore (e-mail: jiancheng an@sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' yuenchau@sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='02360v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='SP] 6 Jan 2023 2 around 210% improvement in capacity compared with the distributed noncooperative algorithm and almost 96% compared with the centralized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Index Terms Cell-free system, reconfigurable intelligent surface, distributed cooperative design, algorithm un- rolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' INTRODUCTION Next-generation wireless communication systems are expected to meet an even greater demand for higher capacity, denser connectivity, and broader coverage with the advent of the internet of everything [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The conventional communication network relies on cellular topology where effective communication paradigms, such as small-cell network and cellular massive multiple- input multiple-output (MIMO) are developed based on cell-centric principles [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, a single base station (BS) serves all users in the same cell while appropriate resource reuse policies are adopted among different cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As a result, users at the cell edge are more likely to be disturbed by the uplink/downlink signals from other adjacent cells, resulting in the common issue of inter-cell interference [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' It has been demonstrated that small-cell network can achieve better energy efficiency than cellular massive MIMO in some typical scenarios by properly reducing the cell size [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' However, as the cell density increases, the inter-cell interference will increase accordingly and become the main bottleneck limiting the capacity of the cellular network [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Although cellular massive MIMO is not affected by the inter-cell interference, the shadow fading due to blocking will become a performance-limiting factor if a large number of antennas are centralizedly configured on a single BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, cellular massive MIMO’s coverage and network capacity may be significantly deteriorated in some harsh environments [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In sharp contrast to the aforementioned cell-centric networks, a user-centric network paradigm known as the cell-free massive MIMO network has recently received significant attention as a potential and cutting-edge substitute [7], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In a cell-free massive MIMO network, a large number of antennas is spread on numerous BSs in a distributed form [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' These BSs provide service to a relatively small number of users within the same time-frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Since cell- free massive MIMO removes the underlying cell edge, it does not cause inter-cell interference as existing cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Although cell-free massive MIMO has many appealing advantages, 3 its application in higher frequency bands of future communication systems still has to overcome issues related to severe transmission attenuation and coverage blind spots [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In addition, deploying a large number of BSs also brings prohibitive hardware cost and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Fortunately, a promising technology, reconfigurable intelligent surface (RIS), has recently been introduced in various communication scenarios to significantly improve the system throughput and spectrum/energy efficiency [17]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, an RIS is a metal panel equipped with many low-cost passive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The phase shifts of these passive elements can be adjusted to achieve intelligent manipulation of the wireless environment and enhance communication quality-of-service (QoS) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In view of the superiority of the RIS and cell-free systems, it is of interest to develop an RIS-assisted cell-free approach for future wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Prior Works Downlink precoding is crucial to unleash the full potential of cell-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Currently, most existing precoding schemes for cell-free systems can be generally classified into non- cooperative [7], [22], [23] and cooperative [13], [24]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Non-cooperative precoding assumes that each BS can only utilize local channel state information (CSI) acquired through uplink channel estimation, without performing any CSI exchange among BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Along this direction, some rather simple strategies such as maximum ratio transmission (MRT) [7], local zero-forcing (ZF) [22], and local minimum mean square error (MMSE) [23] designs have been employed for precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Cooperative precoding including both centralized and distributed cooperative schemes that perform joint precoding across all BSs achieves better system performance compared with its non-cooperative counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, in the centralized scheme, BSs upload their local CSI to the centralized processing unit (CPU) through a specific backhaul link, based on which the precoding matrices of all BSs are jointly designed and then distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Most existing works on centralized precoding concentrate on developing precoding algorithms for the CPU, such as the centralized ZF precoding [24], [25] and the centralized MMSE precoding [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Distributed cooperative precoding distributes the computational load to multiple BSs, thus reducing the computational burden of the CPU [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The precoding of each BS is carried out locally and updated based on the cross-term information exchange among different BSs to approach the optimal performance of the centralized design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Meanwhile, RIS reflection coefficient design has received much attention recently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', [29]– [32], by taking into account of various practical constraints and application backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' How- 4 ever, the research on RIS-assisted cell-free network is still in its infancy stage [33]–[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specif- ically, the authors of [34] used the conjugate beamforming method with the local CSI to design precoding vectors along with randomly adjusted RIS reflection coefficient vector to illustrate the performance gain of the cell-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' By assuming that BSs send their local CSI to the CPU, the authors of [35], [36] adopted alternating optimization algorithms to jointly design the BS precoding and the RIS reflection coefficient vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Note that the works mentioned above are based on either a non-cooperative scheme, which requires no CSI exchange but yields inferior performance, or a centralized scheme, which trades system complexity for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Although recently [37] proposed a distributed cooperative optimization method for RIS-assisted cell-free system, it has to perform a set of iterations at different BSs without fully taking advantage of the distributed parallel computing capabilities of cell-free systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Also, most existing research efforts on RIS-assisted cell-free systems are focused on developing iterative optimization algorithms, which are based on some sophisticated models derived from the underlying physical processes or through handcrafting [38], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' On the contrary, deep learning (DL) methods attempt to automatically infer model information and network parameters directly from training data [19], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, DL is very promising for scenarios where the environment is complex and the system model is challenging to be constructed explicitly [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In addition, the number of layers of most neural networks is much fewer than the number of iterations incurred by typical iterative algorithms, which allows DL methods to attain a faster inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Nevertheless, neural networks are often trained as a “black-box” with poor interpretability and lack essential domain knowledge that is beneficial for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, combining conventional iterative algorithms and raw data-driven DL has become a new surge of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Recently, an appealing concept called algorithm unrolling has been proposed, which unrolls iteration-based algorithms into learning-based neural network structures [42]–[51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Such a unfolding process can not only integrate domain knowledge but also learn complex mapping functions and hyper-parameters from input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, each step in the traditional iterative algorithm is unrolled into a layer or a block of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Different network layers or blocks are cascaded to form a holistic neural network framework for solving the original problem more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The algorithm unrolling methods have shown advantages in many application domains, such as computational imaging [48], [49], speech processing [50], and remote sensing [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 5 TABLE I CONTRAST OF THE PROPOSED D2-ADMM TO EXISTING WORKS FOR RIS-ASSISTED CELL-FREE SYSTEMS Features Proposed [34] [35] [36] [37] Optimization objective WSR ASR EE WSR WSR BS precoding design DL Local MRT IA PDS ADMM RIS passive beamforming design DL Random IA PDS MM Centralized design × × ✓ ✓ × Distributed design Noncooperative × ✓ × × × Cooperative ✓ × × × ✓ Convergence speed Fast N/A Moderate Moderate Slow WSR: weighted sum-rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ASR: average sum-rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' EE: energy efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' IA: inner approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' PDS: primal-dual subgradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' MM: majorization-minimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Contributions Targeting RIS-assisted cell-free systems, we design a fully distributed joint BS precoding and RIS reflection coefficient optimization scheme based on the alternating direction method of mul- tipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Furthermore, we unroll the proposed solver to a learning-based neural network to attain better convergence and system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' More specifically, the main contributions of this paper in contrast with existing works are shown in Table I and further summarized as follows: We propose a distributed RIS-assisted cell-free system, where multiple energy-efficient RISs are deployed to assist in the downlink communications from a set of distributed BSs to multiple users (UEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A distributed cooperative BS precoding and RIS reflection coefficient design scheme is developed to make full use of distributed computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Furthermore, we propose a distributed design based on ADMM that iteratively updates the corresponding auxiliary variables, BS precoding, RIS reflection coefficient vectors, and multipliers involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The proposed design considers the consensus problem when separately designing the RIS reflection coefficients at each BS in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' We unroll the proposed distributed ADMM design into a learning-based deep distributed ADMM (D2-ADMM) neural network structure, which consists of a cascade of multiple neural blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Each neural block is designed by unfolding a single iteration of the proposed distributed ADMM design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moreover, an effective monodirectional information exchange 6 strategy with a small information exchange overhead is proposed for implementing our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In addition to obtaining deterministic variable updating strategies from domain knowledge, D2-ADMM adaptively learns hyper-parameters and non-convex solvers of the RIS design problem through data-driven end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Furthermore, D2-ADMM re- quires only a few neural blocks to reach convergence thanks to the strong inferential capability of DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Finally, we elaborate on the training and implementation of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Numer- ical results demonstrate that the proposed algorithm has faster convergence, less computa- tional complexity, and better performance compared with various traditional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Organization and Notations The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Section II introduces the system model and for- mulates the joint precoding and RIS reflection design problem in RIS-assisted cell-free systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In Sections III, we propose a distributed ADMM-based design by maximizing the weighted sum-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Section IV presents a D2-ADMM neural network structure and a monodirectional information exchange strategy to design the BS precoding and the RIS reflection coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Numerical results are provided in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Finally, we conclude the paper in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Notations: In this paper, scalars are denoted by italic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Vectors and matrices are denoted by bold-face lower-case and upper-case letters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The superscripts (·)T and (·)H represent the operations of transpose and Hermitian transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' |·| denotes the absolute value of a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ∥·∥ denotes the 2-norm of a vector or a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Re {x} and Im {x} denote the real and imaginary parts of the complex number x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' diag (·) denotes the diagonal operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The distribution of a circularly symmetric complex Gaussian (CSCG) with mean v and variance σ is denoted as ∼ CN(v, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' log2(·) represents the logarithmic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' C denotes the set of complex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' S denotes the set of symmetric positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' SYSTEM MODEL AND PROBLEM FORMULATION This section starts by introducing the system model of the RIS-assisted cell-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In order to design the BS precoding and RIS reflection coefficient vector, a practical weighted sum-rate (WSR) maximization problem is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' System Model In this paper, we consider a downlink RIS-assisted cell-free system, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 1, where multiple BSs and RISs are deployed in a distributed arrangement to serve all UEs cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The sets of BSs, RISs, and UEs are defined as B = {1, 2, · · · B}, R = {1, 2, · · · R}, and K = {1, 2, · · · K}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The number of antennas of each BS and UE is Nt and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Each RIS is equipped with a rectangular metasurface having N passive reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 1, each RIS builds one virtual channel consisting of the BS-RIS and RIS-UE channels between a BS and a UE to assist in the downlink communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In this paper, the BS-RIS channel between the b-th BS and the r-th RIS is denoted by Gb,r ∈ CN×Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The RIS-UE channel between the r-th RIS and the k-th UE is denoted by vH r,k ∈ C1×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moreover, the direct channel between the b-th BS and the k-th UE is denoted by hH b,k ∈ C1×Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' We consider that the proposed system operates in the mmWave band, where Gb,r, vr,k, and hb,k are described by the Saleh-Valenzuela model [52], which are expressed as Gb,r = � NtN LG LG � l=1 βb,r,laP (ψb,r,l, ςb,r,l) aH L (χb,r,l) , vr,k = � N Lv Lv � l=1 βr,k,laP (ψr,k,l, ςr,k,l) , hb,k = � Nt Lh Lh � l=1 βb,k,laL (ψb,k,l) (1) respectively, where LG, Lv, and Lh denote the multi-path number of Gb,r, vr,k, and hb,k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ψ∗,∗,l(ς∗,∗,l), and χ∗,∗,l denote the azimuth (elevation) angles of arrival (AoAs), and azimuth angles of departure (AoDs), where ∗ represents the index of the BS, the RIS element, and the UE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' βb,r,l ∼ CN(0, PLb,r,l), βr,k,l ∼ CN(0, PLr,k,l), and βb,k,l ∼ CN(0, PLb,k,l) denote the corresponding complex-valued path gain, where PL represents the path loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Besides, aL (ς) and a (ψ, ς) denote the array response vectors of uniform linear array (ULA) and uniform planar array (UPA), which are defined as aL (ψ) = 1 √NL [1, · · ·, ejπnl sin ψ, · · ·, ejπ(NL−1) sin ψ]T, (2) aP (ψ, ς) = 1 √ NxNy � 1, · · ·, ejπ(nx sin ψ sin ς+ny cos ς), · · ·, ejπ((Nx−1) sin ψ sin ς+(Ny−1) cos ς)�T, (3) 8 BS 1 BS b BS B RIS 1 RIS r RIS R UE 1 UE k UE K , b r G , H b k h , H r k v Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The downlink RIS-assisted cell-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' respectively, where NL and nl denote the total antenna number and the antenna index of ULA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Nx, Ny, nx, and ny represent the horizontal antenna number, the vertical antenna number, the horizontal antenna index, and the vertical antenna index of UPA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In the downlink transmission, the transmitted symbol xb ∈ CNt×1 at the b-th BS is defined as xb = K � k=1 wb,ksk, b ∈ B, (4) where sk is the transmitted symbol for the k-th UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Thus, we have s = [s1, s2, · · · , sK]T ∈ CK×1 representing the transmitted symbol vector that satisfies E[ssH] = IK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' wb,k ∈ CNt×1 denotes the precoding vector at the b-th BS for the k-th UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' At each UE, the received signal component corresponding to one BS includes two parts, one is that directly propagated from the BS to the UE, while the other is that superimposing mutiple signal copies reflected by R RISs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Hence, the received signal component at the k-th UE corresponding to the b-th BS can be expressed as yb,k = � hH b,k + R � r=1 vH r,kΘH r Gb,r � K � k=1 wb,ksk = � hH b,k + θHVH k Gb � xb = ˆhH b,kxb, (5) where Θr = diag([ejϕr,1, ejϕr,2, · · · , ejϕr,N]T) ∈ CN×N is the reflection coefficient matrix of the r-th RIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ϕr,n is the phase shift imposed by the n-th element of the r-th RIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' θ ∆= e � j[ϕ1,1,··· ,ϕ1,N,ϕ2,1,··· ,ϕR,N] T � ∈ CNR×1 denotes the phase shift vector of R RISs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Vk ∆= diag([vT 1,k, vT 2,k, 9 · · , vT R,k]) ∈ CNR×NR represents the equivalent channel from R RISs to the k-th UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Gb = [GT b,1, GT b,2, · · · , GT b,R]T ∈ CNR×Nt denotes the equivalent channel from the b-th BS to R RISs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ˆhb,k is the composite channel from the b-th BS to the k-th UE, incorporating one direct and NR reflected channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' We assume that all BSs are synchronized to ensure joint service for all UEs in the same time-frequency resource block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, the received signal at the k-th UE is the superposition of the signals transmitted from all BSs, which can be expressed as yk = B � b=1 yb,k + zk = B � b=1 K � j=1 ˆhH b,kwb,jsj + zk = B � b=1 ˆhH b,kwb,ksk � �� � Desired signal + B � b=1 K � j=1,j̸=k ˆhH b,kwb,jsj � �� � Interference of other UEs +zk, (6) where zk ∼ CN(0, δ2 k) denotes the additive white Gaussian noise (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Without loss of generality, we assume that all UEs have the same noise power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', δ2 k = δ2, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Problem Formulation Based on the signal model expressed in (6), the signal-to-interference-plus-noise ratio (SINR) ζk of the k-th UE can be written as ζk = ���� B� b=1 ˆhH b,kwb,k ���� 2 K � j=1,j̸=k ���� B� b=1 ˆhH b,kwb,j ���� 2 + δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (7) To evaluate the performance of the RIS-assisted cell-free system, the WSR is given as WSR = K � k=1 ωklog2 (1 + ζk), (8) where ωk > 0 is the weight of the k-th UE, which indicates the priority of different UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 10 In this paper, we endeavor to maximum the WSR of the RIS-assisted cell-free system by designing the BS precoding W = {wb,k|∀b ∈ B, ∀k ∈ K} and the RIS reflection coefficient vector θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Mathematically, the optimization problem can be formulated as P0 : max θ,W WSR = K � k=1 ωklog2 (1 + ζk) (9a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' K � k=1 ∥wb,k∥2 ≤ Pb,max, ∀b ∈ B, (9b) |θr,b| = 1, ∀r ∈ R, ∀n ∈ N, (9c) where (9a) is the WSR objective function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (9b) is the power constraints of BSs, where Pb,max denotes the maximum transmit power budget at the b-th BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Constraint (9c) represents that the amplitude of reflection coefficient of each RIS remains constant in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Remark 1: Although the centralized algorithm can achieve the optimal solution to P0 [36], it requires collecting the local CSI of all BSs for joint optimization at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' This inevitably increases both the CSI feedback and control signaling overhead as well as the computational complexity of the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, we aim to develop a distributed algorithm to solve P0 by spreading the computational load to the distributed BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, the distributed optimation problem of P0 is rewritten as P1 : max θ,W WSR = K � k=1 ωklog2 (1 + ζk) (10a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (9b), (9c), θb = θ¯b, ∀b ∈ B, ∀¯b ∈ Fb, (10b) where (10b) is the consensus constraint, which means that θb optimized at adjacent BSs should be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Fb represents the index set of the adjacent BSs that can exchange information with the b-th BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, the b-th BS requires utilizing the information from the adjacent BSs when designing the RIS reflection coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Then, it sends its local information to the adjacent BSs until the RIS reflection coefficients on all BSs reach a consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Remark 2: Here we highlight that the optimization of Wb and θb in a distributed system are distinctly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, the downlink precoding matrix Wb is unique for different BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' By contrast, θb optimized by different BSs correspond to the same RIS and need to be appropriately fused into a single reflection coefficient vector, which is known as the consensus 11 problem in distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Although the centralized algorithm does not involve the consensus problem, the distributed optimization strategy is more effective and practical considering the distributed deployment of BSs as well as the limited backhaul capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ADMM-BASED DISTRIBUTED OPTIMIZATION In this section, we propose a distributed ADMM-based design for effectively optimizing the precoder and reflection phase shifts in the practical RIS-assisted cell-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, we first convert the non-convex P1 into a tractable form P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Then, we propose a distributed ADMM design to solve P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A Tractable Form of P1 Observe form (10a) that P1 is a non-convex optimization problem due to the coupling of the optimization variables W and θ and the consensus constraint (10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, we transform P1 into a tractable problem by applying the Lagrangian dual transform and the quadratic transform, which are summarized in Lemmas 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Lemma 1 (Lagrangian dual transform): Given a sum-of-logarithmic-ratios problem, expressed as [53] max x D � d=1 ωdlog2 � 1 + Qd (x) Fd (x) � (11a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' x ∈ χ, (11b) where ωd is a nonnegative weight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Qd (x) is a nonnegative function that satisfies Qd (x) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Fd (x) is a positive function with Fd (x) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' x is the optimization variable, and χ denotes a nonempty constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moving the ratio from inside of the logarithm to the outside, (11a) can be rewritten as min x,γ D � d=1 ωd � γd − log2 (1 + γd) − (1 + γd) Qd (x) Qd (x) + Fd (x) � (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' x ∈ χ, γd ≤ Qd(x) Fd(x) , (12b) where γ = [γ1, γ2, · · · γD]T is the auxiliary variable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 12 Lemma 2 (Quadratic transform): Given a sum-of-functions-of-ratio problem for the multidi- mensional and complex cases, expressed as [54] min x D � d=1 ¯fd � Qd (x) F −1 d (x) QH d (x) � (13a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' x ∈ χ, (13b) where function Qd (x) : Cd1 → Cd2, Fd (x) Cd1 → Sd2×d2, x, and constraint χ ⊆ Cd1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Let ¯fd (·) denotes a monotonically nondecreasing function, problem (13) can be transformed to min x,η D � d=1 ¯fd � 2Re {ηdQd (x)} − η2 dFd (x) � (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' x ∈ χ, ηd ∈ Cd1, (14b) where η = [η1, η2, · · · ηD]T denotes the auxiliary variable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, by using the Lagrangian dual transform, P1 can be reformulated as LP1 : min θ,W,γ f1 (θ, W, γ) (15a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (9b), (9c), (10b), where f1 (θ, W, γ) is the new objective function via Lemma 1, which is described in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Besides, γ = [γ1, γ2, · · · γK]T represents the auxiliary variable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' f1 (θ, W, γ) = K � k=1 ωk � � � � �γk − log2 (1 + γk) − (1 + γk) ���� B� b=1 ˆhH b,kwb,k ���� 2 K � k=1 ���� B� b=1 ˆhH b,kwb,k ���� 2 + δ2 � � � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (16) Then we use the quadratic transform shown in Lemma 2 to decouple the numerator and the denominator of the fraction in LP1 to further simplify the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Consequently, LP1 can be transformed as LP2 : min θ,W,γ,η f2 (θ, W, γ, η) (17a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (9b), (9c), (10b), where f2 (θ, W, γ, η) is given in (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' η = [η1, η2, · · · ηK]T denotes the auxiliary variable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' f2 (θ, W, γ, η) = K � k=1 (|ηk|2 K � j=1 ����� B � b=1 ˆhH b,jwb,k ����� 2 + |ηk|2δ2 + ωkγk − 2 � (1 + γk) ωk B � b=1 Re � ηkˆhH b,kwb,k � − ωklog2 (1 + γk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (18) 13 Next, we rewrite problem LP2 in its augmented Lagrangian form, expressed as L (θ, W, γ, η, λ) = f2 (θ, W, γ, η) + B � b=1 µb � K � k=1 ∥wb,k∥2 − Pb,max � � �� � Power constraint + B � b=1 I (θb) � �� � Feasible constraint + B � b=1 ρb 2 ����θb − θ¯b + λb ρb ���� 2 � �� � Consensus constraint .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (19) where λ = � λb ∈ CN×1|∀b ∈ B � is the Lagrange multiplier and ρb > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' I (·) represents a feasible function such that I (θb) = 0 for |θb| = 1 and I (θb) = ∞ for |θb| ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As a result, the final tractable form of P1 is given as P2 : min θ,W,γ,η,λ L (θ, W, γ, η, λ) (20) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Proposed Distributed Design Based on ADMM To solve the problem P2, we propose a distributed design based on ADMM [55], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The proposed design iteratively designs the local BS precoding and RIS reflection coefficient vectors at each BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, the i-th iteration at the b-th BS can be expressed as γi = arg min γ f1 � θi−1, Wi−1, γ � , (21a) ηi = arg min η L � θi−1, Wi−1, γi, η, λi−1� , (21b) Wi b = arg min Wb L � θi−1, ¯ Wi−1 b , Wb, γi, ηi, λi−1� , (21c) θi b = arg min θb L � ¯θ i−1 b , θb, ¯ Wi−1 b , Wi b, γi, ηi, λi−1� , (21d) λi b = λi−1 b + ρb � θi b − θi ¯b � , (21e) where Wb = {wb,k|∀k ∈ K} denotes the local precoding at the b-th BS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ¯ Wb = W\\Wb and ¯θb = θ\\θb are the downlink precoding and the RIS reflection coefficient vector of other BSs except the b-th BS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Observe from (21) that γ, η, Wb, θb, and λb are updated locally in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Next, we give the solutions to problems (21a)-(21d) one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Note that P2 is an equivalence problem to LP1, which means that the optimal γ of LP1 is equal to ones of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Besides, it is easier to solve LP1 than P2 for optimal γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, we solve LP1 for optimal γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 14 1) The Solver of (21a): For problem (21a), f1 (θ, W, γ) is a convex function for γ with fixed θ and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, the optimal γk can be obtained by taking ∂(f1(γ)) ∂γk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Thus, we have γ† k = ���� B� b=1 ˆhH b,kwb,k ���� 2 K � j=1,j̸=k ���� B� b=1 ˆhH b,kwb,j ���� 2 + δ2 = |ϖk,k + ϑk,k|2 K � j=1,j̸=k |ϖk,j + ϑk,j|2 + δ2 , (22) where ϖk,j = B� b=1 hH b,kwb,j and ϑk,j = B� b=1 θHVH k Gbwb,j are two defined cross-term information, which contain the information of all BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Note that {ϖk,j|∀k, j ∈ K} and {ϑk,j|∀k, j ∈ K} are then exchanged among different BSs to achieve the goal of cooperative design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 2) The Solver of (21b): For problem (21b), we note that only f2 (θ, W, γ, η) in (19) is dependent on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, problem (21b) can be reformulated as η = arg min η K � k=1 � �|ηk|2 � � K � j=1 ����� B � b=1 ˆhH b,jwb,k ����� 2 + δ2 � � − � (1 + γk) ωk � B � b=1 � ηkˆhH b,kwb,k + η∗ kwH b,kˆhb,k ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (23) The optimal η† k can also be obtained by taking ∂(f2(η)) ∂η∗ k = 0, which is expressed as η† k = (ϖk,k + ϑk,k)∗� (1 + γk) ωk K � j=1 |ϖk,j + ϑk,j|2 + δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (24) 3) The Solver of (21c): Given a set of tentative values of other variables, we have L (wb,k) = K � k=1 (|ηk|2 K � j=1 � � ������ � � B � b′̸=b ˆhH b′,jwb′,k � � wH b,kˆhb,j ������ 2 + ��hH b,jwb,k ��2 � � − � (1 + γk) ωkRe � ηkˆhH b,kwb,k � ) + µb∥wb,k∥2 + C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (25) where C1 is defined by C1 = K � k=1 (|ηk|2 K � j=1 B � b′̸=b ���ˆhH b′,jwb′,k ��� 2 K � j=1 −2 � (1 + γk) ωk B � b′̸=b Re � ηkˆhH b′,kwb′,k � +|ηk|2δ2 + ωkγk − ωklog2 (1 + γk)) + B � b′̸=b µb′ � K � k=1 ��wb′,k ��2 − Pb,max � + µb � � K � k′̸=k ��wb,k′��2 − Pb,max � � + B � b=1 I (θb) + B � b=1 ρb 2 ����θb − θ¯b + λb ρb ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (26) 15 Similarly, the optimal w† b,k can be obtained by taking ∂(L(wb,k)) ∂wH b,k = 0, which is given as w† b,k = � (1 + γk) ωkη∗ kˆhb,k − Ωb,k � µb + |ηk|2 K � j=1 ˆhb,jˆhH b,j � , (27) where Ωb,k = |ηk|2 K � j=1 ˆhb,k(ϖj,k + ϑj,k − ˆhH b,jwb,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' µb is a normalized factor used to scale wb,k for satisfying the total power constraint, which needs to be dynamically updated in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' We apply the power normalization approach below to bypass the update of µb in D2-ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, wl b,k is scaled by w† b,k = � Pb,maxw† b,k � K � k=1 ���w† b,k ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (28) 4) The Solver of (21d): For problem (21d), we first rewrite (19) in a more intuitive form as L (θb) = θH b Sθb − 2Re � θH b Z � + B � b=1 I (θb) + C2, (29) where S, Z, and C2 are independent of θb and given in (30), (31), and (32) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Note that (29) is a non-convex problem due to the feasible constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To solve this problem, we build a neural block based on DL, and the details will be discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' S = K � k=1 |ηj|2 K � j=1 VH j Gbwb,kwH b,kGH b Vj + ρb 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (30) Z = K � k=1 |ηk|2 K � j=1 VH j Gbwb,k � wH b,kˆhb,j − ϖ∗ j,k − ϑ∗ j,k − wH b,khb,j � + � (1 + γk) ωkηkVH k Gbwb,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (31) C2 = K � k=1 � �|ηk|2 K � j=1 � � B � b′̸=b ���ˆhH b′,jwb′,k ��� 2 + 2Re � � �hH b,jwb,k � � B � b′̸=b wH b′,kˆhb′,j � � � � � + ��hH b,jwb,k ��2 � � − � (1 + γk) ωk B � b′̸=b Re � ηkˆhH b′,kwb′,k � + ωkγk − ωklog2 (1 + γk) + |ηk|2δ2 � � + B � b′̸=b ρb′ 2 ����θb′ − θ¯b′ + λb′ ρb′ ���� 2 +ρb 2 ���� λb ρb − θ¯b ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (32) 16 Ini-Block Mid- Block1 Mid- Block2 Mid- Block3 Out- Block BS b BS b+1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' b k b k G V h 1 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' L L b b k \uf071 w 1 1ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' b b \uf076 \uf04a 1 1ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' l l b b \uf076 \uf04a 2 2ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' l l b b \uf076 \uf04a 3 3ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' l l b b \uf076 \uf04a 3 3 1 1 ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' l l b b \uf076 \uf04a + + 2 2 1 1 ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' l l b b \uf076 \uf04a + + 1 1 1 1 ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' l l b b \uf076 \uf04a + + 1 1 1 1 ˆ ˆ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' b b \uf076 \uf04a + + BS b-1 { B 1 1 b \uf071 + 1 1 l b \uf071 + 1l b \uf071 2l b \uf071 3l b \uf071 3 1 l b \uf071 + 2 1 l b \uf071 + 1 b \uf071 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The structure of D2-ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' D2-ADMM: A LEARNING-BASED ALGORITHM UNROLLING METHOD This section proposes a D2-ADMM neural network structure to design the BS precoding and the RIS reflection coefficient by unfolding the proposed distributed ADMM design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Furthermore, an efficient monodirectional information exchange strategy is proposed to link different BSs to improve the performance of our distributed designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Finally, we elaborate on the training and the implementation of D2-ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Structure of Deep Distributed ADMM The proposed distributed ADMM design iteratively updates auxiliary variables, BS precoding, RIS reflection coefficient vectors, and multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' However, it has high computational complexity since the conventional distributed ADMM may take hundreds or thousands of iterations to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' System performance and convergence are additionally hampered by the requirement to manually choose crucial hyper-parameters, such as the power normalized factor {µb|∀b ∈ B} and the penalty factor {ρb|∀b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To overcome these shortcomings, we unfold the proposed distributed ADMM design into the D2-ADMM to learn the hype-parameters {ρb|∀b ∈ B} auto- matically and bypass {µb|∀b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Besides, we create a neural block called θ-Block to solve the complicated problem (21d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The specific D2-ADMM structure is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 2, a total of B D2-ADMM are respectively implemented at B BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A D2- ADMM is composed of L ≥ B cascaded neural blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Each neural block is designed according to one iteration of the distributed ADMM design, which means that a neural block is equivalent to a single iteration in traditional iterative algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The (l−1)-st neural block’s output constitutes 17 I-Layer A-Layer W-Layer I1-Layer \uf071 Layer \uf06c 0 b \uf071 0 , b k w 1 1 , b b \uf076 \uf04a 2 2 , b b \uf076 \uf04a 1 1 , , , b k b k \uf067 \uf068 In\uf071 1 1 b \uf071 + 1 b \uf06c 1 b \uf071 1 1 1 1 ˆ ˆ , b b \uf076 \uf04a + + 1 1ˆ ˆ , b b \uf076 \uf04a 1 , b k w 1 , b k w 1 b \uf071 Block , , , b k b k G V h Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The structure of Ini-Block relying on algorithm unrolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' the input of the l-th neural block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The input of the first neural block is initialized, and the last neural block outputs the optimized BS precoding matrix and RIS reflection coefficient vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' More specifically, we have five different kinds of neural blocks, namely the initialization neural block (Ini-Block), the middle neural block 1 (Mid-Block1), the middle neural block 2 (Mid- Block2), the middle neural block 3 (Mid-Block3), and the output neural block (Out-Block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' For the sake of illustration, we give the schematic diagram of the Ini-Block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The structures of other neural blocks are based on the Ini-Block by replacing or pruning certain parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Ini-Block initializes the network, which includes a cross-term information initialization layer (I-Layer), an auxiliary variable update layer (A-Layer), a BS precoding update layer (W-Layer), an RIS update block (θ-Block), a multiplier update layer (λ-Layer), and a cross-term information exchange layer 1 (I1-Layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The first neural block of D2-ADMM is an Ini-Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The 2nd to (B−1)-st network blocks of D2-ADMM are created as the Mid-Block1, which contains an A-Layer, a W-Layer, a θ-Block, a λ-Layer, and a I1-Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The B-th neural block of D2-ADMM is Mid-Block2, which has a similar structure as Mid-Block1, except that I1-Layer is replaced with a cross-term information exchange layer 2 (I2-Layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moreover, the (B + 1 ∼ L − 1)-st neural blocks are Mid-Block3, which is constructed similarly as Mid-Block2 with the exception of using a cross- term information exchange layer 3 (I3-Layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The last neural block of D2-ADMM is referred to Out-Block, which consists of an A-Layer, a W-Layer, and a θ-Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Next, we will discuss the structure and function of each layer and the θ-Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ConvCony 2Cony 318 1) Auxiliary Variable Update Layer (A-Layer): A-Layer updates two auxiliary variables, γ and η, according to (22) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To reflect the iteration order, we rewrite (22) and (24) as γl b,k = ��ϖl b,k,k + ϑl b,k,k ��2 K � j=1,j̸=k ��ϖl b,k,j + ϑl b,k,j ��2 + δ2 , (33) ηl b,k = � ϖl b,k,k + ϑl b,k,k �∗�� 1 + γl b,k � ωk K � j=1 ��ϖl b,k,j + ϑl b,k,j ��2 + δ2 , (34) respectively, where ϖl b,k,j and ϑl b,k,j denote the cross-term information of the l-th neural block for the b-th BS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' γl b,k, and ηl b,k are the two auxiliary variables of the l-th neural block for the b-th BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 2) BS Precoding Update Layer (W-Layer): According to (27), W-Layer updates the BS precoding matrix W as wl b,k = �� 1 + γl b,k � ωb,k � ηl k �∗ˆhl−1 b,k − Ωl−1 b,k K � j=1 ��ηl b,j ��2ˆhl−1 b,j � ˆhl−1 b,j �H , (35) where Ωl−1 b,k = K � j=1 ��ηl b,j ��2ˆhl−1 b,k (ϖl b,j,k+ϑl b,j,k−(ˆhl−1 b,j )Hwl−1 b,k ) and ˆhl−1 b,j = (hH b,j + (θl−1 b ) HVH k Gb)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In order to satisfy the power constraint, wl b,k can be rewrited as wl b,k = � Pb,maxwl b,k � K � k=1 ��wl b,k ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (36) 3) RIS Update Block (θ-Block): As previously mentioned, (29) is a non-convex function that is challenging to solve by conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, we introduce the θ-Block, which aims to exploit the inference ability of DL to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' θ-Block is composed of multiple convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, we first rewrite (29) as ∠ (θb) = fθ (S, Z) , (37) where fθ denotes a non-linear function that applied as the solver of problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' We then use multiple convolutional layers to approximate this complicated non-linear function fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Since the neural network is more amenable with real-valued data, we first convert S and Z into real-valued sequences Inθ as the input of the θ-Block, expressed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Inθ = [Re {S} , Im {S} , Re {Z} , Im {Z}] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (38) 19 Therefore, the working principle of θ-Block can be expressed as ∠ (θb) = fC,U (· · · fC,u (· · · fC,1 (Inθ|υ1) |υu) |υU) , (39) where fC,u is the u-th convolutional layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' U denotes the number of convolutional layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' υu is the parameter set of the u-th convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In this paper, we empirically choose U = 3 which is sufficient for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Note that in the proposed architecture, the parameters of each convolutional layer can be automatically learned through end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 4) Multiplier Update Layer (λ-Layer): The multipliers are updated through this layer using the following strategy λl b = λl−1 b + ρb � θl b − θl ¯b � , (40) where ρb is a learnable parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' θl ¯b is the RIS reflection coefficient vector exchanged from the ¯b-th BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 5) Cross-Term Information Initialization Layer (I-Layer): Again, in the cooperative design of distributed RIS-assisted cell-free systems, CSI sharing is necessary among BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' However, considering the security and the excessive overhead associated with direct CSI exchange, we define {ϖb,k,j, ϑb,k,j|∀b ∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ∀k, j ∈ K} as two types of necessary cross-information in A-layer, W-Layer, and θ-Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' I-Layer initializes the local cross-term information, which is expressed as � � � ˆϖ0 b,k,j = 0, ˆϑ0 b,k,j = 0, (41a) � � � ϖ1 b,k,j = hH b,kw0 b,k, ϑ1 b,k,j = � θ0 b �HVH k Gbw0 b,k, (41b) where ˆϖ0 b,k,j and ˆϑ0 b,k,j are two initialized cross-term information, which will be sent to the adjacent BSs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' ϖ1 b,k,j and ϑ1 b,k,j denote two cross-term information, which will be used for updating the next neural block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' w0 b,k and θ0 b are initialized randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6) Cross-Term Information Layer 1 (I1-Layer): I1-Layer includes two processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The process 1 is to send the updated cross-term information to the adjacent BSs, expressed as (42a), where ˆϖl b,k,j and ˆϑl b,k,j are two cross-term information that needs to be shared with the adjacent BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moreover, ˆϖl−1 ¯b,k,j and ˆϑl−1 ¯b,k,j represent two cross-term information symbols that are received from the adjacent BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' θl b and wl b,k denote the l-th update of the RIS reflection coefficient vector and 20 the BS precoding vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In the process 2, the cross-term information required for updating the next neural block will be determined based on the received cross-term information from the adjacent BSs, as demonstrated in (42b), where ϖl+1 b,k,j and ϑl+1 b,k,j denote the two cross- term information symbols that are required for updating the (l + 1)-st neural block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' I1-Layer is configured for the (1 ∼ B−1)-st neural blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' � � � ˆϖl b,k,j = ˆϖl−1 ¯b,k,j + hH b,kwl b,k, ˆϑl b,k,j = ˆϑl−1 ¯b,k,j + � θl b �HVH k Gbwl b,k, (42a) � � � ϖl+1 b,k,j = ˆϖl ¯b,k,j + hH b,kwl b,k, ϑl+1 b,k,j = ˆϑl ¯b,k,j + � θl b �HVH k Gbwl b,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (42b) 7) Cross-Term Information Layer 2 (I2-Layer): I2-Layer has the similar process 1 but distinct process 2 as I1-Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, the updates of the b-th BS in the first neural block is included in the cross-term information needed for updating the (B + 1)-st neural block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Thus, we have to eliminate the obsolete updates from the cross-term information and add the B-th update to guarantee that only the new update is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The specific process 2 is expressed as follows � � � ϖB+1 b,k,j = ˆϖB ¯b,k,j − hH b,kw1 b,j + hH b,kwB b,j, ϑB+1 b,k,j = ˆϑB ¯b,k,j − � θ1 b �HVH k Gbw1 b,j + � θB b �HVH k GbwB b,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (43) Therefore, I2-Layer is only exploited in the B-th neural block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 8) Cross-Term Information Layer 3 (I3-Layer): When l ≥ B + 1, both the cross-term infor- mation to be sent to the adjacent BSs and the cross-term information used for updating the next neural block need to eliminate obsolete updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, the two processes of I3-Layer can be described as� � � ˆϖl b,k,j = ˆϖl−1 ¯b,k,j − hH b,kwl−B+1 b,j + hH b,kwl b,j, ˆϑl b,k,j = ˆϑl−1 ¯b,k,j − � θl−B+1 b �HVH k Gbwl−B+1 b,j + � θl b �HVH k Gbwl b,j, (44a) � � � ϖl+1 b,k,j = ˆϖl ¯b,k,j − hH b,kwl−B+2 b,j + hH b,kwl b,j, ϑl+1 b,k,j = ˆϑl ¯b,k,j − � θl−B+2 b �HVH k Gbwl−B+2 b,j + � θl b �HVH k Gbwl b,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (44b) We deploy I3-Layer in the (B + 1 ∼ L − 1)-st neural blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Information Exchange Strategy Next, we elaborate on the proposed information exchange strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To safeguard the informa- tion privacy of different BSs and reduce the proposed system’s information exchange overhead, 21 BS 1 BS 2 BS b BS B 2 2ˆ ˆ , l l \uf076 \uf04a 2 l \uf071 1 l \uf071 1 1ˆ ˆ , l l \uf076 \uf04a l B \uf071 1 l b \uf071 + 3 l \uf071 3 3ˆ ˆ , l l \uf076 \uf04a ˆ ˆ , l l b b \uf076 \uf04a l b \uf071 1 1 ˆ ˆ , l l b b \uf076 \uf04a + + ˆ ˆ , l l B B \uf076 \uf04a Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The proposed information exchange strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' we define two types of cross-term information used for the update at each BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The updating of each neural block needs to guarantee the integrality and timeliness of the cross-term information, as demonstrated by the updating process of the I1-layer, the I2-layer, and the I3-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In most existing distributed information exchange strategies, each BS often receives information shared by multiple BSs [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' This exchange strategy will reduce the integrality and timeliness of the cross-term information defined in our paper, affecting the convergence and performance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, we propose an effective monodirectional information exchange strategy, assuming all BSs have a monodirectional topology, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Each BS performs a monodirectional information exchange with two adjacent BSs through a dedicated link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' For instance, the b-th BS receives cross-term information from the (b + 1)-st BS and sends its cross-term information to the (b − 1)-st BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Such a strategy requires at least B exchanges to ensure the integrality of the cross-term information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As the iteration proceeds, the timeliness of the cross-term information is guaranteed by replacing the obsolete information with the latest information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The specific cross-term information processing are completed at the I1-Layer, I2-Layer, and I3-Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In addition to exchanging cross-term information, we also need to exchange the RIS reflection coefficient vectors updated by each neural block among various BSs to update the multiplier λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Therefore, the b-th BS needs to send { ˆϖl b,k,j|∀k, j ∈ K} (K2 dimension), {ˆϑl b,k,j|∀k, j ∈ K} (K2 dimension), and θl b (RN dimension) in the l-th neural block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As a consequence, the total dimension of exchanged data in the practical RIS-assisted cell-free system is B(L − 1)(2K2 + RN), which is significantly reduced compared with that exchanging CSI directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 22 RIS (50,-50,3) BS UE (100,-50,3) (150,-50,3) (200,-50,3) (75,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='5) (75,10,6) (125,10,6) x(m) y(m) (0,0,0) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The 3D scenario of the RIS-assisted cell-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Training of D2-ADMM In this section, we give the specific training and practical application methods of the proposed D2-ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The input to D2-ADMM at the b-th BS is its local CSI, the initialized w0 b,k and θ0 b, while the output is the optimized wL b,k and θL b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Then the parameters of θ-Layer and ρb in D2-ADMM are updated through an end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The loss function for training is set as fLoss = 1 Q Q � q=1 B � b=1 ��θq,b − θq,¯b ��2 � �� � Consensus error − WSRq, (45) where Q is the sample number of one training batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' By minimizing the loss function fLoss, the consensus error is minimized while maximizing WSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' It is worth noting that the training process is completed on a single CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' After complet- ing the training, we deploy B D2-ADMMs to the corresponding BSs for practical distributed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' NUMERICAL RESULTS This section provides simulation results to demonstrate the effectiveness of our proposed D2- ADMM framework for the RIS-assisted cell-free system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Simulation Setup We consider a typical RIS-assisted cell-free system 3D scenario shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In this scenario, the b-th BS is deployed at (200× b B, −50, 3) m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Without loss of generality, we consider 23 R = 2 RISs, which are deployed at (75, 10, 6)m and (125, 10, 6) m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' K UEs served by B BSs are randomly distributed in a circular area with a center at (75, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='5) m, a radius of 5m, and a height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The number of antennas at each BS is set to Nt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Given the location information of each device, the corresponding channel can be determined by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In this setup, we assume that the multi-path number of each channel is 3 (1 LoS, 2 NLoS), and their AOAs and AODs are chosen randomly in the range [− π 2, π 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Likewise, all BSs have the same maximum transmit power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', Pb,max = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The received noise power is set to δ2 = −80 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To better demonstrate the performance of the proposed D2-ADMM, we consider several representative benchmarks, as listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Centralized: Assuming that all BSs send their local CSI to the central CPU for the centralized design of the BS precoding matrix and the RIS reflection coefficient vectors [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' MRT Random θ: A distributed design method, where the RIS reflection coefficient vector is randomly configured, and the precoding of BS is designed as the conjugate of local CSI [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' MRT Comb MaxAO: A distributed algorithm that maximizes the channel gain of cascaded channels for configuring the RIS, and the design of BS precoding is the same as MRT Random θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Local ZF Comb MaxAO: This distributed algorithm has the same design of RIS as MRT Comb MaxAO and exploits the local ZF algorithm proposed in [57] for optimizing the BS precoding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Training Performance of D2-ADMM In order to show the convergence of D2-ADMM, we first conduct experiments to evaluate various indicators in the training process of D2-ADMM, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6(a)-6(c), where we set B = 4, N = 50, K = 4, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Specifically, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6(a) illustrates the training loss of the D2-ADMM under the different number of neural blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' It can be seen that the D2-ADMM training loss can converge as the training proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In addition, the final convergent training loss gap for different L is negligible when L ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6(b) shows the fluctuation of the consensus error of D2-ADMM against different L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6(b), we can observe that the consensus error of D2-ADMM can converge to a minimal value as the training proceeds, and varied L does not severely impact the convergence result of the consensus error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The WSR in the training phase against the number 24 0 50 100 150 200 24 22 20 18 16 14 12 10 8 6 (a) 0 50 100 150 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='5 (b) 0 50 100 150 200 6 8 10 12 14 16 18 20 22 24 (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (a) The training loss of D2-ADMM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (b) The consensus error of D2-ADMM in the training process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' (c) The WSR of D2-ADMM in the training process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 0 10 20 30 40 0 5 10 15 20 25 30 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The WSR comparison against transmit power P, where B = 4, Nt = 2, N = 50, K = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' of neural blocks is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6(c), demonstrating that D2-ADMM can gradually converge to performance comparable to the centralized algorithm as the training progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moreover, D2-ADMM converges more quickly as the number of neural blocks increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Again, the final convergence performance reaches saturation when L ≥ 6 in the simulation setups considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' By comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 6(a)-6(c), it can be concluded that the performance of D2-ADMM can converge nearly to that of the centralized algorithm and gradually saturate as the number of neural blocks L grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Considering the tradeoff between the number of neural blocks and system performance, we provide a empirical selection criterion for the number of neural blocks as L = B + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Performance of D2-ADMM under Various Setups This section presents the performance comparison of D2-ADMM and benchmark algorithms under various setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 25 10 20 30 40 50 4 6 8 10 12 14 16 18 20 22 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The WSR comparison against the number of RIS elements N, where B = 4, Nt = 2, K = 4, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 7, we compare the WSR against the transmit power P of different algorithms when B = 4, N = 50, K = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' According to the conclusions given in Section V-A, we choose L = 6 to balance the computational complexity and system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 7, the WSR of all algorithms increases as P increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The centralized algorithm performs the best because it perfectly utilizes the CSI of all BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' D2-ADMM is demonstrated to have comparable performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='6% when P = 30 dBm, to the centralized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The MRT Rand θ algorithm performs the worst because the unoptimized RIS reflection coefficient does not attain any benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Since Local ZF Comb MaxAO and MRT Comb MaxAO algorithms are non- distributed algorithms without incorporating all BSs for system design, they suffer from severe performance penalty compared with the proposed D2-ADMM, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', the WSR by applying the D2-ADMM attains about 213% WSR improvement compared with the Local ZF Comb MaxAO when P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 8 shows the performance comparison between D2-ADMM and benchmarks for different number of RIS elements N, where B = 4, K = 4, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 8 that the centralized algorithm, the D2-ADMM, and the local ZF Comb MaxAO algorithms improve as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' However, MRT Comb MaxAO and MRT rand θ algorithms hardly benefit from increasing the number of RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Besides, D2-ADMM outperforms the other three distributed design algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', 223% compared with the Local ZF Comb MaxAO when N = 30, and can attain comparable performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='6% when N = 30, to the centralized method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Next, we show the WSR of various algorithms versus the number of UEs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 9, where 26 1 2 3 4 5 6 8 10 12 14 16 18 20 22 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The WSR comparison against the number of UE K, where B = 4, Nt = 2, N = 50, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' B = 4, N = 50, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The centralized algorithm, the D2-ADMM, and the Local ZF Comb MaxAO algorithm increase with K thanks to the spatial multiplexing gain brought by the increased number of UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Again, the D2-ADMM can perform as well as the centralized algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', about 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='5% when K = 5, and better than the Local ZF Comb MaxAO algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', about 216% when K = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' When only a single UE is served, the performance of the other four algorithms is the same except for the MRT rand θ algorithm since the inter-user interference disappears in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' However, as a larger number of UEs access into the network, the MRT Comb MaxAO algorithm’s performance declines, due to the fact that the distributed algorithm fails to suppress the inter-user interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Finally, we evaluate the D2-ADMM algorithm’s performance against other benchmarks by considering various numbers of BSs B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 10, where N = 50, K = 4, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Again, the D2-ADMM can achieve comparable performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=', about 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content='2% when B = 5, to the centralized algorithm with different B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The performance of D2-ADMM also increases as B increases since that more BSs can provide more power for UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' CONCLUSION In this paper, we considered a RIS-assisted cell-free system that can boost communication capacity and overcome the drawbacks of conventional cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' To jointly design the downlink precoding of BSs and the reflection phase shifts of RISs, we proposed a distributed cooperative design based on ADMM, which can fully utilize the parallel computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Subsequently, we developed a neural network framework, D2-ADMM, by unrolling each iteration 27 4 6 8 10 12 5 10 15 20 25 30 35 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' The WSR comparison against the number of BS B, where Nt = 2, N = 50, K = 4, P = 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' of the proposed distributed cooperative design, to automatically learn hyper-parameters and non- convex RIS solvers through end-to-end training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Compared with conventional iterative algorithms, D2-ADMM has a faster convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Moreover, we proposed an effective monodirectional information exchange strategy to attain the cooperative design of all BSs with a small exchange overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Finally, numerical results demonstrated that the proposed D2-ADMM achieve around 210% improvement in capacity compared with the distributed noncooperative algorithm and almost 96% compared with the centralized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xNE0T4oBgHgl3EQfcQCC/content/2301.02360v1.pdf'} +page_content=' Samdanis and T.' metadata={'source': 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